%Title: Renormalization and Running of Quark Mass and Field   in the Regularization Invariant and MS-bar Schemes   at Three and Four Loops 
%Author: K. G. Chetyrkin and A. Retey
%Published: <EM> Nucl.Phys. </EM> <B> B583 </B> (2000) 3-3. 
%hep-ph/9910332
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\begin{document}

\title{
 \vskip-3cm{\baselineskip14pt
  \centerline{\normalsize\hfill TTP99--43}
  \centerline{\normalsize\hfill hep-ph/9910332}
 }
 \vskip.7cm
 {\Large\bf{ Renormalization and Running of Quark Mass and Field \\
 in the Regularization Invariant and $\msbar$ Schemes  \\ 
 at Three and Four Loops}
 \vspace{1.5cm}
 }
}

\author{{K.G. Chetyrkin}\thanks{Permanent address:
Institute for Nuclear Research, Russian Academy of Sciences,
60th October Anniversary Prospect 7a, Moscow 117312, Russia.}
\  and 
A. R\'etey
  \\[3em]
  {\it Institut f\"ur Theoretische Teilchenphysik,} \\
  {\it Universit\"at Karlsruhe, D-76128 Karlsruhe, Germany}
}

\date{}
\maketitle

\begin{abstract}
\nin We derive explicit transformation formulae relating the
renormalized quark mass and field as defined in the $\msbar$-scheme
with the corresponding quantities defined in any other scheme. By
analytically computing the three-loop quark propagator in the
high-energy limit (that is keeping only massless terms and terms of
first order in the quark mass) we find the NNNLO conversion factors
transforming the $\msbar$ quark mass and the renormalized quark field
to those defined in a ``Regularization Invariant'' (\RI) scheme which
is more suitable for lattice QCD calculations.  The NNNLO contribution
in the mass conversion factor turns out to be large and comparable to
the previous NNLO contribution at a scale of 2 GeV --- the typical
normalization scale employed in lattice simulations. Thus, in order to
get a precise prediction for the $\msbar$ masses of the light quarks
from lattice calculations the latter should use somewhat higher
scale of around, say, 3 GeV where the (apparent) convergence of the
perturbative series for the mass conversion factor is better.

\ice{
  It leads to a further decrease of the $\msbar$
quark mass by about 10\% in comparison to its NNLO value.
}

We also compute two more terms in the high-energy expansion of the
$\msbar$ renormalized quark propagator. The result is then used to
discuss the uncertainty caused by the use of the high energy limit in
determining the $\msbar$ mass of the charmed quark.  Finally, as a
by-product of our calculations we determine the four-loop anomalous
dimensions of quark mass and field in the Regularization Invariant
scheme.
\end{abstract} 

\ice{ 
 \vspace*{4cm}
 \vfill\eject
 \pagestyle{plain}
 \setcounter{page}{1}
}

\section{Introduction}\label{sec:intro}

Quark masses are fundamental parameters of the QCD Lagrangian.
Nevertheless, their relation to measurable physical quantities is not
direct: the masses depend on the renormalization scheme and, within a
given one, on the renormalization scale $\mu$.

\ice{ The dependence on the latter is usually referred to as
``running'' and is governed by the quark mass anomalous dimension,
$\g_m$.  }

In the realm of pQCD the definition which is most often used is based
on the $\msbar$-scheme \cite{ms,MSbar} which leads to the so-called
short-distance $\msbar$ mass.  Such a definition is of great
convenience for dealing with mass-dependent inclusive physical
observables dominated by short distances (for a review see
\cite{CKKrep}).  Unfortunately it is usually difficult to get precise
information about the quark masses from predictions from these
considerations, as their mass dependence is relatively weak.

To determine the absolute values of quark masses, one necessarily has
to rely on the methods which incorporate the features of
nonperturbative QCD. So far, the only two methods which are based on
QCD from first principles are QCD sum rules and Lattice QCD (for
recent discussions see e.g.
\cite{Prades:1997vn}--\cite{Sharpe:1998hh}).
\ice{
\cite{Prades:1997vn,Jamin:1997sa,Dominguez:1998zs,Narison:1999mv%
Chetyrkin:1998ej,Prades:1998xa,Lubicz:1998kc,Kenway:1998ew,%
Gimenez:1998pu,Sharpe:1998hh}.  
}
Rather accurate determinations of the
ratios of various quarks masses can be obtained within Chiral
Perturbation Theory \cite{leut1}.

Lattice QCD provides a direct way to determine quark masses from first
principles.  Unlike QCD sum rules it does not require model
assumptions.  It is possible to carry out the systematic improvement
of Lattice QCD so that all the discretization errors proportional
to the lattice spacing are eliminated (a comprehensible review is
given in \cite{Luscher:1998pe}). The resulting quark mass is the
(short distance) bare lattice quark mass. The matching of the lattice
quark masses to those defined in a continuum perturbative scheme
requires the calculation of the corresponding multiplicative
renormalization constants.  In the \RI scheme \cite{NMPmethod} the
renormalization conditions are applied to amputated Green functions in
Landau gauge, setting them equal to their tree-level values.  This
allows the non-perturbative calculation of the renormalization
constants. An alternative to the \RI approach is the Schroedinger
functional scheme (SF) which was used in
\cite{Capitani:1997mw,Sint:1998iq}.

An impressive number of various lattice determinations of quark masses
has recently been performed (see Refs~\cite{mq_lanl}--\cite{mq_dwall}). 


Once the RI quark masses are determined from lattice calculations they
can  be related to the $\msbar$ mass by a corresponding conversion
factor. By necessity this factor can be defined and, hence, computed
only {\em perturbatively}. The conversion factor is presently known at
next-to-next-to-leading order (NNLO) from
\cite{Franco:1998bm}. The NNLO contribution happens to be
numerically significant. This makes mandatory to know the NNNLO
$O(\alpha_s^3)$ term in the conversion factor.

In the present article we describe the calculation of this term. It
turns out that the size of the newly computed term is comparable to
the previous one at a renormalization scale of 2 GeV --- the typical
scale currently used in lattice calculations of the light quark
masses. This means that perturbation theory can not be used for a
precise conversion of the presently available \RI quark masses to the
$\msbar$ ones. A simple analysis shows that the convergence gets much
better if the scale is increased to, say, 3 GeV.  Thus, once the
lattice calculations produce the \RI quark masses at this scale our
formulas will allow an accurate conversion to the $\msbar$ masses at
the same scale.

The article is organized as follows.  In section \ref{sec:massfield} we
discuss the scheme dependence of the quark field and mass and present
a general procedure to find corresponding conversion factors from one
scheme to another. The general technique is illustrated by
constructing the conversion factors between the $\msbar$ and the \RI
scheme.  In section \ref{sec:quarkprop3l}, we present the first few
terms of the small mass expansion of the three-loop quark-propagator
in the $\msbar$ scheme. In section \ref{sec:RIvsMSandMOM} we first
present the conversion functions for the quark mass and field, and
then investigate the validity of the massless approximation for these
functions. Then we use these results to calculate the anomalous
dimensions for the quark mass and field in the \RI scheme, the
so-called RG invariant mass $\hat{m}$.
% and, finally, the $\msbar$ and RGI masses for light quarks at NNNLO.  
The final section is devoted to conclusions.

In  Appendices A and B we display our results for the small mass
expansion of the fermion propagator and the various conversion factors
with their full dependence on the group theoretical factors $C_F$,
$C_A$ and $T $. In Appendix C the four loop anomalous dimensions of
the quark mass and field are listed for the case of a $SU(N)$ gauge
group.

\section{Scheme dependence of quark mass and field}
\label{sec:massfield}

\subsection{Generalities}
\label{sub:general}

We start by considering the bare quark propagator (for simplicity we
stick to the Landau gauge in this section and, thus, do not explicitly
display the gauge dependence)
\beq
\label{def:fermi}
S_0(q,\alpha_s^0,m_0)
=
i \int dx e^{iqx}
\langle T[\psi_0 (x) \bar\psi_0 (0)] \rangle
 =  \frac{1}{m_0  - \slash{q} - \Si_0}
{},
\eeq
with the quark mass operator $\Si_0$ being conveniently decomposed
into Lorentz invariant structures according to
\beq
\label{def:sigmas}
\Si_0 = \slash{q} \Si^0_V + m_0 \Si^0_S \nnb {},, 
\eeq 
where $m_0$ and $\psi_0$ are the bare quark mass and field,
respectively.  Additionally we are using the common shortcuts for the
coupling constant throughout this article:
\[
a_s
\equiv  \frac{\alpha_s}{\pi}
= \frac{g^2}{4\pi^2}
\, ,
\]
$g$ being the bare QCD gauge coupling. To be precise we assume that
\re{def:fermi} is dimensionally regulated by going to non-integer
values of the space-time dimension $D=4-2\ep$
\cite{dim.reg-a,dim.reg-b}. The $\msbar$ renormalized counterpart of
the Green function \re{def:fermi} reads 
\beq
\label{def:fermi-ren}
\ba{l}
\dsp
S(q,\alpha_s,m,\mu) = i \int dx e^{iqx}
\langle T[\psi (x) \bar\psi (0)] \rangle
=  \frac{1}{m  - \slash{q} - \Si} 
\\
\dsp
=Z_2^{-1} S_0(q,\alpha_s^0,m_0)|_{{}_{
\scriptstyle
m_0=Z_m m,\, \alpha_s^0= \mu^\ep Z_\alpha \alpha_s 
 	} 
	}
{},
\ea
\eeq
where the renormalized quark field is
\[
\psi= Z_2^{-1/2} \psi_0
{}
\]
and the {}'Hooft mass parameter $\mu$ is the scale at which the
renormalized quark mass is defined.  The renormalization
constants $Z_2,Z_\alpha$ and $Z_m$ are series of the generic form 
\beq
Z_? = 1 + \sum_{i>0} Z_?^{(i)} \frac{1}{\ep^i}, \, \, \, \, \, , \, \,
Z_?^{(i)} = \sum_{ j \ge i } Z_?^{(i,j)} \left( \aspi \right )^j {}.
\label{def:Z-expansions}
\end{equation}

A general theorem (first rigorously proven for minimal subtractions in
\cite{Breitenlohner:1977hr,Breitenlohner:1977te}) states that there is
a unique choice of renormalization constants of the form
\re{def:Z-expansions} which makes the propagator\footnote{In fact all
Green functions if proper renormalization constants for gluon and
ghost fields are introduced.}  finite in the limit of $D \to 4$.

The independence of the bare coupling constant, mass and quark field
on $\mu$ leads in the standard way to the following renormalization
group equations for their renormalized counterparts
\begin{equation}
\mu^2\frac{d}{d\mu^2} \left[ \frac{\alpha_s(\mu)}{\pi}
\right]|{{}_{\displaystyle g_{\scriptscriptstyle {0}},
                           m_{\scriptscriptstyle {0}}
        }   }
= \beta(\alpha_s) \equiv
-\sum_{i\geq 0}\beta_{{i}}\left(\frac{\alpha_s}{\pi}\right)^{i+2}
= \sum_{i > 0} (i) Z_\alpha^{(1,i)} 
{},
\label{def:beta}
\end{equation}
\begin{equation}
\mu^2\frac{d}{d\mu^2} \overline{m}(\mu)|{{}_{\displaystyle g_{0},
 m_{0} }}
 =  m(\mu)\gamma_m(\alpha_s) \equiv
-\overline{m}\sum_{i\geq0}\gamma_m^{(i)}
{}
\left(\frac{\alpha_s}{\pi}\right)^{i+1}
= m(\mu) \as \frac{\partial Z_m^{(1)}}{\partial \as}
\label{def:anomal-mass}
\end{equation}
and
\begin{equation}
2\mu^2\frac{d}{d\mu^2} \psi(\mu)|{{}_{\displaystyle 
\psi_0, 
g_{0},
 m_{0} }}
 =  \psi(\mu)\gamma_2(\alpha_s) \equiv
\psi(\mu)\sum_{i\geq0} 
%convention \gamma_2 = \sum \a_s^i (- \gamma_2^{(i)} ????
- 
\gamma_2^{(i)}
{}
\left(\frac{\alpha_s}{\pi}\right)^{i+1}
= \psi(\mu) \as \frac{\partial Z_2^{(1)}}{\partial \as}{}.
% Factor 2 because we consider Z_2^{1/2} ?!!! (cf. Eq.~for Z_m^1)
\label{def:anomal-wave}
\end{equation}

Now let us consider the quark propagator renormalized according to a
different subtraction procedure. Marking with a prime parameters of
the second scheme one can write
\beq
\label{def:fermi-ren-prime}
\ba{l}
\dsp
S(q,\alpha'_s,m',\mu) = i \int dx e^{iqx}
\langle T[\psi' (x) \bar\psi' (0)] \rangle
=  \frac{1}{m'  - \slash{q} - \Si'}
\\
=(Z'_2)^{-1} S_0(q,\alpha_s^0,m_0)|_{{}_{
\scriptstyle
m_0=Z'_m m, \,  \alpha_s^0= \mu^\ep Z_\alpha' \alpha_s 
                                     }
                                }
{},
\ea
\eeq
where without essential loss of generality we have set $\mu' = \mu$.
The finiteness of the renormalized fields and parameters in both
schemes implies that, within the perturbation theory framework, the
relation between them can be uniquely described as follows
\bea
m = C_m \cdot m'
\label{def:Cm}
\\
\psi = \sqrt{ C_2} \cdot \psi'  
\label{def:C2}
{},
\eea
with the ``conversion functions'' being themselves {\em finite} series
in $\alpha_s'$, i.e.
\begin{equation}
C_?  \equiv        
1 + \sum_{i>0} C_?^{(i)} \left( \frac{\alpha_s'}{\pi}\right)^i 
{}
\label{def:C-expansions-ap}
\end{equation}
for $?=m$ or $2$.

Note that in general the coefficients $C_?^i$ may depend on the ratio
$m'/\mu$.  If such a dependence is absent then the corresponding
subtraction scheme is referred to as a ``mass independent'' one.  In
what follows we mainly limit ourselves to considering this latter
case. In addition, being only interested in the conversion functions
$C_2$ and $C_m$, we will assume that the function $C_\alpha$ has
already been determined and, thus, will deal with the following
representation of $C_2$ and $C_m$ in terms of the $\msbar$ coupling
constant $\alpha_s$:
\begin{equation}
C_?  \equiv        
1 + \sum_{i>0} C_?^{(i)} \left( \aspi\right)^i 
{}.
\label{def:C-expansions-a}
\end{equation}

The running of $m'$ and $\psi'$ is governed by the corresponding
anomalous dimensions $\gamma_m(\as)$ and $\gamma_2(\as)$. A direct use
of Eqs.~(\ref{def:Cm},\ref{def:C2}) gives
\bea
\g'_m = { \g_m - \beta \frac{\prd}{\prd \as} \ln C_m }
{},
\label{form:gamma-pm}
\\
\g'_2 = { \g_2 - \beta\frac{\prd}{\prd\as} \ln C_2 }
{}.
\label{form:gamma-p2}
\eea
At last, from Eq.~(\ref{def:fermi}) it is easy to see that 
\beq
S(q) = C_2 \cdot  F'(q) = \frac{C_2}{ m' ( 1 - \Si'_S) - \slash{q} (1 + \Si'_V) } 
\eeq
or, equivalently,
\bea
C_2 \cdot ( 1 + \Si_V ) = 1 + \Si'_V  
\label{form:C2}
\\
C_2 \cdot C_m \cdot ( 1 - \Si_S ) = 1 - \Si'_S 
\label{form:Cm}
\eea

The renormalization conditions for the non-$\msbar$ scheme should then
be used to provide the necessary information about the right hand side
to calculate the conversion factors $C_m$ and $C_2$ given the $\msbar$
renormalized $\Si_V$ and $\Si_S$.

\subsection{Regularization Invariant scheme versus $\msbar$}
\label{sub:RIvsMS}

The $\msbar$ subtraction scheme is intimately connected to
dimensional regularization and, thus, can not be directly used with
other regularizations, including the lattice one. In addition, the
physical meaning of its normalization parameter $\mu$ is not
transparent and leads to the well-known ambiguities when considering
the decoupling of heavy particles.


It is well-known that the above shortcomings are absent for a wide
class of so-called momentum subtraction ($\MOM$) schemes\footnote{ In
a sense the oldest subtraction scheme --- the on-shell one for QED ---
can also be considered as an example of a $\MOM$ scheme.}.  The $\MOM$
schemes require the values of properly chosen Green functions with
predefined $\mu$ dependent configurations of external momenta to be
fixed (usually to their tree values) independently on the considered
order. Practical calculations can then be performed with any
regulator (or even without it in the regulator-free approach of
\cite{Lowenstein:1974wu,Lowenstein:1974uk}).  A shortcoming of $\MOM$
schemes is that they are in general not mass-independent which leads
to a complicated running of coupling constant(s) and mass(es).


A general analysis of the problem of constructing of mass-independent
subtraction schemes was performed long ago in
\cite{Weinberg:1973ss}. Following essentially Weinberg's ideas, a
specific example of a mass independent $\MOM$ scheme for QCD has
recently been considered in \cite{NMPmethod} under the name of $\RI$
(``Regularization Invariant'') scheme (see also
\cite{Gockeler:1998ye}).  The corresponding renormalization conditions
for $\Si^{\RI}_S$ and $\Si^{\RI}_V$ read
\beq
\begin{array}{l}
\lim \limits_{m\rightarrow 0} \frac{1}{48} Tr \left[ \gamma_\mu
    \frac{\displaystyle \partial\left( \slash{q}
    (1+\Sigma^{\RI}_V)\right)}{\displaystyle \partial q_\mu}
\right]_{q^2=-\mu^2}= 1
\\
\lim \limits_{m\rightarrow 0} \frac{1}{12} Tr \left[ 1 -  \Sigma^{\RI}_S \right]_{q^2=-\mu^2}
= 1 {},
\\
\end{array}
\label{def:RI-rencond}
\eeq 
where the trace is to be taken over Dirac, Lorentz and colour
indices.  Note that the zero mass limit in \re{def:RI-rencond} means
that both $\Si^{\RI}_V$ and $\Si^{\RI}_S$ are effectively {\em
massless} functions only depending on the QCD coupling constant, the
normalization point $\mu$ and $q^2$. This also implies that it is also
sufficient to compute the $\msbar$ functions $\Si_V$ and $\Si_S$ in
massless QCD when computing the conversion factors from relations
\re{form:C2} and \re{form:Cm}.

