%Title:  Five-loop vacuum polarization in pQCD:  O(alpha_s^4 n_f^2) results 
%Author: P.A. Baikov, K.G. Chetyrkin and J.H. K\"uhn
%Published: <EM> Nucl.Phys.B(Proc.Suppl.) </EM> <B> 116 </B> (2003 <EM> Proceedings of RADCOR 2002</EM> 8-13 September 2002, Kloster Banz, Germany. ) 78-82. 
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\title{
\hfill {\normalsize  TTP02-44}
%\\
%\hfill {\normalsize  hep/ph}
\vspace{2cm} 
\\
Five-loop vacuum polarization in pQCD:
$O(\alpha_s^4 N_f^2)$ results\thanks{
Talk presented at RADCOR/Loops and Legs 2002, Kloster Banz, Germany}
%Five-loop polarization functions in pQCD:
%$O(\alpha_s^4 N_f^2)$ results 
%%and applications
}

\author{P.~A.~Baikov\address{Institute of Nuclear Physics,
        Moscow State University, \\
        Moscow~~119992, Russia}%
       ,
        K.G.Chetyrkin\address{
        Fakult{\"a}t f{\"u}r Physik  Albert-Ludwigs-Universit{\"a}t Freiburg, \\
        D-79104 Freiburg, Germany}%
\thanks{On leave from Institute for Nuclear Research
of the Russian Academy of Sciences, Moscow, 117312, Russia.}
        and
        J.H.~K\"uhn\address{Institut f\"ur Theoretische Teilchenphysik, \\
        Universit\"at Karlsruhe, D-76128 Karlsruhe, Germany
        }}



\begin{document}

\begin{abstract}
We give a short presentation of the first genuine QCD five-loop
calculations, viz. analytical contributions of order $\alpha_s^4 n_f^2$ and
$m_q^2 \alpha_s^4 n_f^2$ to the vacuum polarization function of
vector  currents. These corrections form an important gauge-invariant
subset of the full ${\cal O}(\als^4)$ correction to the total cross-section 
of $e^+ e^-$ annihilation  into hadrons.
\end{abstract}

% typeset front matter (including abstract)
\maketitle

\section{Introduction}

In QCD the correlator of two currents is a central object from which
important physical consequences can be deduced (for a detailed review
see, e.g.~\cite{phys_report}).  In particular, important physical
observables like the cross-section of $e^+ e^-$ annihilation into
hadrons and the decay rate of the $Z$ boson are related to the vector
and axial-vector current correlators.  Furthermore, total decay rates
of CP even or CP odd Higgs bosons can be obtained by considering the
scalar and pseudo-scalar current densities, respectively.

From the theoretical viewpoint the two-point correlators are especially
suited for pQCD evaluations. Indeed, due to a simple kinematic (only
one external momentum) even multiloop calculations can be analytically
performed (whereas the non-perturbative contributions can be
effectively put under control with the help of the operator product
expansion \cite{svz,OPE:Wilson}).  As a consequence, the results for
practically  all physically interesting correlators (vector,
axial-vector, scalar and pseudo-scalar) are available up to order
$\alpha_s^2$ taking into account the {\em full} quark mass dependence
\cite{CheKueSte96,CheHarSte98}.

In many important cases (with the Z decay rate being a prominent
example) the momentum transfer is much larger than the masses of
(active) quarks involved. This leads to the possibility to use the
massless approximation which significantly simplifies the calculation.
As a result, in massless QCD the vector and scalar correlators are
analytically known to $\als^3$ \cite{GorKatLar91SurSam91,gvvq,gssq}.
The residual quarks mass effects can be taken into account via the
expansion in quark masses.  At present this has been done for the
quadratic and quartic terms to the same $\als^3$ order
\cite{ChetKuhn90,mq2as3,mq4as3}.


During the past years, in particular through the analysis of $Z$
decays at LEP and of $\tau$ decays, an enormous reduction of the
experimental uncertainty (down to 3\%) in $R(s)$ has been achieved
with the perspective of a further reduction by a factor of four at a
future linear collider \cite{TESLA}.

Inclusion of the ${\cal O}(\als^3)$ corrections
\cite{GorKatLar91SurSam91} is thus mandatory already now. Quark mass
effects as well as corrections specific to the axial current
\cite{KniKue90b} must be included for the case of $Z$-decays.  The
remaining theoretical uncertainty from uncalculated higher orders is
at present comparable to the experimental one
\cite{phys_report}. 
Thus, the full calculation of the next ${\cal O}(\als^4)$ contributions 
to $R(s)$ is on agenda.


A few words about technical aspects: Due to the
well-known ``optical theorem'' $R(s)$ is related to the absorptive
part of the vector vacuum polarization function.  As was first
demonstrated in \cite{me84} the absorptive part of the arbitrary
(L+1)-loop  p-integral (that is a massless Feynman integral
depending on exactly one external momentum) is expressible in terms
the corresponding (L+1)-loop UV counterterm along with some L-loop
p-integrals (the former have be to known {\em including} their
non-divergent finite parts).

The calculation of UV counterterms is simpler than that of the very
integral, because in the MS scheme any UV counterterm is polynomial in
momenta and masses \cite{Collins75}.  This observation was effectively
employed in Ref.~\cite{Vladimirov78} to simplify considerably the
calculation of UV counterterms.  The method was further developed and
named InfraRed Rearrangement (IRR) in Ref.~\cite{me79}.  It
essentially amounts to an appropriate transformation of the IR
structure of Feynman amplitudes by setting to zero some external
momenta and masses (in some cases after some differentiation is
performed with respect to the latter).  As a result the calculation of
UV counterterms is much simplified by reducing the problem to
the evaluation  of p-integrals.  The method of IRR was extended and
generalized in Ref.