Application of the renormalization conditions \re{def:RI-rencond} to
the conversion formulae \re{form:Cm} and \re{form:Cm} leads to
equations that can simply be solved for $C^{\RI}_m$ and $C^{\RI}_2$. All
the dependence of $\Si_V$ on $q^2$ is of the form of $\ell =
log(-\frac{q^2}{\mu^2})$, which simplifies the trace and derivative
w.r.t. $q_\mu$ and leads to
\bea
C^{\RI}_2 = \left[ 
1 + \Si_V + \frac{1}{2}\cdot \frac{\partial\Si_V(\ell)}{\partial \ell} 
\right]^{-1}_{q^2=-\mu^2,\, m=0} 
\label{form:C2RI}
\\
C^{\RI}_m = \left[ 
\frac{ 1 + \Si_V + \frac{1}{2} \cdot \frac{\displaystyle \partial
\Si_V(\ell)}{\displaystyle \partial \ell}}
     { 1 - \Si_S } \right]_{q^2=-\mu^2\, m=0} {}.
\label{form:CmRI}
\eea
Another quark field renormalization that is useful in numerical lattice 
simulations is defined by the condition
\beq
\lim \limits_{m\rightarrow 0} 
\frac{1}{12} Tr \left[ 1+\Sigma_V^{\RIp} \right]_{q^2=-\mu^2}= 1 \, {},
\label{def:RIp-rencond}
\eeq
which results in the even simpler conversion factors
\bea
C^{\RIp}_2 = \left[ 1 + \Si_V \right]^{-1}_{q^2=-\mu^2, \, m=0} 
\label{form:C2RIp}
 \\
C^{\RIp}_m = \left[
		\frac{ 1 + \Si_V }{ 1 - \Si_S }
	     \right]_{q^2=-\mu^2, \, m=0} \, {}.
\label{form:CmRIp}
\eea

Using these equations all conversion factors can easily be obtained,
once the $\msbar$ renormalized expressions $\Si_S$ and $\Si_V$ are
known.

It should be noted that in practical lattice calculations the massless
limit on the left hand side of \re{def:RI-rencond} and
\re{def:RIp-rencond} is implemented by choosing $\mu \gg m$. On
the other hand, $\mu$ should be much less than the inverse lattice
spacing $1/a$.  Typically $\mu$ is taken around $2$ GeV. 
This means that lattice determinations do
not lead directly to the $\RI$ quark mass but rather to the mass in a
different, mass-dependent, scheme.  The difference between both
schemes can be numerically non-negligible for the case of the charmed
quark.

Thus, for both $\RI$ and $\RIp$ it is suggestive to introduce their
mass-dependent counterparts $\MOM$ and $\MOMp$ as defined by the same
Eqs.~(\ref{def:RI-rencond}) and (\ref{def:RIp-rencond}) but without the $m
\to 0$ requirement.  The corresponding conversion factors to the
$\msbar$ scheme read
\bea
C^{\MOM}_2 = \left[ 
       1 + \Si_V + \frac{\displaystyle 1}{\displaystyle 2}\cdot q^2 
       \frac{\partial\Si_V}{\partial q^2} \right]^{-1}_{q^2=-\mu^2}  \\
C^{\MOM}_m = \left[ \frac{ 
       1 + \Si_V + \frac{1}{2} \cdot q^2 
       \frac{\displaystyle \partial\Si_V}{\displaystyle \partial q^2}}
       { 1 - \Si_S } \right]_{q^2=-\mu^2} 
\eea
and
\bea
C^{\MOMp}_2 = \left[ 1 + \Si_V \right]^{-1}_{q^2=-\mu^2}  \\
C^{\MOMp}_m = \left[ \frac{ 1 + \Si_V }{ 1 - \Si_S } \right]_{q^2=-\mu^2}
{}\, \, .
\eea
respectively.

\section{Three-loop quark propagator in $\msbar$-scheme}
\label{sec:quarkprop3l}
 
To find the conversion factors for $\MOM$ and $\MOMp$ one needs to
compute the functions $\Si_V$ and $\Si_S$ including their full mass
dependence. A full analytical result at two-loop level has been
obtained only recently in \cite{Fleischer:1998dw}. An extension of
this calculation up to three-loops is out of reach of present
calculational technologies. Fortunately for the $\RI$ and $\RIp$
schemes one effectively only needs to compute {\em massless }
three-loop diagrams -- a problem which in principle was solved long
ago in \cite{me81b}. A promising approach to recover the full mass
dependence of the quark propagator seems to be to employ an expansion
in $(m^2/q^2)$ \cite{Kuehn:Barcelona}. Indeed, as has been
demonstrated in
\cite{Chetyrkin:1994ex,Chetyrkin:1997qi,Harlander:1997xa,Harlander:1997kw}
small mass expansions can be a very effective tool for accurate
predictions of mass dependences provided one is not too close to the
threshold ($q^2=m^2$ in our case). The exact two-loop result for the
quark propagator can provide some insight into the accuracy of such an
expansion.

We have analytically computed three terms in the small mass expansion
of the quark propagator to order $\alpha_s^3$.  The calculation has
been done with intensive use of computer algebra programs.  In
particular, we have used {\tt QGRAF}~\cite{qgraf} for the generation
of diagrams and {\tt LMP} \cite{rhdiss} for the diagrammatic small
mass expansion.  The small mass expansion results in products of
massless propagators and massive tadpoles. These have been evaluated
with the help of the FORM packages {MATAD}~\cite{Stediss} and {\tt
MINCER}~\cite{mincer2} (A detailed description of the status of these
algebraic programs can be found in~\cite{HarSte:review} ).  It is
convenient to write the functions $\Si_{S/V}$ in the following way:
\beq
\label{def:sigma-expansion}
\Si_{S/V} = 1 
+ \sum_{i\ge 1} \Si_{S/V}^{(i)} \, \, \as^i(\mu)
% + \sum_{i\ge 1,n \ge 0} \Si_{S/V}^{(i),n} \, \, \as^i(\mu) z^n
\, {}.
\eeq 
Our results for the separate contributions in the Landau gauge and in
the $\msbar$ scheme read \footnote{Below we keep only the $n_f$
dependence; the expressions including the full dependence on the gauge
parameter and the group theoretical factors $C_A$, $C_F$ and $T$ are
given in  Appendix A. Note also that the result for three loop
massless quark propagator can  be also found in the source code of the
updated version of the FORM program MINCER \cite{JosVer}.} 



($\logqmms \equiv \ln(-q^2/\bar{m}^2)$,
$\logqmums \equiv \ln(-q^2/\mu^2)$, $z=m^2/q^2$, also note that in the
Landau gauge $\Si_V^{(1)} \equiv 0$):

% \input{Formulae/SigmaV1M.lo}

\begin{eqnarray}\Sigma_V^{(2)} = & & 
\left[
\frac{359}{144} 
-\frac{7}{48}  \,n_f 
-\frac{3}{4}  \,\zeta_{3}
-\frac{67}{48} l_{q \mu}
+\frac{1}{12}  \,n_f l_{q \mu}
%zero == 0
\right]
\nonumber\\
 &{+}& z
\left[
\frac{79}{24} 
-\frac{1}{6}  \,n_f 
-\frac{9}{4}  \,\zeta_{3}
- l_{q \mu}
-\frac{1}{2} l_{q m}
%zero == 0
\right]
\nonumber\\
 &{+}& z^{2}
\left[
-\frac{331}{216} 
-\frac{1}{24}  \,n_f 
+\frac{10}{3}  \,\zeta_{3}
-\frac{209}{72} l_{q \mu}
-\frac{1}{6}  \,n_f l_{q \mu}
-\frac{19}{12} l_{q \mu}^{2}
-\frac{209}{144} l_{q m}
\Break
\phantom{+z^{2}}
-\frac{1}{12}  \,n_f l_{q m}
-\frac{19}{12} l_{q \mu}l_{q m}
-\frac{19}{48} l_{q m}^{2}
%zero == 0
\right]
\nonumber\\
 &{+}& z^{3}
\left[
\frac{123}{32} 
+\frac{1}{18}  \,n_f 
+\frac{109}{54} l_{q \mu}
+\frac{7}{36} l_{q \mu}^{2}
+\frac{109}{108} l_{q m}
+\frac{7}{36} l_{q \mu}l_{q m}
+\frac{7}{144} l_{q m}^{2}
%zero == 0
\right]
\nonumber\\
 &{+}& z^{4}
\left[
\frac{668909}{172800} 
+\frac{1}{48}  \,n_f 
+\frac{5093}{2160} l_{q \mu}
-\frac{317}{72} l_{q \mu}^{2}
+\frac{5093}{4320} l_{q m}
-\frac{317}{72} l_{q \mu}l_{q m}
-\frac{317}{288} l_{q m}^{2}
%zero == 0
\right]
\nonumber\\
 &{+}& z^{5}
\left[
\frac{23155073}{2592000} 
+\frac{1}{90}  \,n_f 
+\frac{249353}{10800} l_{q \mu}
-\frac{25553}{1080} l_{q \mu}^{2}
+\frac{249353}{21600} l_{q m}
\Break
\phantom{+z^{5}}
-\frac{25553}{1080} l_{q \mu}l_{q m}
-\frac{25553}{4320} l_{q m}^{2}
%zero == 0
\right]
{},
\label{SigmaV2M}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\Sigma_V^{(3)} = & & 
\left[
\frac{439543}{10368} 
-\frac{12361}{2592}  \,n_f 
+\frac{785}{7776}  \, n_f^{2}
-\frac{8009}{384}  \,\zeta_{3}
+\frac{55}{72}  \,n_f  \,\zeta_{3}
-\frac{79}{256}  \,\zeta_{4}
+\frac{1165}{192}  \,\zeta_{5}
\Break
\phantom{+}
-\frac{52321}{2304} l_{q \mu}
+\frac{559}{216}  \,n_f l_{q \mu}
-\frac{13}{216}  \, n_f^{2}l_{q \mu}
+\frac{607}{128}  \,\zeta_{3}l_{q \mu}
-\frac{1}{4}  \,n_f  \,\zeta_{3}l_{q \mu}
+\frac{737}{192} l_{q \mu}^{2}
\Break
\phantom{+}
-\frac{133}{288}  \,n_f l_{q \mu}^{2}
+\frac{1}{72}  \, n_f^{2}l_{q \mu}^{2}
%zero == 0
\right]
\nonumber\\
 &{+}& z
\left[
\frac{5461}{96} 
-\frac{667}{144}  \,n_f 
+\frac{5}{54}  \, n_f^{2}
-\frac{10025}{288}  \,\zeta_{3}
+\frac{65}{36}  \,n_f  \,\zeta_{3}
+\frac{3}{2}  \,\zeta_{4}
-\frac{6235}{576}  \,\zeta_{5}
-\frac{857}{24} l_{q \mu}
\Break
\phantom{+z}
+\frac{185}{72}  \,n_f l_{q \mu}
-\frac{1}{18}  \, n_f^{2}l_{q \mu}
+\frac{123}{8}  \,\zeta_{3}l_{q \mu}
-\frac{3}{4}  \,n_f  \,\zeta_{3}l_{q \mu}
+\frac{9}{16} l_{q \mu}^{2}
-\frac{481}{96} l_{q m}
+\frac{1}{9}  \,n_f l_{q m}
\Break
\phantom{+z}
-\frac{3}{4}  \,\zeta_{3}l_{q m}
-\frac{51}{16} l_{q \mu}l_{q m}
+\frac{1}{6}  \,n_f l_{q \mu}l_{q m}
-\frac{111}{64} l_{q m}^{2}
+\frac{1}{12}  \,n_f l_{q m}^{2}
%zero == 0
\right]
\nonumber\\
 &{+}& z^{2}
\left[
-\frac{389057}{82944} 
-\frac{397}{1296}  \,n_f 
+\frac{11}{216}  \, n_f^{2}
+\frac{196475}{3456}  \,\zeta_{3}
-\frac{719}{432}  \,n_f  \,\zeta_{3}
+\frac{35}{36}  \,\zeta_{4}
-\frac{8815}{432}  \,\zeta_{5}
\Break
\phantom{+z^{2}}
+\frac{19}{216}  \,B_{4}
-\frac{8303}{384} l_{q \mu}
-\frac{1697}{648}  \,n_f l_{q \mu}
+\frac{17}{216}  \, n_f^{2}l_{q \mu}
-\frac{5177}{144}  \,\zeta_{3}l_{q \mu}
+\frac{47}{18}  \,n_f  \,\zeta_{3}l_{q \mu}
\Break
\phantom{+z^{2}}
-\frac{4739}{288} l_{q \mu}^{2}
+\frac{553}{432}  \,n_f l_{q \mu}^{2}
-\frac{1}{18}  \, n_f^{2}l_{q \mu}^{2}
+\frac{973}{288} l_{q \mu}^{3}
-\frac{23}{54}  \,n_f l_{q \mu}^{3}
-\frac{115007}{6912} l_{q m}
\Break
\phantom{+z^{2}}
-\frac{3029}{2592}  \,n_f l_{q m}
+\frac{5}{108}  \, n_f^{2}l_{q m}
-\frac{617}{288}  \,\zeta_{3}l_{q m}
+\frac{3}{4}  \,n_f  \,\zeta_{3}l_{q m}
-\frac{4127}{144} l_{q \mu}l_{q m}
\Break
\phantom{+z^{2}}
+\frac{35}{36}  \,n_f l_{q \mu}l_{q m}
-\frac{1}{36}  \, n_f^{2}l_{q \mu}l_{q m}
-\frac{157}{64} l_{q \mu}^{2}l_{q m}
-\frac{3}{8}  \,n_f l_{q \mu}^{2}l_{q m}
-\frac{3923}{384} l_{q m}^{2}
\Break
\phantom{+z^{2}}
+\frac{287}{1728}  \,n_f l_{q m}^{2}
-\frac{1915}{384} l_{q \mu}l_{q m}^{2}
-\frac{1}{18}  \,n_f l_{q \mu}l_{q m}^{2}
-\frac{3359}{2304} l_{q m}^{3}
+\frac{11}{864}  \,n_f l_{q m}^{3}
%zero == 0
\right]
{},
\label{SigmaV3M}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\Sigma_S^{(1)} = & & 
\left[
-\frac{4}{3} 
+ l_{q \mu}
- z
-2 l_{q \mu}z
- l_{q m}z
+\frac{1}{2} z^{2}
+\frac{1}{6} z^{3}
+\frac{1}{12} z^{4}
+\frac{1}{20} z^{5}
%zero == 0
\right]
{},
\label{SigmaS1M}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\Sigma_S^{(2)} = & & 
\left[
-\frac{5009}{288} 
+\frac{13}{18}  \,n_f 
+\frac{47}{12}  \,\zeta_{3}
+\frac{509}{48} l_{q \mu}
-\frac{4}{9}  \,n_f l_{q \mu}
-\frac{15}{8} l_{q \mu}^{2}
+\frac{1}{12}  \,n_f l_{q \mu}^{2}
%zero == 0
\right]
\nonumber\\
 &{+}& z
\left[
-\frac{791}{144} 
+\frac{5}{18}  \,n_f 
-\frac{43}{12}  \,\zeta_{3}
-\frac{319}{24} l_{q \mu}
+\frac{7}{18}  \,n_f l_{q \mu}
+\frac{7}{2} l_{q \mu}^{2}
-\frac{1}{3}  \,n_f l_{q \mu}^{2}
\Break
\phantom{+z}
-\frac{409}{48} l_{q m}
+\frac{5}{18}  \,n_f l_{q m}
-\frac{9}{4} l_{q \mu}l_{q m}
-\frac{1}{6}  \,n_f l_{q \mu}l_{q m}
-2 l_{q m}^{2}
%zero == 0
\right]
\nonumber\\
 &{+}& z^{2}
\left[
\frac{13}{12} 
-\frac{7}{72}  \,n_f 
-\frac{95}{24} l_{q \mu}
+\frac{1}{4}  \,n_f l_{q \mu}
-\frac{49}{12} l_{q \mu}^{2}
-\frac{1}{24} l_{q m}
+\frac{1}{12}  \,n_f l_{q m}
\Break
\phantom{+z^{2}}
-\frac{49}{12} l_{q \mu}l_{q m}
-\frac{49}{48} l_{q m}^{2}
%zero == 0
\right]
\nonumber\\
 &{+}& z^{3}
\left[
-\frac{451}{324} 
-\frac{5}{54}  \,n_f 
+\frac{829}{216} l_{q \mu}
+\frac{1}{12}  \,n_f l_{q \mu}
+\frac{527}{108} l_{q \mu}^{2}
+\frac{295}{108} l_{q m}
\Break
\phantom{+z^{3}}
+\frac{1}{36}  \,n_f l_{q m}
+\frac{527}{108} l_{q \mu}l_{q m}
+\frac{527}{432} l_{q m}^{2}
%zero == 0
\right]
\nonumber\\
 &{+}& z^{4}
\left[
-\frac{211333}{20736} 
-\frac{53}{864}  \,n_f 
-\frac{4831}{432} l_{q \mu}
+\frac{1}{24}  \,n_f l_{q \mu}
+\frac{4843}{216} l_{q \mu}^{2}
-\frac{551}{108} l_{q m}
\Break
\phantom{+z^{4}}
+\frac{1}{72}  \,n_f l_{q m}
+\frac{4843}{216} l_{q \mu}l_{q m}
+\frac{4843}{864} l_{q m}^{2}
%zero == 0
\right]
\nonumber\\
 &{+}& z^{5}
\left[
-\frac{24966073}{864000} 
-\frac{77}{1800}  \,n_f 
-\frac{349147}{3600} l_{q \mu}
+\frac{1}{40}  \,n_f l_{q \mu}
+\frac{3539}{40} l_{q \mu}^{2}
\Break
\phantom{+z^{5}}
-\frac{21667}{450} l_{q m}
+\frac{1}{120}  \,n_f l_{q m}
+\frac{3539}{40} l_{q \mu}l_{q m}
+\frac{3539}{160} l_{q m}^{2}
%zero == 0
\right]
{},
\label{SigmaS2M}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\Sigma_S^{(3)} = & & 
\left[
-\frac{1612847}{6912} 
+\frac{79621}{3888}  \,n_f 
-\frac{191}{729}  \, n_f^{2}
+\frac{150265}{1728}  \,\zeta_{3}
-\frac{755}{216}  \,n_f  \,\zeta_{3}
-\frac{1}{54}  \, n_f^{2} \,\zeta_{3}
\Break
\phantom{+}
+\frac{79}{256}  \,\zeta_{4}
+\frac{5}{12}  \,n_f  \,\zeta_{4}
-\frac{6455}{576}  \,\zeta_{5}
+\frac{40329}{256} l_{q \mu}
-\frac{6083}{432}  \,n_f l_{q \mu}
+\frac{73}{324}  \, n_f^{2}l_{q \mu}
\Break
\phantom{+}
-\frac{10013}{384}  \,\zeta_{3}l_{q \mu}
+\frac{17}{36}  \,n_f  \,\zeta_{3}l_{q \mu}
-\frac{2589}{64} l_{q \mu}^{2}
+\frac{119}{32}  \,n_f l_{q \mu}^{2}
-\frac{2}{27}  \, n_f^{2}l_{q \mu}^{2}
+\frac{65}{16} l_{q \mu}^{3}
\Break
\phantom{+}
-\frac{7}{18}  \,n_f l_{q \mu}^{3}
+\frac{1}{108}  \, n_f^{2}l_{q \mu}^{3}
%zero == 0
\right]
\nonumber\\
 &{+}& z
\left[
-\frac{303803}{6912} 
+\frac{2969}{648}  \,n_f 
-\frac{25}{324}  \, n_f^{2}
-\frac{178745}{3456}  \,\zeta_{3}
+\frac{313}{54}  \,n_f  \,\zeta_{3}
-\frac{1}{9}  \, n_f^{2} \,\zeta_{3}
-\frac{13}{3}  \,\zeta_{4}
\Break
\phantom{+z}
-\frac{9635}{1728}  \,\zeta_{5}
+\frac{2}{9}  \,B_{4}
-\frac{172685}{1152} l_{q \mu}
+\frac{4703}{432}  \,n_f l_{q \mu}
-\frac{5}{81}  \, n_f^{2}l_{q \mu}
+\frac{7493}{192}  \,\zeta_{3}l_{q \mu}
\Break
\phantom{+z}
+\frac{17}{36}  \,n_f  \,\zeta_{3}l_{q \mu}
+\frac{367}{6} l_{q \mu}^{2}
-\frac{193}{24}  \,n_f l_{q \mu}^{2}
+\frac{17}{108}  \, n_f^{2}l_{q \mu}^{2}
-\frac{135}{8} l_{q \mu}^{3}
+\frac{14}{9}  \,n_f l_{q \mu}^{3}
\Break
\phantom{+z}
-\frac{1}{18}  \, n_f^{2}l_{q \mu}^{3}
-\frac{73547}{768} l_{q m}
+\frac{839}{108}  \,n_f l_{q m}
-\frac{25}{324}  \, n_f^{2}l_{q m}
+\frac{1645}{384}  \,\zeta_{3}l_{q m}
\Break
\phantom{+z}
+\frac{5}{6}  \,n_f  \,\zeta_{3}l_{q m}
-\frac{431}{32} l_{q \mu}l_{q m}
-\frac{391}{144}  \,n_f l_{q \mu}l_{q m}
+\frac{5}{54}  \, n_f^{2}l_{q \mu}l_{q m}
-\frac{167}{16} l_{q \mu}^{2}l_{q m}
\Break
\phantom{+z}
+\frac{1}{3}  \,n_f l_{q \mu}^{2}l_{q m}
-\frac{1}{36}  \, n_f^{2}l_{q \mu}^{2}l_{q m}
-\frac{2359}{96} l_{q m}^{2}
+\frac{17}{18}  \,n_f l_{q m}^{2}
-10 l_{q \mu}l_{q m}^{2}
\Break
\phantom{+z}
-\frac{9}{2} l_{q m}^{3}
+\frac{1}{9}  \,n_f l_{q m}^{3}
%zero == 0
\right]
\nonumber\\
 &{+}& z^{2}
\left[
\frac{882619}{82944} 
-\frac{3151}{1728}  \,n_f 
-\frac{35}{648}  \, n_f^{2}
-\frac{34153}{1728}  \,\zeta_{3}
+\frac{91}{144}  \,n_f  \,\zeta_{3}
-3  \,\zeta_{4}
-\frac{55}{54}  \,\zeta_{5}
\Break
\phantom{+z^{2}}
-\frac{131681}{6912} l_{q \mu}
+\frac{3293}{432}  \,n_f l_{q \mu}
-\frac{13}{72}  \, n_f^{2}l_{q \mu}
-\frac{3251}{576}  \,\zeta_{3}l_{q \mu}
-\frac{1203}{32} l_{q \mu}^{2}
\Break
\phantom{+z^{2}}
+\frac{35}{432}  \,n_f l_{q \mu}^{2}
+\frac{5}{72}  \, n_f^{2}l_{q \mu}^{2}
+\frac{5401}{432} l_{q \mu}^{3}
-\frac{49}{54}  \,n_f l_{q \mu}^{3}
+\frac{47095}{13824} l_{q m}
\Break
\phantom{+z^{2}}
+\frac{2285}{864}  \,n_f l_{q m}
-\frac{2}{27}  \, n_f^{2}l_{q m}
-\frac{3251}{1152}  \,\zeta_{3}l_{q m}
-\frac{1303}{24} l_{q \mu}l_{q m}
+\frac{887}{432}  \,n_f l_{q \mu}l_{q m}
\Break
\phantom{+z^{2}}
+\frac{1}{36}  \, n_f^{2}l_{q \mu}l_{q m}
-\frac{773}{288} l_{q \mu}^{2}l_{q m}
-\frac{49}{72}  \,n_f l_{q \mu}^{2}l_{q m}
-\frac{2431}{192} l_{q m}^{2}
+\frac{1271}{1728}  \,n_f l_{q m}^{2}
\Break
\phantom{+z^{2}}
-\frac{6947}{576} l_{q \mu}l_{q m}^{2}
-\frac{13121}{3456} l_{q m}^{3}
+\frac{49}{864}  \,n_f l_{q m}^{3}
%zero == 0
\right]
{}.
\label{SigmaS3M}
\end{eqnarray}
%zero == 0

\section{$\RI$ scheme versus $\msbar$ and $\MOM$ schemes}
\label{sec:RIvsMSandMOM}

\subsection{Three Loop Conversion functions}
\label{sub:conversion3l}

A direct use of Eqs.~(\ref{form:C2RI}, \ref{form:CmRI}, \ref{form:C2RIp})
and \re{form:CmRIp} leads to the following analytical expressions for
the conversion factors between the $\msbar$ and $\RI$ schemes. The
results are shown for QCD ($SU(3)$) and Landau gauge as functions of
$n_f$ ( the $\zeta_i$ are the values $\zeta(i)$ of Riemann's Zeta
function ):


\begin{eqnarray}C_2^{\RI} = & & 
1 \quad + \quad \left(\frac{\alpha_s}{4\pi}\right)^{2}
\left[
-\frac{517}{18} 
+12  \,\zeta_{3}
+\frac{5}{3}  \,n_f 
%zero == 0
\right]
\nonumber\\
 &{+}& \left(\frac{\alpha_s}{4\pi}\right)^{3}
\left[
-\frac{1287283}{648} 
+\frac{14197}{12}  \,\zeta_{3}
+\frac{79}{4}  \,\zeta_{4}
-\frac{1165}{3}  \,\zeta_{5}
\Break
\phantom{+\left(\frac{\alpha_s}{4\pi}\right)^{3}}
+\frac{18014}{81}  \,n_f 
-\frac{368}{9}  \,\zeta_{3} \,n_f 
-\frac{1102}{243}  \, n_f^{2}
%zero == 0
\right]
{},
\label{C2RIqcd}
\end{eqnarray}
%zero == 0

\begin{eqnarray}C_m^{\RI} = & & 
1 \quad + \quad  \frac{\alpha_s}{4\pi}
\left[ 
-\frac{16}{3}\right] 
 \quad + \quad \left(\frac{\alpha_s}{4\pi}\right)^{2}
\left[
-\frac{1990}{9} 
+\frac{152}{3}  \,\zeta_{3}
+\frac{89}{9}  \,n_f 
%zero == 0
\right]
\nonumber\\
 &{+}& \left(\frac{\alpha_s}{4\pi}\right)^{3}
\left[
-\frac{6663911}{648} 
+\frac{408007}{108}  \,\zeta_{3}
-\frac{2960}{9}  \,\zeta_{5}
+\frac{236650}{243}  \,n_f 
\Break
\phantom{+\left(\frac{\alpha_s}{4\pi}\right)^{3}}
-\frac{4936}{27}  \,\zeta_{3} \,n_f 
+\frac{80}{3}  \,\zeta_{4} \,n_f 
-\frac{8918}{729}  \, n_f^{2}
-\frac{32}{27}  \,\zeta_{3} \, n_f^{2}
%zero == 0
\right]
{},
\label{CmRIqcd}
\end{eqnarray}
%zero == 0

\begin{eqnarray}C_2^{\RIp} = & & 
1 \quad + \quad \left(\frac{\alpha_s}{4\pi}\right)^{2}
\left[
-\frac{359}{9} 
+12  \,\zeta_{3}
+\frac{7}{3}  \,n_f 
%zero == 0
\right]
\nonumber\\
 &{+}& \left(\frac{\alpha_s}{4\pi}\right)^{3}
\left[
-\frac{439543}{162} 
+\frac{8009}{6}  \,\zeta_{3}
+\frac{79}{4}  \,\zeta_{4}
-\frac{1165}{3}  \,\zeta_{5}
\Break
\phantom{+\left(\frac{\alpha_s}{4\pi}\right)^{3}}
+\frac{24722}{81}  \,n_f 
-\frac{440}{9}  \,\zeta_{3} \,n_f 
-\frac{1570}{243}  \, n_f^{2}
%zero == 0
\right]
{},
\label{C2RIpqcd}
\end{eqnarray}
%zero == 0

\begin{eqnarray}C_m^{\RIp} = & & 
1 \quad + \quad  \frac{\alpha_s}{4\pi}
\left[ 
-\frac{16}{3}\right] 
 \quad + \quad \left(\frac{\alpha_s}{4\pi}\right)^{2}
\left[
-\frac{3779}{18} 
+\frac{152}{3}  \,\zeta_{3}
+\frac{83}{9}  \,n_f 
%zero == 0
\right]
\nonumber\\
 &{+}& \left(\frac{\alpha_s}{4\pi}\right)^{3}
\left[
-\frac{3115807}{324} 
+\frac{195809}{54}  \,\zeta_{3}
-\frac{2960}{9}  \,\zeta_{5}
+\frac{217390}{243}  \,n_f 
\Break
\phantom{+\left(\frac{\alpha_s}{4\pi}\right)^{3}}
-\frac{4720}{27}  \,\zeta_{3} \,n_f 
+\frac{80}{3}  \,\zeta_{4} \,n_f 
-\frac{7514}{729}  \, n_f^{2}
-\frac{32}{27}  \,\zeta_{3} \, n_f^{2}
%zero == 0
\right]
{}.
\label{CmRIpqcd}
\end{eqnarray}
%zero == 0

At a scale $\mu = 2$ GeV and $n_f=4$, the numerical contributions of
the leading order to NNNLO terms are as follows (for simplicity we
inserted $\alpha_s/\pi = 0.1$):
\begin{equation}
C_2^{\RI} = 1.0 + 0.0 - 0.00476 - 0.00508 \, {},
\label{C2RInum}
\end{equation}
\begin{equation}
C_m^{\RI} = 1. - 0.1333 - 0.0754 - 0.0495  
\label{CmRInum}
\end{equation}
and
\begin{equation}
C_2^{\RIp} = 1.0 + 0.0 - 0.0101 - 0.0095 \, {},
\label{C2RIpnum}
\end{equation}
\begin{equation}
C_m^{\RIp} = 1.0 - 0.1333 - 0.0701 - 0.0458 \, {}.
\label{CmRIpnum}
\end{equation}
One observes that the sizes of the NNLO and NNNLO contributions to
$C_m^{\RI}$ at this scale amount to about 7.5\% and 5\%
respectively. This shows that perturbation theory can not be used for
a precise conversion of the \RI quark masses to the $\msbar$ ones at
the renormalization scale $\mu = $ 2 GeV. The convergence can be
improved if one increases $\mu$ to, say, 3 GeV.  Indeed, with this
choice of $\mu$ the standard three-loop evolution gives $\alpha_s(3
\mbox{ GeV}) = 0.262$ and Eqs.~(\ref{C2RInum},\ref{CmRInum}) transform
to
\beq
C_2^{RI} = 1.0 + 0.0 - 0.00333 - 0.00296 
\label{C2_3}
{}
\eeq
and
\beq
C_m^{RI} =  1.0 - 0.111 - 0.0526 - 0.0289 
{}.
\label{Cm_3}
\eeq

\begin{figure}
\begin{center}
\epsfig{file=figure1a.ps,height=11.0cm,width=7.5cm}
\epsfig{file=figure1b.ps,height=11.0cm,width=7.5cm}
\vspace{-1.5cm}
\end{center}
\caption{The ratio of the RI and $\MOM$ scheme conversion functions
$C_{2,l}^{\MOM}/C_{2,l}^{\RI}$ as functions of $-\mu^2/m^2$. Shown are
the coefficients in the expansion in $a_s/\pi$ as expansions to order
$z$ to $z^5$ ($z^2$ for 3 loops). Note that in Landau gauge $C_2^? =
0$ , for 2 loops some numeric values for the exact mass dependence are
shown as well.}
\label{plot:C2}
\end{figure}

\begin{figure}
\begin{center}
\epsfig{file=figure2a.ps,height=11.0cm,width=7.5cm}
\epsfig{file=figure2b.ps,height=11.0cm,width=7.5cm}

\vspace{-2.5cm}

\epsfig{file=figure2c.ps,height=11.0cm,width=7.5cm}
\vspace{-1.5cm}
\end{center}
\caption{The ratio of the RI and $\MOM$ scheme conversion functions
$C_{m,l}^{\MOM}/C_{m,l}^{\RI}$ as functions of $-\mu^2/m^2$. Shown are the
coefficients in the expansion in $a_s/\pi$ as expansions to order $z$
to $z^5$ ($z^2$ for 3 loops). For 1 and 2 loops some numeric values
for the exact mass dependence are shown as well.}
\label{plot:Cm}
\end{figure}

The accuracy of the massless approximation can be tested by computing the
ratio $C_?^{\RI}/C_?^{\MOM}$ as a series in $z = m_q^2/(-\mu^2)$.
Using the results of the previous section one obtains


\begin{eqnarray}\frac{C_m^{\RI}}{C_m^{\MOM}} = & & 
1.\nonumber\\
 &{+}&  a_s 
\left[
-1. z
-0.5 z^{2}
+0.16667 z^{3}
-0.083333 z^{4}
+0.05 z^{5}
-1. zl_z
%zero == 0.*z + 0.*lmm*z + 0.*z^2 + 0.*z^3 + 0.*z^4 + 0.*z^5
\right]
\nonumber\\
 &{+}&  a_s^{2} 
\left[
-7.6458 z
+0.86458 z^{2}
-3.5129 z^{3}
+13.913 z^{4}
-36.828 z^{5}
\Break
\phantom{+ a_s^{2} }
-6.3264 zl_z
+0.10417 z^{2}l_z
+2.3866 z^{3}l_z
+7.3259 z^{4}l_z
-71.347 z^{5}l_z
\Break
\phantom{+ a_s^{2} }
-2. zl_z^{2}
+1.0208 z^{2}l_z^{2}
+1.1956 z^{3}l_z^{2}
-6.706 z^{4}l_z^{2}
+30.991 z^{5}l_z^{2}
%zero == 0.*z + 0.*lmm*z + 0.*lmm^2*z + 0.*z^2 + 0.*lmm*z^2 + 0.*lmm^2*z^2 + 0.*z^3 + 0.*lmm*z^3 + 0.*lmm^2*z^3 + 0.*z^4 + 0.*lmm*z^4 + 0.*lmm^2*z^4 + 0.*z^5 + 0.*lmm*z^5 + 0.*lmm^2*z^5
\right]
\nonumber\\
 &{+}&  a_s^{3} 
\left[
-59.008 z
+48.743 z^{2}
-41.464 zl_z
-5.4235 z^{2}l_z
-18.829 zl_z^{2}
\Break
\phantom{+ a_s^{3} }
+9.1024 z^{2}l_z^{2}
-4.0556 zl_z^{3}
+3.5697 z^{2}l_z^{3}
%zero == 0.*z + 0.*lmm*z + 0.*lmm^2*z + 0.*lmm^3*z + 0.*z^2 + 0.*lmm*z^2 + 0.*lmm^2*z^2 + 0.*lmm^3*z^2
\right]
{},
\label{CmRIbyCMOM}
\end{eqnarray}
%zero == 0. + 0.*as*z + 0.*as^2*z + 0.*as^3*z + 0.*as*lmm*z + 0.*as^2*lmm*z + 0.*as^3*lmm*z + 0.*as^2*lmm^2*z + 0.*as^3*lmm^2*z + 0.*as^3*lmm^3*z + 0.*as*z^2 + 0.*as^2*z^2 + 0.*as^3*z^2 + 0.*as^2*lmm*z^2 + 0.*as^3*lmm*z^2 + 0.*as^2*lmm^2*z^2 + 0.*as^3*lmm^2*z^2 + 0.*as^3*lmm^3*z^2 + 0.*as*z^3 + 0.*as^2*z^3 + 0.*as^2*lmm*z^3 + 0.*as^2*lmm^2*z^3 + 0.*as*z^4 + 0.*as^2*z^4 + 0.*as^2*lmm*z^4 + 0.*as^2*lmm^2*z^4 + 0.*as*z^5 + 0.*as^2*z^5 + 0.*as^2*lmm*z^5 + 0.*as^2*lmm^2*z^5

\begin{eqnarray}\frac{C_m^{\RI}}{C_m^{\MOM}} = & & 
1.\nonumber\\
 &{+}&  a_s^{2} 
\left[
0.28981 z
-0.89236 z^{2}
+1.5284 z^{3}
-3.3649 z^{4}
+7.6945 z^{5}
\Break
\phantom{+ a_s^{2} }
+0.25 zl_z
-0.39583 z^{2}l_z
+0.45602 z^{3}l_z
-2.2796 z^{4}l_z
+23.231 z^{5}l_z
\Break
\phantom{+ a_s^{2} }
+0.024306 z^{3}l_z^{2}
+1.1007 z^{4}l_z^{2}
-8.8726 z^{5}l_z^{2}
%zero == 0.*z + 0.*lmm*z + 0.*z^2 + 0.*lmm*z^2 + 0.*z^3 + 0.*lmm*z^3 + 0.*lmm^2*z^3 + 0.*z^4 + 0.*lmm*z^4 + 0.*lmm^2*z^4 + 0.*z^5 + 0.*lmm*z^5 + 0.*lmm^2*z^5
\right]
\nonumber\\
 &{+}&  a_s^{3} 
\left[
4.4496 z
-20.979 z^{2}
+2.5931 zl_z
-3.0558 z^{2}l_z
+0.70052 zl_z^{2}
\Break
\phantom{+ a_s^{3} }
-0.49414 z^{2}l_z^{2}
%zero == 0.*z + 0.*lmm*z + 0.*lmm^2*z + 0.*z^2 + 0.*lmm*z^2 + 0.*lmm^2*z^2
\right]
{},
\label{C2RIbyCMOM}
\end{eqnarray}
%zero == 0. + 0.*as^2*z + 0.*as^3*z + 0.*as^2*lmm*z + 0.*as^3*lmm*z + 0.*as^3*lmm^2*z + 0.*as^2*z^2 + 0.*as^3*z^2 + 0.*as^2*lmm*z^2 + 0.*as^3*lmm*z^2 + 0.*as^3*lmm^2*z^2 + 0.*as^2*z^3 + 0.*as^2*lmm*z^3 + 0.*as^2*lmm^2*z^3 + 0.*as^2*z^4 + 0.*as^2*lmm*z^4 + 0.*as^2*lmm^2*z^4 + 0.*as^2*z^5 + 0.*as^2*lmm*z^5 + 0.*as^2*lmm^2*z^5
where $l_z = log(-m^2/\mu^2)$ and we have evaluated 
the coefficients in the series in $z$ with $n_f=4$.

To illustrate the quality of these expansions, we have plotted (see
figures \ref{plot:C2} and \ref{plot:Cm}) the ratio of the 1, 2 and 3
loop coefficients of $C_m^{\MOM}$ and $C_m^{\RI}$ as functions of
$1/z=-\mu^2/m^2$ in the Landau gauge and for simplicity with $n_f=4$
for all values of $z$. The circles in the plots for 1 and 2 loops
correspond to the exact results from \cite{Fleischer:1998dw}. The
convergence of the small mass (corresponding to large negative values
of $1/z$) expansions is good for $1/z < -4$, where the expansions for
higher orders of $z$ are almost indistinguishable as well among each
other as well as from the numbers received from the exact 2 loop
propagator. On the other hand, due to the $z\,\ln(z)^i , i=1,\dots,l$
( where $l$ is the number of loops) terms, the $\MOM$ coefficients are
approaching the corresponding values in the $\RI$-scheme for
increasing $|z|$ only very slowly. This makes the $\RI$-scheme as an
approximation to the $\MOM$-scheme for the c quark useless.

\subsection{Four Loop Quark Anomalous Dimensions}
\label{sub:anomaldims4l}

We start from the $\msbar$ scheme. The quark mass anomalous dimension
was computed at four loops quite recently in
~\cite{Chetyrkin:1997dh,Vermaseren:1997fq} and reads
\[
\gamma_m(a_s) \equiv                                           
-\sum_{i\geq0}\gamma_m^{(i)}                                         
a_s^{i+1}                                                             
{},        
\]


\begin{eqnarray}\gamma_{m}^{(0)} = & & 
1{},
\label{MassAnomalDimMS0qcd}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\gamma_{m}^{(1)} = \frac{1}{16}\Bigg\{ & & 
 \frac{202}{3} \quad + \quad  \,n_f 
\left[ 
-\frac{20}{9}\right] 
\qquad\Bigg\}{},
\label{MassAnomalDimMS1qcd}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\gamma_{m}^{(2)} = \frac{1}{64}\Bigg\{ & & 
 1249 +  \,n_f 
\left[
-\frac{2216}{27} 
-\frac{160}{3}  \,\zeta_{3}
%zero == 0
\right]
 +  \, n_f^{2}
\left[ 
-\frac{140}{81}\right] 
\qquad\Bigg\}{},
\label{MassAnomalDimMS2qcd}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\gamma_{m}^{(3)} = \frac{1}{256}\Bigg\{ & & 
\left[
\frac{4603055}{162} 
+\frac{135680}{27}  \,\zeta_{3}
-8800  \,\zeta_{5}
%zero == 0
\right]
\nonumber\\
 &{+}&  \,n_f 
\left[
-\frac{91723}{27} 
-\frac{34192}{9}  \,\zeta_{3}
+880  \,\zeta_{4}
+\frac{18400}{9}  \,\zeta_{5}
%zero == 0
\right]
\nonumber\\
 &{+}&  \, n_f^{2}
\left[
\frac{5242}{243} 
+\frac{800}{9}  \,\zeta_{3}
-\frac{160}{3}  \,\zeta_{4}
%zero == 0
\right]
 \quad + \quad  \, n_f^{3}
\left[
-\frac{332}{243} 
+\frac{64}{27}  \,\zeta_{3}
%zero == 0
\right]
\qquad\Bigg\}{}.
\label{MassAnomalDimMS3qcd}
\end{eqnarray}
%zero == 0

The result for the quark field anomalous dimension was found by one of
the authors in the course of computing $\gamma_m$ 
\cite{Chetyrkin:1997dh} and read for
the QCD case in Landau gauge\footnote{$\gamma_{2}$ is
gauge dependent and in the Landau gauge $\gamma_{2}^{(0)} = 0$; the 
results for SU(N) group in general covariant gauge are given  in Appendix C.} 
\[
\gamma_2(a_s) \equiv                                           
-\sum_{i\geq0}\gamma_2^{(i)}                                         
a_s^{i+1}                                                             
{},        
\]


\begin{eqnarray}\gamma_{2}^{(1)} = \frac{1}{16}\Bigg\{ & & 
 \frac{67}{3} \quad + \quad  \,n_f 
\left[ 
-\frac{4}{3}\right] 
\qquad\Bigg\}{},
\label{FieldAnomalDimMS1qcd}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\gamma_{2}^{(2)} = \frac{1}{64}\Bigg\{ & & 
\left[
\frac{20729}{36} 
-\frac{79}{2}  \,\zeta_{3}
%zero == 0
\right]
 +  \,n_f 
\left[ 
-\frac{550}{9}\right] 
 +  \, n_f^{2}
\left[ 
 \frac{20}{27}\right] 
\qquad\Bigg\}{},
\label{FieldAnomalDimMS2qcd}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\gamma_{2}^{(3)} = \frac{1}{256}\Bigg\{ & & 
\left[
\frac{2109389}{162} 
-\frac{565939}{324}  \,\zeta_{3}
+\frac{2607}{4}  \,\zeta_{4}
-\frac{761525}{1296}  \,\zeta_{5}
%zero == 0
\right]
\nonumber\\
 &{+}&  \,n_f 
\left[
-\frac{162103}{81} 
-\frac{2291}{27}  \,\zeta_{3}
-\frac{79}{2}  \,\zeta_{4}
-\frac{160}{3}  \,\zeta_{5}
%zero == 0
\right]
\nonumber\\
 &{+}&  \, n_f^{2}
\left[
\frac{3853}{81} 
+\frac{160}{9}  \,\zeta_{3}
%zero == 0
\right]
 \quad + \quad  \, n_f^{3}
\left[ 
 \frac{140}{243}\right] 
\qquad\Bigg\}{}.
\label{FieldAnomalDimMS3qcd}
\end{eqnarray}
%zero == 0

For completeness 
we also give the field
anomalous dimension for the  case of QED with $n_f$ 
different fermion species\footnote{For the QED
case, all gauge dependence is in $\gamma_{2}^{(0)}$, where $\xi_L$
is defined in Appendix \ref{app:quarkprop3l} ($\xi_L=0$ for Landau
gauge)}:

\begin{eqnarray}\gamma_{2}^{QED\,(0)} = & & \frac{1}{4}
\left[
  \,\xi_{L} 
%zero == 0
\right]
{},
\label{FieldAnomalDimMS0qed}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\gamma_{2}^{QED\,(1)} = \frac{1}{16}\Bigg\{ & & 
-\frac{3}{2} \quad + \quad  \,n_f 
\left[ 
-2\right] 
\qquad\Bigg\}{},
\label{FieldAnomalDimMS1qed}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\gamma_{2}^{QED\,(2)} = \frac{1}{64}\Bigg\{ & & 
 \frac{3}{2} +  \,n_f 
\left[ 
 3\right] 
 +  \, n_f^{2}
\left[ 
 \frac{20}{9}\right] 
\qquad\Bigg\}{},
\label{FieldAnomalDimMS2qed}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\gamma_{2}^{QED\,(3)} = \frac{1}{256}\Bigg\{ & & 
\left[
-\frac{1027}{8} 
-400  \,\zeta_{3}
+640  \,\zeta_{5}
%zero == 0
\right]
 \quad + \quad  \,n_f 
\left[
\frac{460}{3} 
-64  \,\zeta_{3}
%zero == 0
\right]
\nonumber\\
 &{+}&  \, n_f^{2}
\left[
\frac{304}{9} 
-32  \,\zeta_{3}
%zero == 0
\right]
 \quad + \quad  \, n_f^{3}
\left[ 
 \frac{280}{81}\right] 
\qquad\Bigg\}{}.
\label{FieldAnomalDimMS3qed}
\end{eqnarray}
%zero == 0

In order to compute the corresponding anomalous dimensions for the
$\RI$ and $\RIp$ schemes, one just needs to make use of
Eqs.~(\ref{MassAnomalDimMS0qcd}) to (\ref{MassAnomalDimMS3qcd}) in
combination with the three loop conversion functions from section
\ref{sub:conversion3l}.  As a result for the QCD case we get for the
$\RI$ scheme:


\begin{eqnarray}\gamma_{m}^{\RI (0)} = & & 
1{},
\label{MassAnomalDimRI0qcd}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\gamma_{m}^{\RI (1)} = \frac{1}{16}\Bigg\{ & & 
 126 +  \,n_f 
\left[ 
-\frac{52}{9}\right] 
\qquad\Bigg\}{},
\label{MassAnomalDimRI1qcd}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\gamma_{m}^{\RI (2)} = \frac{1}{64}\Bigg\{ & & 
\left[
\frac{20911}{3} 
-\frac{3344}{3}  \,\zeta_{3}
%zero == 0
\right]
 +  \,n_f 
\left[
-\frac{18386}{27} 
+\frac{128}{9}  \,\zeta_{3}
%zero == 0
\right]
 +  \, n_f^{2}
\left[ 
 \frac{928}{81}\right] 
\qquad\Bigg\}{},
\label{MassAnomalDimRI2qcd}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\gamma_{m}^{\RI (3)} = \frac{1}{256}\Bigg\{ & & 
\left[
\frac{300665987}{648} 
-\frac{15000871}{108}  \,\zeta_{3}
+\frac{6160}{3}  \,\zeta_{5}
%zero == 0
\right]
\nonumber\\
 &{+}&  \,n_f 
\left[
-\frac{7535473}{108} 
+\frac{627127}{54}  \,\zeta_{3}
+\frac{4160}{3}  \,\zeta_{5}
%zero == 0
\right]
\nonumber\\
 &{+}&  \, n_f^{2}
\left[
\frac{670948}{243} 
-\frac{6416}{27}  \,\zeta_{3}
%zero == 0
\right]
 \quad + \quad  \, n_f^{3}
\left[ 
-\frac{18832}{729}\right] 
\qquad\Bigg\}{}.
\label{MassAnomalDimRI3qcd}
\end{eqnarray}
%zero == 0
\ice{
and 
   }
The corresponding equations for the $\RIp$ scheme are:


\begin{eqnarray}\gamma_{m}^{\RIp (0)} = & & 
1{},
\label{MassAnomalDimRIp0qcd}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\gamma_{m}^{\RIp (1)} = \frac{1}{16}\Bigg\{ & & 
 126 +  \,n_f 
\left[ 
-\frac{52}{9}\right] 
\qquad\Bigg\}{},
\label{MassAnomalDimRIp1qcd}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\gamma_{m}^{\RIp (2)} = \frac{1}{64}\Bigg\{ & & 
\left[
\frac{20174}{3} 
-\frac{3344}{3}  \,\zeta_{3}
%zero == 0
\right]
 +  \,n_f 
\left[
-\frac{17588}{27} 
+\frac{128}{9}  \,\zeta_{3}
%zero == 0
\right]
 +  \, n_f^{2}
\left[ 
 \frac{856}{81}\right] 
\qquad\Bigg\}{},
\label{MassAnomalDimRIp2qcd}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\gamma_{m}^{\RIp (3)} = \frac{1}{256}\Bigg\{ & & 
\left[
\frac{141825253}{324} 
-\frac{7230017}{54}  \,\zeta_{3}
+\frac{6160}{3}  \,\zeta_{5}
%zero == 0
\right]
\nonumber\\
 &{+}&  \,n_f 
\left[
-\frac{3519059}{54} 
+\frac{298241}{27}  \,\zeta_{3}
+\frac{4160}{3}  \,\zeta_{5}
%zero == 0
\right]
\nonumber\\
 &{+}&  \, n_f^{2}
\left[
\frac{611152}{243} 
-\frac{5984}{27}  \,\zeta_{3}
%zero == 0
\right]
 \quad + \quad  \, n_f^{3}
\left[ 
-\frac{16024}{729}\right] 
\qquad\Bigg\}{}.
\label{MassAnomalDimRIp3qcd}
\end{eqnarray}
%zero == 0
\ice{
and 

\begin{eqnarray}\gamma_{m}^{\RIp (3)} = \frac{1}{256}\Bigg\{ & & 
\left[
\frac{141825253}{324} 
-\frac{7230017}{54}  \,\zeta_{3}
+\frac{6160}{3}  \,\zeta_{5}
%zero == 0
\right]
\nonumber\\
 &{+}&  \,n_f 
\left[
-\frac{3519059}{54} 
+\frac{298241}{27}  \,\zeta_{3}
+\frac{4160}{3}  \,\zeta_{5}
%zero == 0
\right]
\nonumber\\
 &{+}&  \, n_f^{2}
\left[
\frac{611152}{243} 
-\frac{5984}{27}  \,\zeta_{3}
%zero == 0
\right]
 \quad + \quad  \, n_f^{3}
\left[ 
-\frac{16024}{729}\right] 
\qquad\Bigg\}{}.
\label{MassAnomalDimRIp3qcd}
\end{eqnarray}
%zero == 0
    }

The quark field anomalous dimensions for the  
\RI and {\RIp\ }  schemes are given in Appendix C.

\subsection{NNNLO relation for the $\RI$ quark mass and the RGI mass
  $\hat{m}_q$}\label{sub:RIandRGmasses}

It is customary to solve the RG equation \re{def:anomal-mass} for the
quark running mass $m(\mu)$ as follows
\beq \frac{m(\mu)}{m(\mu_0)} = \frac{c(a_s(\mu))}{c(a_s(\mu_0))}
\label{def:c-functions}
{},
\eeq
where  ($x$ stands for either $\alpha_s(\mu)/\pi$ or $\alpha_s(\mu_0)/
\pi$)
\bea \nonumber
c(x) &=& 
\exp \,\left\{
\int^{x}
     {\rm d} x' \frac{\gamma_m(x')}{\beta(x')}\right\}
\\ \nonumber
 &=&
(x)^{\bar{\g_0}}  \left\{ 1 + (\bar{\g_1} - \bar{\beta_1}\bar{\g_0})x
\right.
\\ \nonumber
&+&
\frac{1}{2}
\left[
(\bar{\g_1} - \bar{\beta_1}\bar{\g_0})^2 
+
\bar{\g_2} + \bar{\beta_1}^2\bar{\g_0}
- \bar{\beta_1}\bar{\g_1} -\bar{\beta_2}\bar{\g_0}
\right] x^2
\\ \label{c(x)}
&+&
\left[ 
\frac{1}{6}(\bar{\g_1} - \bar{\beta_1}\bar{\g_0})^3
+
\frac{1}{2}(\bar{\g_1} - \bar{\beta_1}\bar{\g_0})
(
\bar{\g_2} + \bar{\beta_1}^2\bar{\g_0}
- \bar{\beta_1}\bar{\g_1} -\bar{\beta_2}\bar{\g_0}
)
\right.
\\ \nonumber
&&
+\frac{1}{3}\left(
\left.\left.
\bar{\g_3}
-\bar{\beta_1}^3\bar{\g_0} + 2\bar{\beta_1} \bar{\beta_2}\bar{\g_0}
-\bar{\beta_3}\bar{\g_0} + \bar{\beta_1}^2\bar{\g_1}
- \bar{\beta_2}\bar{\g_1} - \bar{\beta_1}\bar{\g_2}
\right)
\right] x^3 + {\cal O}(x^4)
\right\}
{}.
\eea

Here $\bar{\g_i} = \g_m^{(i)}/\beta_0$, $\bar{\beta_i} = \beta_i/\beta_0$,
($i$=1,2,3) and $\beta_i$ are the coefficients of the QCD beta-function
as defined in Eq.~(\re{def:beta}).  The four-loop beta-function has
recently been computed  in \cite{vanRitbergen:1997va}
with the result
\begin{equation}\label{form:beta}
\begin{array}{ll}\displaystyle
\beta_0 = &\displaystyle\frac{1}{4}\left(11-\frac{2}{3}n_f\right),
  \;\;\;\;  
\beta_1=\frac{1}{16}\left(102-\frac{38}{3}n_f\right), 
\\[3ex] \displaystyle
\beta_2  = & \displaystyle \frac{1}{64}\left(\frac{2857}{2}
-\frac{5033}{18}n_f+ 
\frac{325}{54}n_f^2\right),
\\[3ex]
\beta_3  = & \dsp\frac{1}{256}\left( 
                 \frac{149753}{6} + 3564\zeta(3)
\right.
\\[3ex]
\rule{0cm}{.4cm} &
\left.
 \dsp
                 -\left[\frac{1078361}{162} + \frac{6508}{27}\zeta(3)\right]n_f
                 +\left[\frac{50065}{162}+ \frac{6472}{81}\zeta(3)\right]n_f^2 
                 + \frac{1093}{729}n_f^3
                  \right)
{}\, .
\end{array} 
\end{equation} 

Eq.~(\ref{def:c-functions}) directly leads to the RG invariant $\mu$
independent mass $\hat{m}_q$
\beq
\hat{m}_q = \frac{m_q(\mu) }{c(a_s(\mu))}
\label{def1:RG-mass}
{}.
\eeq
An important property of $\hat{m}_q$ is its $\mu$ and scheme
independence. The latter follows from the fact that $\hat{m}_q$ could
be alternatively defined as follows
\beq
\hat{m}_q = \lim_{\mu \to \infty} 
m_q(\mu) 
\left( \frac{\alpha_s(\mu )}{\pi}\right )^{-\frac{\gamma_m^{0}}{\beta_0}}
\label{def2:RG-mass}
{}
\eeq
and from the well-known universality of the one loop coefficients of
the quark mass anomalous dimension and the $\beta$-function.

Evaluating the four loop approximation of the $c$-function in the
$\RI$ and $\RIp$ schemes, we can state our results for the conversion
functions as a relation between the RG invariant mass $\hat{m}$ and
the masses $m^{\RI}$ and $m^{\RIp}$:

For $n_f=3,4,5$  Eq.~(\ref{def1:RG-mass}) assumes the form


\begin{eqnarray}\frac{\hat{m}^{(3)}}{m^{\RI}} = \left(\frac{\alpha_s}{\pi}\right)^{-\frac{4}{9}}\Bigg\{ & & 
1 \quad + \quad  \frac{\alpha_s}{4\pi}
\left[ 
-\frac{722}{81}\right] 
 \quad + \quad \left(\frac{\alpha_s}{4\pi}\right)^{2}
\left[
-\frac{2521517}{13122} 
+\frac{536}{9}  \,\zeta_{3}
%zero == 0
\right]
\nonumber\\
 &{+}& \left(\frac{\alpha_s}{4\pi}\right)^{3}
\left[
-\frac{88484924345}{12754584} 
+\frac{3089567}{972}  \,\zeta_{3}
-\frac{18640}{81}  \,\zeta_{5}
%zero == 0
\right]
\qquad\Bigg\}{},
\label{MRGI3}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\frac{\hat{m}^{(4)}}{m^{\RI}} = \left(\frac{\alpha_s}{\pi}\right)^{-\frac{12}{25}}\Bigg\{ & & 
1 \quad + \quad  \frac{\alpha_s}{4\pi}
\left[ 
-\frac{17606}{1875}\right] 
 \quad + \quad \left(\frac{\alpha_s}{4\pi}\right)^{2}
\left[
-\frac{3819632767}{21093750} 
+\frac{952}{15}  \,\zeta_{3}
%zero == 0
\right]
\nonumber\\
 &{+}& \left(\frac{\alpha_s}{4\pi}\right)^{3}
\left[
-\frac{8512503162869851}{1423828125000} 
+\frac{1035345331}{337500}  \,\zeta_{3}
-304  \,\zeta_{5}
%zero == 0
\right]
\qquad\Bigg\}{},
\label{MRGI4}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\frac{\hat{m}^{(5)}}{m^{\RI}} = \left(\frac{\alpha_s}{\pi}\right)^{-\frac{12}{23}}\Bigg\{ & & 
1 \quad + \quad  \frac{\alpha_s}{4\pi}
\left[ 
-\frac{15926}{1587}\right] 
 \quad + \quad \left(\frac{\alpha_s}{4\pi}\right)^{2}
\left[
-\frac{2559841211}{15111414} 
+\frac{4696}{69}  \,\zeta_{3}
%zero == 0
\right]
\nonumber\\
 &{+}& \left(\frac{\alpha_s}{4\pi}\right)^{3}
\left[
-\frac{4334826270205387}{863345304648} 
+\frac{3889063057}{1314036}  \,\zeta_{3}
-\frac{26960}{69}  \,\zeta_{5}
%zero == 0
\right]
\qquad\Bigg\}{}.
\label{MRGI5}
\end{eqnarray}
%zero == 0

\section{Conclusions}\label{sec:conclusion}

In this paper we have analytically computed the first few terms of the
high-energy expansion of the three-loop quark propagator.  These
results have been used to find the NNNLO conversion factors
transforming the $\msbar$ quark mass and the renormalized quark field
to those defined in the $\RI$ scheme which is more suitable for
lattice QCD calculations.  The newly computed NNNLO corrections are
numerically significant and should be taken into account when
transforming the \RI quark masses to the $\msbar$ ones.

We also have presented the four loop results for the quark mass and
wave function anomalous dimensions in the $\RI$ and $\RIp$ schemes.

In principle, the knowledge of N${}^4$LO conversion factors would be
useful to even better control the convergence  of the perturbation series.
Unfortunately, such a calculation requires the knowledge of the quark
propagator at four loops --- a problem certainly out of the range of
present calculational techniques.
\vspace{1cm}


{\bf Acknowledgments} 
\vspace{1cm}

We would like to thank Oleg Veretin for his valuable comments on the
work \cite{Fleischer:1998dw} as well as for his help in the numerical
evaluation of the two loop fermion propagator.  
One of us (K.G.Ch.) is grateful to  Vladimir Smirnov and  Damir
Becirevic for discussions and advice. This work was supported by DFG
under Contract Ku 502/8-1 ({\it DFG-Forschergruppe
``Quantenfeldtheorie, Computeralgebra und Monte-Carlo-Simulationen''})
and the {\it Graduiertenkolleg ``Elementarteilchenphysik an
Beschleunigern''}.
%\newpage
\vspace{1cm}

{\bf Note added} 
\vspace{1cm}

The main new findings of the present work have been communicated
before publication to David Broadhurst (the four loop quark wave
function anomalous dimension in the $\msbar$ scheme) and to Damir
Becirevic (the NNNLO transformation formulae as well as the four loop
quark mass  anomalous dimensions in $\RI$ and $\RIp$ schemes). As
a result, in \cite{Becirevic:1999kb} the NNNLO $\msbar$ --- $\RI$
conversion relation have been used to transform the lattice results for
the $\RI$ light quark masses into those for the $\msbar$ ones. In
\cite{Broadhurst:1999zi} the results for the (QED) fermion mass and
field anomalous dimensions 
(
Eqs.~(\ref{FieldAnomalDimMS1qed},\ref{FieldAnomalDimMS2qed},
\ref{FieldAnomalDimMS3qed})
and Eqs.~(7-9) of
\cite{Chetyrkin:1997dh}) have been reproduced within an entirely
different approach.


\newpage

\noindent
{\Large\bf Appendixes}

\renewcommand {\theequation}{\Alph{section}.\arabic{equation}}

\begin{appendix}

\setcounter{equation}{0}

\section{Quark Propagators}\label{app:quarkprop3l}




Below we list the full three loop results for the quark
propagator\footnote{for the exact definition of the $\Sigma$'s see
Eqs.~(\ref{def:sigmas}) and (\ref{def:sigma-expansion})} computed in general
covariant gauge with the tree gluon propagator
\[
\frac{1}{q^2}(g_{{\mu\nu}}  - (1-\xi_L)q_\mu q_\nu/q^2)
{}.
\] 

For $SU(N)$ gauge group colour factors have the values $C_A = N$,
$C_F = (N^2-1)/(2N)$ and T = 1/2. The QED case is obtained with
substitutions $C_A \to 0$, $C_F \to 1$ and $T \to 1$.


\begin{eqnarray}\Sigma_s^{(1)} = & &  \,C_F 
\left[
-1 
-\frac{1}{2}  \,\xi_{L} 
+\frac{3}{4} l_{q \mu}
+\frac{1}{4}  \,\xi_{L} l_{q \mu}
%zero == 0
\right]
{},
\label{SigmaS1}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\Sigma_s^{(2)} = & &  C_F^{2}
\left[
-\frac{13}{16} 
-\frac{3}{4}  \,\zeta_{3}
-\frac{1}{2}  \,\xi_{L} 
-\frac{1}{16}  \xi_{L}^{2}
+\frac{3}{4} l_{q \mu}
+\frac{5}{8}  \,\xi_{L} l_{q \mu}
+\frac{1}{8}  \xi_{L}^{2}l_{q \mu}
\Break
\phantom{+ C_F^{2}}
-\frac{9}{32} l_{q \mu}^{2}
-\frac{3}{16}  \,\xi_{L} l_{q \mu}^{2}
-\frac{1}{32}  \xi_{L}^{2}l_{q \mu}^{2}
%zero == 0
\right]
\nonumber\\
 &{+}&  \,C_F \,C_A 
\left[
-\frac{1531}{384} 
+\frac{21}{16}  \,\zeta_{3}
-\frac{5}{8}  \,\xi_{L} 
+\frac{3}{16}  \,\zeta_{3} \,\xi_{L} 
-\frac{15}{128}  \xi_{L}^{2}
+\frac{445}{192} l_{q \mu}
\Break
\phantom{+ \,C_F \,C_A }
+\frac{5}{16}  \,\xi_{L} l_{q \mu}
+\frac{5}{64}  \xi_{L}^{2}l_{q \mu}
-\frac{11}{32} l_{q \mu}^{2}
-\frac{3}{64}  \,\xi_{L} l_{q \mu}^{2}
-\frac{1}{64}  \xi_{L}^{2}l_{q \mu}^{2}
%zero == 0
\right]
\nonumber\\
 &{+}&  \,C_F \,T \,n_f 
\left[
\frac{13}{12} 
-\frac{2}{3} l_{q \mu}
+\frac{1}{8} l_{q \mu}^{2}
%zero == 0
\right]
{},
\label{SigmaS2}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\Sigma_s^{(3)} = & &  C_F^{3}
\left[
-\frac{229}{48} 
-\frac{19}{32}  \,\zeta_{3}
+\frac{15}{8}  \,\zeta_{5}
-\frac{29}{64}  \,\xi_{L} 
-\frac{21}{32}  \,\zeta_{3} \,\xi_{L} 
-\frac{1}{16}  \xi_{L}^{2}
+\frac{3}{32}  \,\zeta_{3} \xi_{L}^{2}
\Break
\phantom{+ C_F^{3}}
+\frac{1}{96}  \,\zeta_{3} \xi_{L}^{3}
+\frac{105}{64} l_{q \mu}
+\frac{9}{16}  \,\zeta_{3}l_{q \mu}
+\frac{37}{64}  \,\xi_{L} l_{q \mu}
+\frac{3}{16}  \,\zeta_{3} \,\xi_{L} l_{q \mu}
\Break
\phantom{+ C_F^{3}}
+\frac{11}{64}  \xi_{L}^{2}l_{q \mu}
+\frac{1}{64}  \xi_{L}^{3}l_{q \mu}
-\frac{9}{32} l_{q \mu}^{2}
-\frac{21}{64}  \,\xi_{L} l_{q \mu}^{2}
-\frac{1}{8}  \xi_{L}^{2}l_{q \mu}^{2}
-\frac{1}{64}  \xi_{L}^{3}l_{q \mu}^{2}
\Break
\phantom{+ C_F^{3}}
+\frac{9}{128} l_{q \mu}^{3}
+\frac{9}{128}  \,\xi_{L} l_{q \mu}^{3}
+\frac{3}{128}  \xi_{L}^{2}l_{q \mu}^{3}
+\frac{1}{384}  \xi_{L}^{3}l_{q \mu}^{3}
%zero == 0
\right]
\nonumber\\
 &{+}&  C_F^{2}\,C_A 
\left[
-\frac{3005}{1152} 
-\frac{313}{96}  \,\zeta_{3}
-\frac{3}{32}  \,\zeta_{4}
+\frac{5}{8}  \,\zeta_{5}
-\frac{1861}{768}  \,\xi_{L} 
+\frac{43}{64}  \,\zeta_{3} \,\xi_{L} 
-\frac{5}{16}  \,\zeta_{5} \,\xi_{L} 
\Break
\phantom{+ C_F^{2}\,C_A }
-\frac{35}{128}  \xi_{L}^{2}
-\frac{5}{256}  \xi_{L}^{3}
-\frac{1}{64}  \,\zeta_{3} \xi_{L}^{3}
+\frac{2467}{512} l_{q \mu}
+\frac{37}{64}  \,\zeta_{3}l_{q \mu}
+\frac{4511}{1536}  \,\xi_{L} l_{q \mu}
\Break
\phantom{+ C_F^{2}\,C_A }
-\frac{15}{32}  \,\zeta_{3} \,\xi_{L} l_{q \mu}
+\frac{221}{512}  \xi_{L}^{2}l_{q \mu}
-\frac{3}{64}  \,\zeta_{3} \xi_{L}^{2}l_{q \mu}
+\frac{27}{512}  \xi_{L}^{3}l_{q \mu}
-\frac{533}{256} l_{q \mu}^{2}
\Break
\phantom{+ C_F^{2}\,C_A }
-\frac{793}{768}  \,\xi_{L} l_{q \mu}^{2}
-\frac{45}{256}  \xi_{L}^{2}l_{q \mu}^{2}
-\frac{7}{256}  \xi_{L}^{3}l_{q \mu}^{2}
+\frac{33}{128} l_{q \mu}^{3}
+\frac{31}{256}  \,\xi_{L} l_{q \mu}^{3}
\Break
\phantom{+ C_F^{2}\,C_A }
+\frac{3}{128}  \xi_{L}^{2}l_{q \mu}^{3}
+\frac{1}{256}  \xi_{L}^{3}l_{q \mu}^{3}
%zero == 0
\right]
\nonumber\\
 &{+}&  C_F^{2}\,T \,n_f 
\left[
\frac{4699}{1152} 
-\frac{11}{12}  \,\zeta_{3}
-\frac{3}{8}  \,\zeta_{4}
+\frac{23}{48}  \,\xi_{L} 
+\frac{1}{8}  \,\zeta_{3} \,\xi_{L} 
-\frac{167}{64} l_{q \mu}
+\frac{1}{4}  \,\zeta_{3}l_{q \mu}
\Break
\phantom{+ C_F^{2}\,T \,n_f }
-\frac{29}{48}  \,\xi_{L} l_{q \mu}
+\frac{23}{32} l_{q \mu}^{2}
+\frac{11}{48}  \,\xi_{L} l_{q \mu}^{2}
-\frac{3}{32} l_{q \mu}^{3}
-\frac{1}{32}  \,\xi_{L} l_{q \mu}^{3}
%zero == 0
\right]
\nonumber\\
 &{+}&  \,C_F C_A^{2}
\left[
-\frac{4315565}{248832} 
+\frac{20305}{2304}  \,\zeta_{3}
+\frac{69}{1024}  \,\zeta_{4}
-\frac{405}{256}  \,\zeta_{5}
-\frac{66301}{36864}  \,\xi_{L} 
+\frac{193}{256}  \,\zeta_{3} \,\xi_{L} 
\Break
\phantom{+ \,C_F C_A^{2}}
-\frac{3}{512}  \,\zeta_{4} \,\xi_{L} 
-\frac{5}{128}  \,\zeta_{5} \,\xi_{L} 
-\frac{1377}{4096}  \xi_{L}^{2}
+\frac{3}{32}  \,\zeta_{3} \xi_{L}^{2}
-\frac{3}{1024}  \,\zeta_{4} \xi_{L}^{2}
\Break
\phantom{+ \,C_F C_A^{2}}
-\frac{5}{256}  \,\zeta_{5} \xi_{L}^{2}
-\frac{191}{3072}  \xi_{L}^{3}
+\frac{1}{192}  \,\zeta_{3} \xi_{L}^{3}
+\frac{294793}{27648} l_{q \mu}
-\frac{1301}{512}  \,\zeta_{3}l_{q \mu}
\Break
\phantom{+ \,C_F C_A^{2}}
+\frac{6509}{6144}  \,\xi_{L} l_{q \mu}
-\frac{59}{256}  \,\zeta_{3} \,\xi_{L} l_{q \mu}
+\frac{467}{2048}  \xi_{L}^{2}l_{q \mu}
-\frac{9}{512}  \,\zeta_{3} \xi_{L}^{2}l_{q \mu}
\Break
\phantom{+ \,C_F C_A^{2}}
+\frac{43}{1024}  \xi_{L}^{3}l_{q \mu}
-\frac{5507}{2304} l_{q \mu}^{2}
-\frac{715}{3072}  \,\xi_{L} l_{q \mu}^{2}
-\frac{61}{1024}  \xi_{L}^{2}l_{q \mu}^{2}
-\frac{3}{256}  \xi_{L}^{3}l_{q \mu}^{2}
\Break
\phantom{+ \,C_F C_A^{2}}
+\frac{121}{576} l_{q \mu}^{3}
+\frac{31}{1536}  \,\xi_{L} l_{q \mu}^{3}
+\frac{3}{512}  \xi_{L}^{2}l_{q \mu}^{3}
+\frac{1}{768}  \xi_{L}^{3}l_{q \mu}^{3}
%zero == 0
\right]
\nonumber\\
 &{+}&  \,C_F \,C_A \,T \,n_f 
\left[
\frac{16381}{1944} 
-\frac{193}{144}  \,\zeta_{3}
+\frac{3}{8}  \,\zeta_{4}
+\frac{2773}{4608}  \,\xi_{L} 
-\frac{3}{16}  \,\zeta_{3} \,\xi_{L} 
-\frac{5081}{864} l_{q \mu}
\Break
\phantom{+ \,C_F \,C_A \,T \,n_f }
+\frac{1}{8}  \,\zeta_{3}l_{q \mu}
-\frac{251}{768}  \,\xi_{L} l_{q \mu}
+\frac{1}{16}  \,\zeta_{3} \,\xi_{L} l_{q \mu}
+\frac{887}{576} l_{q \mu}^{2}
+\frac{25}{384}  \,\xi_{L} l_{q \mu}^{2}
\Break
\phantom{+ \,C_F \,C_A \,T \,n_f }
-\frac{11}{72} l_{q \mu}^{3}
-\frac{1}{192}  \,\xi_{L} l_{q \mu}^{3}
%zero == 0
\right]
\nonumber\\
 &{+}&  \,C_F  T^{2} \, n_f^{2}
\left[
-\frac{191}{243} 
-\frac{1}{18}  \,\zeta_{3}
+\frac{73}{108} l_{q \mu}
-\frac{2}{9} l_{q \mu}^{2}
+\frac{1}{36} l_{q \mu}^{3}
%zero == 0
\right]
{},
\label{SigmaS3}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\Sigma_v^{(1)} = & &  \,C_F 
\left[
\frac{1}{4}  \,\xi_{L} 
-\frac{1}{4}  \,\xi_{L} l_{q \mu}
%zero == 0
\right]
{},
\label{SigmaV1}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\Sigma_v^{(2)} = & &  C_F^{2}
\left[
-\frac{5}{128} 
+\frac{3}{32} l_{q \mu}
-\frac{1}{16}  \xi_{L}^{2}l_{q \mu}
+\frac{1}{32}  \xi_{L}^{2}l_{q \mu}^{2}
%zero == 0
\right]
\nonumber\\
 &{+}&  \,C_F \,C_A 
\left[
\frac{41}{64} 
-\frac{3}{16}  \,\zeta_{3}
+\frac{13}{32}  \,\xi_{L} 
-\frac{3}{16}  \,\zeta_{3} \,\xi_{L} 
+\frac{9}{128}  \xi_{L}^{2}
-\frac{25}{64} l_{q \mu}
\Break
\phantom{+ \,C_F \,C_A }
-\frac{7}{32}  \,\xi_{L} l_{q \mu}
-\frac{3}{64}  \xi_{L}^{2}l_{q \mu}
+\frac{3}{64}  \,\xi_{L} l_{q \mu}^{2}
+\frac{1}{64}  \xi_{L}^{2}l_{q \mu}^{2}
%zero == 0
\right]
\nonumber\\
 &{+}&  \,C_F \,T \,n_f 
\left[
-\frac{7}{32} 
+\frac{1}{8} l_{q \mu}
%zero == 0
\right]
{},
\label{SigmaV2}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\Sigma_v^{(3)} = & &  C_F^{3}
\left[
-\frac{73}{768} 
+\frac{7}{512}  \,\xi_{L} 
-\frac{1}{96}  \,\zeta_{3} \xi_{L}^{3}
-\frac{3}{128} l_{q \mu}
+\frac{17}{512}  \,\xi_{L} l_{q \mu}
-\frac{3}{128}  \,\xi_{L} l_{q \mu}^{2}
\Break
\phantom{+ C_F^{3}}
+\frac{1}{128}  \xi_{L}^{3}l_{q \mu}^{2}
-\frac{1}{384}  \xi_{L}^{3}l_{q \mu}^{3}
%zero == 0
\right]
\nonumber\\
 &{+}&  C_F^{2}\,C_A 
\left[
-\frac{997}{1536} 
+\frac{11}{16}  \,\zeta_{3}
+\frac{3}{32}  \,\zeta_{4}
-\frac{5}{16}  \,\zeta_{5}
+\frac{1}{16}  \,\xi_{L} 
-\frac{17}{64}  \,\zeta_{3} \,\xi_{L} 
+\frac{5}{16}  \,\zeta_{5} \,\xi_{L} 
\Break
\phantom{+ C_F^{2}\,C_A }
+\frac{3}{128}  \xi_{L}^{2}
-\frac{1}{512}  \xi_{L}^{3}
+\frac{1}{64}  \,\zeta_{3} \xi_{L}^{3}
+\frac{121}{192} l_{q \mu}
-\frac{3}{16}  \,\zeta_{3}l_{q \mu}
-\frac{33}{128}  \,\xi_{L} l_{q \mu}
\Break
\phantom{+ C_F^{2}\,C_A }
+\frac{3}{64}  \,\zeta_{3} \,\xi_{L} l_{q \mu}
-\frac{17}{128}  \xi_{L}^{2}l_{q \mu}
+\frac{3}{64}  \,\zeta_{3} \xi_{L}^{2}l_{q \mu}
-\frac{11}{512}  \xi_{L}^{3}l_{q \mu}
-\frac{11}{128} l_{q \mu}^{2}
\Break
\phantom{+ C_F^{2}\,C_A }
+\frac{25}{256}  \,\xi_{L} l_{q \mu}^{2}
+\frac{17}{256}  \xi_{L}^{2}l_{q \mu}^{2}
+\frac{1}{64}  \xi_{L}^{3}l_{q \mu}^{2}
-\frac{3}{256}  \xi_{L}^{2}l_{q \mu}^{3}
-\frac{1}{256}  \xi_{L}^{3}l_{q \mu}^{3}
%zero == 0
\right]
\nonumber\\
 &{+}&  C_F^{2}\,T \,n_f 
\left[
-\frac{79}{384} 
+\frac{1}{4}  \,\zeta_{3}
-\frac{3}{128}  \,\xi_{L} 
-\frac{7}{96} l_{q \mu}
+\frac{11}{128}  \,\xi_{L} l_{q \mu}
+\frac{1}{32} l_{q \mu}^{2}
-\frac{1}{32}  \,\xi_{L} l_{q \mu}^{2}
%zero == 0
\right]
\nonumber\\
 &{+}&  \,C_F C_A^{2}
\left[
\frac{159257}{41472} 
-\frac{3139}{1536}  \,\zeta_{3}
-\frac{69}{1024}  \,\zeta_{4}
+\frac{165}{256}  \,\zeta_{5}
+\frac{39799}{36864}  \,\xi_{L} 
-\frac{35}{64}  \,\zeta_{3} \,\xi_{L} 
\Break
\phantom{+ \,C_F C_A^{2}}
+\frac{3}{512}  \,\zeta_{4} \,\xi_{L} 
+\frac{5}{128}  \,\zeta_{5} \,\xi_{L} 
+\frac{787}{4096}  \xi_{L}^{2}
-\frac{39}{512}  \,\zeta_{3} \xi_{L}^{2}
+\frac{3}{1024}  \,\zeta_{4} \xi_{L}^{2}
\Break
\phantom{+ \,C_F C_A^{2}}
+\frac{5}{256}  \,\zeta_{5} \xi_{L}^{2}
+\frac{55}{1536}  \xi_{L}^{3}
-\frac{1}{192}  \,\zeta_{3} \xi_{L}^{3}
-\frac{19979}{9216} l_{q \mu}
+\frac{245}{512}  \,\zeta_{3}l_{q \mu}
\Break
\phantom{+ \,C_F C_A^{2}}
-\frac{4393}{6144}  \,\xi_{L} l_{q \mu}
+\frac{59}{256}  \,\zeta_{3} \,\xi_{L} l_{q \mu}
-\frac{295}{2048}  \xi_{L}^{2}l_{q \mu}
+\frac{9}{512}  \,\zeta_{3} \xi_{L}^{2}l_{q \mu}
\Break
\phantom{+ \,C_F C_A^{2}}
-\frac{27}{1024}  \xi_{L}^{3}l_{q \mu}
+\frac{275}{768} l_{q \mu}^{2}
+\frac{529}{3072}  \,\xi_{L} l_{q \mu}^{2}
+\frac{43}{1024}  \xi_{L}^{2}l_{q \mu}^{2}
+\frac{1}{128}  \xi_{L}^{3}l_{q \mu}^{2}
\Break
\phantom{+ \,C_F C_A^{2}}
-\frac{31}{1536}  \,\xi_{L} l_{q \mu}^{3}
-\frac{3}{512}  \xi_{L}^{2}l_{q \mu}^{3}
-\frac{1}{768}  \xi_{L}^{3}l_{q \mu}^{3}
%zero == 0
\right]
\nonumber\\
 &{+}&  \,C_F \,C_A \,T \,n_f 
\left[
-\frac{11887}{5184} 
+\frac{13}{48}  \,\zeta_{3}
-\frac{1723}{4608}  \,\xi_{L} 
+\frac{1}{8}  \,\zeta_{3} \,\xi_{L} 
+\frac{191}{144} l_{q \mu}
-\frac{1}{8}  \,\zeta_{3}l_{q \mu}
\Break
\phantom{+ \,C_F \,C_A \,T \,n_f }
+\frac{175}{768}  \,\xi_{L} l_{q \mu}
-\frac{1}{16}  \,\zeta_{3} \,\xi_{L} l_{q \mu}
-\frac{47}{192} l_{q \mu}^{2}
-\frac{19}{384}  \,\xi_{L} l_{q \mu}^{2}
+\frac{1}{192}  \,\xi_{L} l_{q \mu}^{3}
%zero == 0
\right]
\nonumber\\
 &{+}&  \,C_F  T^{2} \, n_f^{2}
\left[
\frac{785}{2592} 
-\frac{13}{72} l_{q \mu}
+\frac{1}{24} l_{q \mu}^{2}
%zero == 0
\right]
{}.
\label{SigmaV3}
\end{eqnarray}
%zero == 0



\setcounter{equation}{0}

\section{Conversion Functions}\label{app:conversion}

The full  gauge dependent conversion
factors read\footnote{Note that these results are expanded in $a_s =
\frac{\alpha_s}{\pi}$ and not in $\frac{\alpha_s}{4 \pi}$ as in
Eqs.~(\ref{C2RIqcd}) to (\ref{CmRIpqcd}).}
 

\begin{eqnarray}C_2^{\RI (1)} = & &  \,C_F 
\left[
-\frac{1}{8}  \,\xi_{L} 
%zero == 0
\right]
{},
\label{C2RI1}
\end{eqnarray}
%zero == 0

\begin{eqnarray}C_2^{\RI (2)} = & & \,C_A  \,C_F 
\left[
-\frac{57}{128} 
+\frac{3}{16}  \,\zeta_{3}
-\frac{19}{64}  \,\xi_{L} 
+\frac{3}{16}  \,\zeta_{3} \,\xi_{L} 
-\frac{3}{64}  \xi_{L}^{2}
%zero == 0
\right]
\nonumber\\
 &{+}&  C_F^{2}
\left[
-\frac{1}{128} 
+\frac{3}{64}  \xi_{L}^{2}
%zero == 0
\right]
 \quad + \quad  \,C_F  \,n_f \,T
\left[ 
 \frac{5}{32}\right] 
{},
\label{C2RI2}
\end{eqnarray}
%zero == 0

\begin{eqnarray}C_2^{\RI (3)} = & & C_A^{2} \,C_F 
\left[
-\frac{457217}{165888} 
+\frac{5543}{3072}  \,\zeta_{3}
+\frac{69}{1024}  \,\zeta_{4}
-\frac{165}{256}  \,\zeta_{5}
-\frac{6655}{9216}  \,\xi_{L} 
\Break
\phantom{+C_A^{2} \,C_F }
+\frac{221}{512}  \,\zeta_{3} \,\xi_{L} 
-\frac{3}{512}  \,\zeta_{4} \,\xi_{L} 
-\frac{5}{128}  \,\zeta_{5} \,\xi_{L} 
-\frac{123}{1024}  \xi_{L}^{2}
+\frac{69}{1024}  \,\zeta_{3} \xi_{L}^{2}
\Break
\phantom{+C_A^{2} \,C_F }
-\frac{3}{1024}  \,\zeta_{4} \xi_{L}^{2}
-\frac{5}{256}  \,\zeta_{5} \xi_{L}^{2}
-\frac{139}{6144}  \xi_{L}^{3}
+\frac{1}{192}  \,\zeta_{3} \xi_{L}^{3}
%zero == 0
\right]
\nonumber\\
 &{+}& \,C_A  C_F^{2}
\left[
\frac{171}{512} 
-\frac{19}{32}  \,\zeta_{3}
-\frac{3}{32}  \,\zeta_{4}
+\frac{5}{16}  \,\zeta_{5}
+\frac{91}{512}  \,\xi_{L} 
+\frac{25}{128}  \,\zeta_{3} \,\xi_{L} 
\Break
\phantom{+\,C_A  C_F^{2}}
-\frac{5}{16}  \,\zeta_{5} \,\xi_{L} 
+\frac{15}{128}  \xi_{L}^{2}
-\frac{9}{128}  \,\zeta_{3} \xi_{L}^{2}
+\frac{25}{1024}  \xi_{L}^{3}
-\frac{1}{64}  \,\zeta_{3} \xi_{L}^{3}
%zero == 0
\right]
\nonumber\\
 &{+}&  C_F^{3}
\left[
\frac{41}{384} 
-\frac{29}{1024}  \,\xi_{L} 
-\frac{5}{512}  \xi_{L}^{3}
+\frac{1}{96}  \,\zeta_{3} \xi_{L}^{3}
%zero == 0
\right]
\nonumber\\
 &{+}& \,C_A  \,C_F  \,n_f \,T
\left[
\frac{8449}{5184} 
-\frac{5}{24}  \,\zeta_{3}
+\frac{599}{2304}  \,\xi_{L} 
-\frac{3}{32}  \,\zeta_{3} \,\xi_{L} 
%zero == 0
\right]
\nonumber\\
 &{+}&  C_F^{2} \,n_f \,T
\left[
\frac{31}{128} 
-\frac{1}{4}  \,\zeta_{3}
-\frac{15}{256}  \,\xi_{L} 
%zero == 0
\right]
 \quad + \quad  \,C_F  \, n_f^{2} T^{2}
\left[ 
-\frac{551}{2592}\right] 
{},
\label{C2RI3}
\end{eqnarray}
%zero == 0

\begin{eqnarray}C_m^{\RI (1)} = & &  \,C_F 
\left[
-1 
-\frac{3}{8}  \,\xi_{L} 
%zero == 0
\right]
{},
\label{CmRI1}
\end{eqnarray}
%zero == 0

\begin{eqnarray}C_m^{\RI (2)} = & & \,C_A  \,C_F 
\left[
-\frac{85}{24} 
+\frac{9}{8}  \,\zeta_{3}
-\frac{21}{64}  \,\xi_{L} 
-\frac{9}{128}  \xi_{L}^{2}
%zero == 0
\right]
\nonumber\\
 &{+}&  C_F^{2}
\left[
\frac{25}{128} 
-\frac{3}{4}  \,\zeta_{3}
+\frac{3}{8}  \,\xi_{L} 
+\frac{3}{32}  \xi_{L}^{2}
%zero == 0
\right]
 \quad + \quad  \,C_F  \,n_f \,T
\left[ 
 \frac{89}{96}\right] 
{},
\label{CmRI2}
\end{eqnarray}
%zero == 0

\begin{eqnarray}C_m^{\RI (3)} = & & C_A^{2} \,C_F 
\left[
-\frac{7259479}{497664} 
+\frac{64591}{9216}  \,\zeta_{3}
-\frac{15}{16}  \,\zeta_{5}
-\frac{4409}{4096}  \,\xi_{L} 
+\frac{165}{512}  \,\zeta_{3} \,\xi_{L} 
\Break
\phantom{+C_A^{2} \,C_F }
-\frac{885}{4096}  \xi_{L}^{2}
+\frac{27}{1024}  \,\zeta_{3} \xi_{L}^{2}
-\frac{81}{2048}  \xi_{L}^{3}
%zero == 0
\right]
\nonumber\\
 &{+}& \,C_A  C_F^{2}
\left[
\frac{21133}{4608} 
-\frac{245}{48}  \,\zeta_{3}
+\frac{5}{16}  \,\zeta_{5}
+\frac{1771}{1024}  \,\xi_{L} 
-\frac{13}{16}  \,\zeta_{3} \,\xi_{L} 
\Break
\phantom{+\,C_A  C_F^{2}}
+\frac{69}{256}  \xi_{L}^{2}
-\frac{3}{64}  \,\zeta_{3} \xi_{L}^{2}
+\frac{3}{64}  \xi_{L}^{3}
%zero == 0
\right]
\nonumber\\
 &{+}&  C_F^{3}
\left[
-\frac{409}{96} 
+\frac{29}{32}  \,\zeta_{3}
+\frac{15}{8}  \,\zeta_{5}
-\frac{93}{1024}  \,\xi_{L} 
-\frac{3}{32}  \xi_{L}^{2}
+\frac{3}{32}  \,\zeta_{3} \xi_{L}^{2}
\Break
\phantom{+ C_F^{3}}
-\frac{3}{128}  \xi_{L}^{3}
%zero == 0
\right]
\nonumber\\
 &{+}& \,C_A  \,C_F  \,n_f \,T
\left[
\frac{105701}{15552} 
-\frac{163}{144}  \,\zeta_{3}
+\frac{3}{8}  \,\zeta_{4}
+\frac{175}{512}  \,\xi_{L} 
-\frac{3}{32}  \,\zeta_{3} \,\xi_{L} 
%zero == 0
\right]
\nonumber\\
 &{+}&  C_F^{2} \,n_f \,T
\left[
\frac{263}{144} 
-\frac{2}{3}  \,\zeta_{3}
-\frac{3}{8}  \,\zeta_{4}
-\frac{95}{256}  \,\xi_{L} 
+\frac{1}{8}  \,\zeta_{3} \,\xi_{L} 
%zero == 0
\right]
\nonumber\\
 &{+}&  \,C_F  \, n_f^{2} T^{2}
\left[
-\frac{4459}{7776} 
-\frac{1}{18}  \,\zeta_{3}
%zero == 0
\right]
{},
\label{CmRI3}
\end{eqnarray}
%zero == 0

\begin{eqnarray}C_2^{\RIp (1)} = & &  \,C_F 
\left[
-\frac{1}{4}  \,\xi_{L} 
%zero == 0
\right]
{},
\label{C2RIp1}
\end{eqnarray}
%zero == 0

\begin{eqnarray}C_2^{\RIp (2)} = & & \,C_A  \,C_F 
\left[
-\frac{41}{64} 
+\frac{3}{16}  \,\zeta_{3}
-\frac{13}{32}  \,\xi_{L} 
+\frac{3}{16}  \,\zeta_{3} \,\xi_{L} 
-\frac{9}{128}  \xi_{L}^{2}
%zero == 0
\right]
\nonumber\\
 &{+}&  C_F^{2}
\left[
\frac{5}{128} 
+\frac{1}{16}  \xi_{L}^{2}
%zero == 0
\right]
 \quad + \quad  \,C_F  \,n_f \,T
\left[ 
 \frac{7}{32}\right] 
{},
\label{C2RIp2}
\end{eqnarray}
%zero == 0

\begin{eqnarray}C_2^{\RIp (3)} = & & \,C_A  C_F^{2}
\left[
\frac{997}{1536} 
-\frac{11}{16}  \,\zeta_{3}
-\frac{3}{32}  \,\zeta_{4}
+\frac{5}{16}  \,\zeta_{5}
+\frac{33}{128}  \,\xi_{L} 
+\frac{11}{64}  \,\zeta_{3} \,\xi_{L} 
\Break
\phantom{+\,C_A  C_F^{2}}
-\frac{5}{16}  \,\zeta_{5} \,\xi_{L} 
+\frac{23}{128}  \xi_{L}^{2}
-\frac{3}{32}  \,\zeta_{3} \xi_{L}^{2}
+\frac{19}{512}  \xi_{L}^{3}
-\frac{1}{64}  \,\zeta_{3} \xi_{L}^{3}
%zero == 0
\right]
\nonumber\\
 &{+}& C_A^{2} \,C_F 
\left[
-\frac{159257}{41472} 
+\frac{3139}{1536}  \,\zeta_{3}
+\frac{69}{1024}  \,\zeta_{4}
-\frac{165}{256}  \,\zeta_{5}
\Break
\phantom{+C_A^{2} \,C_F }
-\frac{39799}{36864}  \,\xi_{L} 
+\frac{35}{64}  \,\zeta_{3} \,\xi_{L} 
-\frac{3}{512}  \,\zeta_{4} \,\xi_{L} 
-\frac{5}{128}  \,\zeta_{5} \,\xi_{L} 
\Break
\phantom{+C_A^{2} \,C_F }
-\frac{787}{4096}  \xi_{L}^{2}
+\frac{39}{512}  \,\zeta_{3} \xi_{L}^{2}
-\frac{3}{1024}  \,\zeta_{4} \xi_{L}^{2}
-\frac{5}{256}  \,\zeta_{5} \xi_{L}^{2}
\Break
\phantom{+C_A^{2} \,C_F }
-\frac{55}{1536}  \xi_{L}^{3}
+\frac{1}{192}  \,\zeta_{3} \xi_{L}^{3}
%zero == 0
\right]
\nonumber\\
 &{+}&  C_F^{3}
\left[
\frac{73}{768} 
-\frac{17}{512}  \,\xi_{L} 
-\frac{1}{64}  \xi_{L}^{3}
+\frac{1}{96}  \,\zeta_{3} \xi_{L}^{3}
%zero == 0
\right]
\nonumber\\
 &{+}& \,C_A  \,C_F  \,n_f \,T
\left[
\frac{11887}{5184} 
-\frac{13}{48}  \,\zeta_{3}
+\frac{1723}{4608}  \,\xi_{L} 
-\frac{1}{8}  \,\zeta_{3} \,\xi_{L} 
%zero == 0
\right]
\nonumber\\
 &{+}&  C_F^{2} \,n_f \,T
\left[
\frac{79}{384} 
-\frac{1}{4}  \,\zeta_{3}
-\frac{11}{128}  \,\xi_{L} 
%zero == 0
\right]
 \quad + \quad  \,C_F  \, n_f^{2} T^{2}
\left[ 
-\frac{785}{2592}\right] 
{},
\label{C2RIp3}
\end{eqnarray}
%zero == 0

\begin{eqnarray}C_m^{\RIp (1)} = & &  \,C_F 
\left[
-1 
-\frac{1}{4}  \,\xi_{L} 
%zero == 0
\right]
{},
\label{CmRIp1}
\end{eqnarray}
%zero == 0

\begin{eqnarray}C_m^{\RIp (2)} = & & \,C_A  \,C_F 
\left[
-\frac{1285}{384} 
+\frac{9}{8}  \,\zeta_{3}
-\frac{7}{32}  \,\xi_{L} 
-\frac{3}{64}  \xi_{L}^{2}
%zero == 0
\right]
\nonumber\\
 &{+}&  C_F^{2}
\left[
\frac{19}{128} 
-\frac{3}{4}  \,\zeta_{3}
+\frac{1}{4}  \,\xi_{L} 
+\frac{1}{16}  \xi_{L}^{2}
%zero == 0
\right]
 \quad + \quad  \,C_F  \,n_f \,T
\left[ 
 \frac{83}{96}\right] 
{},
\label{CmRIp2}
\end{eqnarray}
%zero == 0

\begin{eqnarray}C_m^{\RIp (3)} = & & C_A^{2} \,C_F 
\left[
-\frac{3360023}{248832} 
+\frac{31193}{4608}  \,\zeta_{3}
-\frac{15}{16}  \,\zeta_{5}
-\frac{4417}{6144}  \,\xi_{L} 
+\frac{53}{256}  \,\zeta_{3} \,\xi_{L} 
\Break
\phantom{+C_A^{2} \,C_F }
-\frac{295}{2048}  \xi_{L}^{2}
+\frac{9}{512}  \,\zeta_{3} \xi_{L}^{2}
-\frac{27}{1024}  \xi_{L}^{3}
%zero == 0
\right]
\nonumber\\
 &{+}& \,C_A  C_F^{2}
\left[
\frac{18781}{4608} 
-\frac{481}{96}  \,\zeta_{3}
+\frac{5}{16}  \,\zeta_{5}
+\frac{1771}{1536}  \,\xi_{L} 
-\frac{43}{64}  \,\zeta_{3} \,\xi_{L} 
\Break
\phantom{+\,C_A  C_F^{2}}
+\frac{23}{128}  \xi_{L}^{2}
-\frac{3}{64}  \,\zeta_{3} \xi_{L}^{2}
+\frac{1}{32}  \xi_{L}^{3}
%zero == 0
\right]
\nonumber\\
 &{+}&  C_F^{3}
\left[
-\frac{3227}{768} 
+\frac{29}{32}  \,\zeta_{3}
+\frac{15}{8}  \,\zeta_{5}
-\frac{31}{512}  \,\xi_{L} 
-\frac{3}{32}  \,\zeta_{3} \,\xi_{L} 
-\frac{1}{16}  \xi_{L}^{2}
\Break
\phantom{+ C_F^{3}}
+\frac{3}{32}  \,\zeta_{3} \xi_{L}^{2}
-\frac{1}{64}  \xi_{L}^{3}
%zero == 0
\right]
\nonumber\\
 &{+}& \,C_A  \,C_F  \,n_f \,T
\left[
\frac{95387}{15552} 
-\frac{77}{72}  \,\zeta_{3}
+\frac{3}{8}  \,\zeta_{4}
+\frac{175}{768}  \,\xi_{L} 
-\frac{1}{16}  \,\zeta_{3} \,\xi_{L} 
%zero == 0
\right]
\nonumber\\
 &{+}&  C_F^{2} \,n_f \,T
\left[
\frac{1109}{576} 
-\frac{2}{3}  \,\zeta_{3}
-\frac{3}{8}  \,\zeta_{4}
-\frac{95}{384}  \,\xi_{L} 
+\frac{1}{8}  \,\zeta_{3} \,\xi_{L} 
%zero == 0
\right]
\nonumber\\
 &{+}&  \,C_F  \, n_f^{2} T^{2}
\left[
-\frac{3757}{7776} 
-\frac{1}{18}  \,\zeta_{3}
%zero == 0
\right]
{}.
\label{CmRIp3}
\end{eqnarray}
%zero == 0



\setcounter{equation}{0}

\section{Anomalous Dimensions}\label{app:anomaldims4l}

The quark mass and field anomalous dimensions for general $SU(N)$ and
for the $\msbar$, $\RI$ and $\RIp$ schemes are listed below. See
Eq.~(\ref{MassAnomalDimMS0qcd}) for the conventions. For the $\msbar$
case also the field anomalous dimension $\gamma_{2}$ is given for
general gauge and $SU(N)$, while for the $\RI$ and $\RIp$ only the
Landau-gauge formulae are given (for which all $\gamma_2^{(0)}$ are
$0$ ):


\begin{eqnarray}\gamma_{m}^{(0)} = \frac{N^2 - 1}{N}\Bigg\{ & & 
 \frac{3}{8}\qquad\Bigg\}{},
\label{MassAnomalDimMS0}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\gamma_{m}^{(1)} = \frac{N^2-1}{16 N^2}\Bigg\{ & & 
\left[
-\frac{3}{8} 
+\frac{203}{24}  N^{2}
%zero == 0
\right]
 \quad + \quad  \,n_f 
\left[
-\frac{5}{6}  N 
%zero == 0
\right]
\qquad\Bigg\}{},
\label{MassAnomalDimMS1}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\gamma_{m}^{(2)} = \frac{N^2-1}{64 N^3}\Bigg\{ & & 
\left[
\frac{129}{16} 
-\frac{129}{16}  N^{2}
+\frac{11413}{216}  N^{4}
%zero == 0
\right]
\nonumber\\
 &{+}&  \,n_f 
\left[
\frac{23}{4}  N 
-\frac{1177}{108}  N^{3}
-6  N  \,\zeta_{3}
-6  N^{3} \,\zeta_{3}
%zero == 0
\right]
\nonumber\\
 &{+}&  \, n_f^{2}
\left[
-\frac{35}{54}  N^{2}
%zero == 0
\right]
\qquad\Bigg\}{},
\label{MassAnomalDimMS2}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\gamma_{m}^{(3)} = \frac{N^2-1}{256 N^4}\Bigg\{ & & 
\left[
\frac{1261}{128} 
+\frac{50047}{384}  N^{2}
-\frac{66577}{1152}  N^{4}
+\frac{460151}{1152}  N^{6}
+21  \,\zeta_{3}
\Break
\phantom{+}
-\frac{47}{2}  N^{2} \,\zeta_{3}
+52  N^{4} \,\zeta_{3}
+\frac{1157}{18}  N^{6} \,\zeta_{3}
-110  N^{4} \,\zeta_{5}
-110  N^{6} \,\zeta_{5}
%zero == 0
\right]
\nonumber\\
 &{+}&  \,n_f 
\left[
\frac{37}{6}  N 
+\frac{10475}{216}  N^{3}
-\frac{11908}{81}  N^{5}
-\frac{111}{2}  N  \,\zeta_{3}
-85  N^{3} \,\zeta_{3}
-\frac{889}{6}  N^{5} \,\zeta_{3}
\Break
\phantom{+ \,n_f }
+33  N^{3} \,\zeta_{4}
+33  N^{5} \,\zeta_{4}
-30  N  \,\zeta_{5}
+50  N^{3} \,\zeta_{5}
+80  N^{5} \,\zeta_{5}
%zero == 0
\right]
\nonumber\\
 &{+}&  \, n_f^{2}
\left[
-\frac{19}{27}  N^{2}
+\frac{899}{324}  N^{4}
+10  N^{2} \,\zeta_{3}
+10  N^{4} \,\zeta_{3}
-6  N^{2} \,\zeta_{4}
-6  N^{4} \,\zeta_{4}
%zero == 0
\right]
\nonumber\\
 &{+}&  \, n_f^{3}
\left[
-\frac{83}{162}  N^{3}
+\frac{8}{9}  N^{3} \,\zeta_{3}
%zero == 0
\right]
\qquad\Bigg\}{},
\label{MassAnomalDimMS3}
\end{eqnarray}
%zero == 0


\begin{eqnarray}\gamma_{m}^{\RI (0)} = \frac{N^2-1}{2 N}\Bigg\{ & & 
 \frac{3}{4}\qquad\Bigg\}{},
\label{MassAnomalDimRI0}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\gamma_{m}^{\RI (1)} = \frac{N^2-1}{16 N^2}\Bigg\{ & & 
\left[
-\frac{3}{8} 
+\frac{379}{24}  N^{2}
%zero == 0
\right]
 \quad + \quad  \,n_f 
\left[
-\frac{13}{6}  N 
%zero == 0
\right]
\qquad\Bigg\}{},
\label{MassAnomalDimRI1}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\gamma_{m}^{\RI (2)} = \frac{N^2-1}{64 N^3}\Bigg\{ & & 
\left[
\frac{129}{16} 
-17  N^{2}
-22  \,\zeta_{3} N^{2}
+\frac{126239}{432}  N^{4}
-44  \,\zeta_{3} N^{4}
%zero == 0
\right]
\nonumber\\
 &{+}&  \,n_f 
\left[
\frac{75}{8}  N 
-2  \,\zeta_{3} N 
-\frac{18611}{216}  N^{3}
+2  \,\zeta_{3} N^{3}
%zero == 0
\right]
\nonumber\\
 &{+}&  \, n_f^{2}
\left[
\frac{116}{27}  N^{2}
%zero == 0
\right]
\qquad\Bigg\}{},
\label{MassAnomalDimRI2}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\gamma_{m}^{\RI (3)} = \frac{N^2-1}{256 N^4}\Bigg\{ & & 
\left[
\frac{1261}{128} 
+\frac{198679}{384}  N^{2}
-\frac{778421}{1152}  N^{4}
+\frac{202580155}{31104}  N^{6}
+21  \,\zeta_{3}
\Break
\phantom{+}
-\frac{149}{4}  N^{2} \,\zeta_{3}
-\frac{4133}{6}  N^{4} \,\zeta_{3}
-\frac{59269}{32}  N^{6} \,\zeta_{3}
-165  N^{2} \,\zeta_{5}
+275  N^{4} \,\zeta_{5}
%zero == 0
\right]
\nonumber\\
 &{+}&  \,n_f 
\left[
-\frac{3175}{48}  N 
+\frac{206575}{432}  N^{3}
-\frac{7671073}{2592}  N^{5}
-59  N  \,\zeta_{3}
-\frac{22}{3}  N^{3} \,\zeta_{3}
\Break
\phantom{+ \,n_f }
+\frac{7767}{16}  N^{5} \,\zeta_{3}
-20  N^{3} \,\zeta_{5}
+60  N^{5} \,\zeta_{5}
%zero == 0
\right]
\nonumber\\
 &{+}&  \, n_f^{2}
\left[
-\frac{5767}{108}  N^{2}
+\frac{113747}{324}  N^{4}
+\frac{62}{3}  N^{2} \,\zeta_{3}
-32  N^{4} \,\zeta_{3}
%zero == 0
\right]
\nonumber\\
 &{+}&  \, n_f^{3}
\left[
-\frac{2354}{243}  N^{3}
%zero == 0
\right]
\qquad\Bigg\}{},
\label{MassAnomalDimRI3}
\end{eqnarray}
%zero == 0


\begin{eqnarray}\gamma_{m}^{\RIp (0)} = \frac{N^2-1}{2 N}\Bigg\{ & & 
 \frac{3}{4}\qquad\Bigg\}{},
\label{MassAnomalDimRIp0}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\gamma_{m}^{\RIp (1)} = \frac{N^2-1}{16 N^2}\Bigg\{ & & 
\left[
-\frac{3}{8} 
+\frac{379}{24}  N^{2}
%zero == 0
\right]
 \quad + \quad  \,n_f 
\left[
-\frac{13}{6}  N 
%zero == 0
\right]
\qquad\Bigg\}{},
\label{MassAnomalDimRIp1}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\gamma_{m}^{\RIp (2)} = \frac{N^2-1}{64 N^3}\Bigg\{ & & 
\left[
\frac{129}{16} 
-\frac{147}{8}  N^{2}
-22  \,\zeta_{3} N^{2}
+\frac{121883}{432}  N^{4}
-44  \,\zeta_{3} N^{4}
%zero == 0
\right]
\nonumber\\
 &{+}&  \,n_f 
\left[
\frac{77}{8}  N 
-2  \,\zeta_{3} N 
-\frac{17819}{216}  N^{3}
+2  \,\zeta_{3} N^{3}
%zero == 0
\right]
\nonumber\\
 &{+}&  \, n_f^{2}
\left[
\frac{107}{27}  N^{2}
%zero == 0
\right]
\qquad\Bigg\}{},
\label{MassAnomalDimRIp2}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\gamma_{m}^{\RIp (3)} = \frac{N^2-1}{256 N^4}\Bigg\{ & & 
\left[
\frac{1261}{128} 
+21  \,\zeta_{3}
+\frac{198283}{384}  N^{2}
-\frac{149}{4}  \,\zeta_{3} N^{2}
-165  \,\zeta_{5} N^{2}
\Break
\phantom{+}
-\frac{844829}{1152}  N^{4}
-\frac{2017}{3}  \,\zeta_{3} N^{4}
+275  \,\zeta_{5} N^{4}
\Break
\phantom{+}
+\frac{191436121}{31104}  N^{6}
-\frac{28551}{16}  \,\zeta_{3} N^{6}
%zero == 0
\right]
\nonumber\\
 &{+}&  \,n_f 
\left[
-\frac{199}{3}  N 
-59  \,\zeta_{3} N 
+\frac{211669}{432}  N^{3}
-\frac{31}{3}  \,\zeta_{3} N^{3}
-20  \,\zeta_{5} N^{3}
\Break
\phantom{+ \,n_f }
-\frac{1794277}{648}  N^{5}
+\frac{3697}{8}  \,\zeta_{3} N^{5}
+60  \,\zeta_{5} N^{5}
%zero == 0
\right]
\nonumber\\
 &{+}&  \, n_f^{2}
\left[
-\frac{1444}{27}  N^{2}
+\frac{62}{3}  \,\zeta_{3} N^{2}
+\frac{25946}{81}  N^{4}
-30  \,\zeta_{3} N^{4}
%zero == 0
\right]
\nonumber\\
 &{+}&  \, n_f^{3}
\left[
-\frac{2003}{243}  N^{3}
%zero == 0
\right]
\qquad\Bigg\}{},
\label{MassAnomalDimRIp3}
\end{eqnarray}
%zero == 0


\begin{eqnarray}\gamma_{2}^{(0)} = \frac{N^2-1}{N}\Bigg\{ & & 
\left[
\frac{1}{8}  \,\xi_{L} 
%zero == 0
\right]
\qquad\Bigg\}{},
\label{FieldAnomalDimMS0}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\gamma_{2}^{(1)} = \frac{N^2-1}{16 N^2}\Bigg\{ & & 
\left[
\frac{3}{8} 
+\frac{11}{4}  N^{2}
+  N^{2} \,\xi_{L} 
+\frac{1}{8}  N^{2} \xi_{L}^{2}
%zero == 0
\right]
 \quad + \quad  \,n_f 
\left[
-\frac{1}{2}  N 
%zero == 0
\right]
\qquad\Bigg\}{},
\label{FieldAnomalDimMS1}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\gamma_{2}^{(2)} = \frac{N^2 - 1}{64 N^3}\Bigg\{ & & 
\left[
\frac{3}{16} 
+\frac{137}{16}  N^{2}
+\frac{6635}{288}  N^{4}
+\frac{263}{64}  N^{4} \,\xi_{L} 
+\frac{39}{64}  N^{4} \xi_{L}^{2}
\Break
\phantom{+}
+\frac{5}{32}  N^{4} \xi_{L}^{3}
-3  N^{2} \,\zeta_{3}
-\frac{21}{16}  N^{4} \,\zeta_{3}
+\frac{3}{8}  N^{4} \,\xi_{L}  \,\zeta_{3}
+\frac{3}{16}  N^{4} \xi_{L}^{2} \,\zeta_{3}
%zero == 0
\right]
\nonumber\\
 &{+}&  \,n_f 
\left[
-\frac{3}{8}  N 
-\frac{547}{72}  N^{3}
-\frac{17}{16}  N^{3} \,\xi_{L} 
%zero == 0
\right]
 +  \, n_f^{2}
\left[
\frac{5}{18}  N^{2}
%zero == 0
\right]
\qquad\Bigg\}{},
\label{FieldAnomalDimMS2}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\gamma_{2}^{(3)} = \frac{N^2 - 1}{256 N^4}\Bigg\{ & & 
\left[
\frac{1027}{128} 
+\frac{11281}{384}  N^{2}
+\frac{86017}{1152}  N^{4}
+\frac{1785121}{10368}  N^{6}
-\frac{5}{8}  N^{4} \,\xi_{L} 
+\frac{1644899}{62208}  N^{6} \,\xi_{L} 
\Break
\phantom{+}
+\frac{6467}{2304}  N^{6} \xi_{L}^{2}
+\frac{149}{192}  N^{6} \xi_{L}^{3}
+\frac{19}{128}  N^{6} \xi_{L}^{4}
+25  \,\zeta_{3}
+31  N^{2} \,\zeta_{3}
\Break
\phantom{+}
-\frac{4287}{64}  N^{4} \,\zeta_{3}
-\frac{1103}{64}  N^{6} \,\zeta_{3}
+\frac{115}{16}  N^{4} \,\xi_{L}  \,\zeta_{3}
+\frac{2761}{192}  N^{6} \,\xi_{L}  \,\zeta_{3}
+\frac{11}{32}  N^{4} \xi_{L}^{2} \,\zeta_{3}
\Break
\phantom{+}
+\frac{289}{96}  N^{6} \xi_{L}^{2} \,\zeta_{3}
-\frac{3}{16}  N^{4} \xi_{L}^{3} \,\zeta_{3}
+\frac{19}{64}  N^{6} \xi_{L}^{3} \,\zeta_{3}
-\frac{7}{64}  N^{4} \xi_{L}^{4} \,\zeta_{3}
-\frac{1}{64}  N^{6} \xi_{L}^{4} \,\zeta_{3}
\Break
\phantom{+}
+\frac{33}{2}  N^{4} \,\zeta_{4}
+\frac{231}{32}  N^{6} \,\zeta_{4}
-\frac{97}{64}  N^{6} \,\xi_{L}  \,\zeta_{4}
-\frac{17}{32}  N^{6} \xi_{L}^{2} \,\zeta_{4}
-\frac{3}{64}  N^{6} \xi_{L}^{3} \,\zeta_{4}
\Break
\phantom{+}
-40  \,\zeta_{5}
-60  N^{2} \,\zeta_{5}
+\frac{1945}{64}  N^{4} \,\zeta_{5}
-\frac{1375}{128}  N^{6} \,\zeta_{5}
-\frac{65}{8}  N^{4} \,\xi_{L}  \,\zeta_{5}
\Break
\phantom{+}
-\frac{95}{8}  N^{6} \,\xi_{L}  \,\zeta_{5}
-\frac{35}{32}  N^{4} \xi_{L}^{2} \,\zeta_{5}
-\frac{75}{64}  N^{6} \xi_{L}^{2} \,\zeta_{5}
+\frac{5}{64}  N^{4} \xi_{L}^{4} \,\zeta_{5}
+\frac{5}{128}  N^{6} \xi_{L}^{4} \,\zeta_{5}
%zero == 0
\right]
\nonumber\\
 &{+}&  \,n_f 
\left[
\frac{307}{12}  N 
-\frac{1555}{144}  N^{3}
-\frac{35641}{432}  N^{5}
+\frac{767}{96}  N^{3} \,\xi_{L} 
-\frac{161347}{15552}  N^{5} \,\xi_{L} 
\Break
\phantom{+ \,n_f }
-\frac{109}{288}  N^{5} \xi_{L}^{2}
-4  N  \,\zeta_{3}
+8  N^{3} \,\zeta_{3}
-\frac{35}{8}  N^{5} \,\zeta_{3}
-\frac{11}{2}  N^{3} \,\xi_{L}  \,\zeta_{3}
\Break
\phantom{+ \,n_f }
-\frac{15}{4}  N^{5} \,\xi_{L}  \,\zeta_{3}
-\frac{7}{24}  N^{5} \xi_{L}^{2} \,\zeta_{3}
-3  N^{3} \,\zeta_{4}
-\frac{21}{16}  N^{5} \,\zeta_{4}
-\frac{3}{2}  N^{3} \,\xi_{L}  \,\zeta_{4}
\Break
\phantom{+ \,n_f }
-\frac{1}{2}  N^{5} \,\xi_{L}  \,\zeta_{4}
+\frac{1}{16}  N^{5} \xi_{L}^{2} \,\zeta_{4}
-20  N^{3} \,\zeta_{5}
%zero == 0
\right]
\nonumber\\
 &{+}&  \, n_f^{2}
\left[
-\frac{19}{9}  N^{2}
+\frac{445}{72}  N^{4}
-\frac{269}{486}  N^{4} \,\xi_{L} 
+2  N^{2} \,\zeta_{3}
+2  N^{4} \,\zeta_{3}
+\frac{2}{3}  N^{4} \,\xi_{L}  \,\zeta_{3}
%zero == 0
\right]
\nonumber\\
 &{+}&  \, n_f^{3}
\left[
\frac{35}{162}  N^{3}
%zero == 0
\right]
\qquad\Bigg\}{},
\label{FieldAnomalDimMS3}
\end{eqnarray}
%zero == 0


\begin{eqnarray}\gamma_{2}^{\RI (1)} = \frac{N^2-1}{16 N^2}\Bigg\{ & & 
\left[
\frac{3}{8} 
+\frac{11}{4}  N^{2}
%zero == 0
\right]
 \quad + \quad  \,n_f 
\left[
-\frac{1}{2}  N 
%zero == 0
\right]
\qquad\Bigg\}{},
\label{FieldAnomalDimRI1}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\gamma_{2}^{\RI (2)} = \frac{N^2 - 1}{64 N^3}\Bigg\{ & & 
\left[
\frac{3}{16} 
+\frac{25}{3}  N^{2}
+\frac{14225}{288}  N^{4}
-3  N^{2} \,\zeta_{3}
-\frac{197}{16}  N^{4} \,\zeta_{3}
%zero == 0
\right]
\nonumber\\
 &{+}&  \,n_f 
\left[
-\frac{1}{3}  N 
-\frac{611}{36}  N^{3}
+2  N^{3} \,\zeta_{3}
%zero == 0
\right]
 +  \, n_f^{2}
\left[
\frac{10}{9}  N^{2}
%zero == 0
\right]
\qquad\Bigg\}{},
\label{FieldAnomalDimRI2}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\gamma_{2}^{\RI (3)} = \frac{N^2 - 1}{256 N^4}\Bigg\{ & & 
\left[
\frac{1027}{128} 
+\frac{7673}{384}  N^{2}
+\frac{174565}{1152}  N^{4}
+\frac{3993865}{3456}  N^{6}
+25  \,\zeta_{3}
+31  N^{2} \,\zeta_{3}
\Break
\phantom{+}
-\frac{10975}{64}  N^{4} \,\zeta_{3}
-\frac{111719}{192}  N^{6} \,\zeta_{3}
-40  \,\zeta_{5}
-60  N^{2} \,\zeta_{5}
+\frac{5465}{64}  N^{4} \,\zeta_{5}
\Break
\phantom{+}
+\frac{20625}{128}  N^{6} \,\zeta_{5}
%zero == 0
\right]
\nonumber\\
 &{+}&  \,n_f 
\left[
\frac{1307}{48}  N 
+\frac{557}{144}  N^{3}
-\frac{172793}{288}  N^{5}
-4  N  \,\zeta_{3}
+2  N^{3} \,\zeta_{3}
\Break
\phantom{+ \,n_f }
+\frac{7861}{48}  N^{5} \,\zeta_{3}
-30  N^{3} \,\zeta_{5}
-\frac{125}{4}  N^{5} \,\zeta_{5}
%zero == 0
\right]
\nonumber\\
 &{+}&  \, n_f^{2}
\left[
-\frac{521}{72}  N^{2}
+\frac{259}{3}  N^{4}
+6  N^{2} \,\zeta_{3}
-\frac{26}{3}  N^{4} \,\zeta_{3}
%zero == 0
\right]
\nonumber\\
 &{+}&  \, n_f^{3}
\left[
-\frac{86}{27}  N^{3}
%zero == 0
\right]
\qquad\Bigg\}{},
\label{FieldAnomalDimRI3}
\end{eqnarray}
%zero == 0


\begin{eqnarray}\gamma_{2}^{\RIp (1)} = \frac{N^2-1}{16 N^2}\Bigg\{ & & 
\left[
\frac{3}{8} 
+\frac{11}{4}  N^{2}
%zero == 0
\right]
 \quad + \quad  \,n_f 
\left[
-\frac{1}{2}  N 
%zero == 0
\right]
\qquad\Bigg\}{},
\label{FieldAnomalDimRIp1}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\gamma_{2}^{\RIp (2)} = \frac{N^2 - 1}{64 N^3}\Bigg\{ & & 
\left[
\frac{3}{16} 
+\frac{233}{24}  N^{2}
+\frac{17129}{288}  N^{4}
-3  N^{2} \,\zeta_{3}
-\frac{197}{16}  N^{4} \,\zeta_{3}
%zero == 0
\right]
\nonumber\\
 &{+}&  \,n_f 
\left[
-\frac{7}{12}  N 
-\frac{743}{36}  N^{3}
+2  N^{3} \,\zeta_{3}
%zero == 0
\right]
 +  \, n_f^{2}
\left[
\frac{13}{9}  N^{2}
%zero == 0
\right]
\qquad\Bigg\}{},
\label{FieldAnomalDimRIp2}
\end{eqnarray}
%zero == 0

\begin{eqnarray}\gamma_{2}^{\RIp (3)} = \frac{N^2 - 1}{256 N^4}\Bigg\{ & & 
\left[
\frac{1027}{128} 
+\frac{8069}{384}  N^{2}
+\frac{240973}{1152}  N^{4}
+\frac{5232091}{3456}  N^{6}
+25  \,\zeta_{3}
+31  N^{2} \,\zeta_{3}
\Break
\phantom{+}
-\frac{12031}{64}  N^{4} \,\zeta_{3}
-\frac{124721}{192}  N^{6} \,\zeta_{3}
-40  \,\zeta_{5}
-60  N^{2} \,\zeta_{5}
+\frac{5465}{64}  N^{4} \,\zeta_{5}
\Break
\phantom{+}
+\frac{20625}{128}  N^{6} \,\zeta_{5}
%zero == 0
\right]
\nonumber\\
 &{+}&  \,n_f 
\left[
\frac{329}{12}  N 
-\frac{1141}{144}  N^{3}
-\frac{113839}{144}  N^{5}
-4  N  \,\zeta_{3}
+5  N^{3} \,\zeta_{3}
\Break
\phantom{+ \,n_f }
+\frac{2245}{12}  N^{5} \,\zeta_{3}
-30  N^{3} \,\zeta_{5}
-\frac{125}{4}  N^{5} \,\zeta_{5}
%zero == 0
\right]
\nonumber\\
 &{+}&  \, n_f^{2}
\left[
-\frac{515}{72}  N^{2}
+\frac{1405}{12}  N^{4}
+6  N^{2} \,\zeta_{3}
-\frac{32}{3}  N^{4} \,\zeta_{3}
%zero == 0
\right]
\nonumber\\
 &{+}&  \, n_f^{3}
\left[
-\frac{125}{27}  N^{3}
%zero == 0
\right]
\qquad\Bigg\}{}.
\label{FieldAnomalDimRIp3}
\end{eqnarray}
%zero == 0

\end{appendix}

\newpage

\def\app#1#2#3{{\it Act.~Phys.~Pol.~}{\bf B #1} (#2) #3}
\def\apa#1#2#3{{\it Act.~Phys.~Austr.~}{\bf#1} (#2) #3}
\def\cmp#1#2#3{{\it Comm.~Math.~Phys.~}{\bf #1} (#2) #3}
\def\cpc#1#2#3{{\it Comp.~Phys.~Commun.~}{\bf #1} (#2) #3}
\def\epjc#1#2#3{{\it Eur.\ Phys.\ J.\ }{\bf C #1} (#2) #3}
\def\fortp#1#2#3{{\it Fortschr.~Phys.~}{\bf#1} (#2) #3}
\def\ijmpa#1#2#3{{\it Int.~J.~Mod.~Phys.~}{\bf A #1} (#2) #3}
\def\jcp#1#2#3{{\it J.~Comp.~Phys.~}{\bf #1} (#2) #3}
\def\jetp#1#2#3{{\it JETP~Lett.~}{\bf #1} (#2) #3}
\def\mpl#1#2#3{{\it Mod.~Phys.~Lett.~}{\bf A #1} (#2) #3}
\def\nima#1#2#3{{\it Nucl.~Inst.~Meth.~}{\bf A #1} (#2) #3}
\def\npb#1#2#3{{\it Nucl.~Phys.~}{\bf B #1} (#2) #3}
\def\nca#1#2#3{{\it Nuovo~Cim.~}{\bf #1A} (#2) #3}
\def\plb#1#2#3{{\it Phys.~Lett.~}{\bf B #1} (#2) #3}
\def\prc#1#2#3{{\it Phys.~Reports }{\bf #1} (#2) #3}
\def\prd#1#2#3{{\it Phys.~Rev.~}{\bf D #1} (#2) #3}
\def\pR#1#2#3{{\it Phys.~Rev.~}{\bf #1} (#2) #3}
\def\prl#1#2#3{{\it Phys.~Rev.~Lett.~}{\bf #1} (#2) #3}
\def\pr#1#2#3{{\it Phys.~Reports }{\bf #1} (#2) #3}
\def\ptp#1#2#3{{\it Prog.~Theor.~Phys.~}{\bf #1} (#2) #3}
\def\sovnp#1#2#3{{\it Sov.~J.~Nucl.~Phys.~}{\bf #1} (#2) #3}
\def\tmf#1#2#3{{\it Teor.~Mat.~Fiz.~}{\bf #1} (#2) #3}
\def\tmph#1#2#3{{\it Theor.~Math.~Phys.}{\bf #1} (#2) #3}
\def\yadfiz#1#2#3{{\it Yad.~Fiz.~}{\bf #1} (#2) #3}
\def\zpc#1#2#3{{\it Z.~Phys.~}{\bf C #1} (#2) #3}
\def\ibid#1#2#3{{ibid.~}{\bf #1} (#2) #3}

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\end{thebibliography}

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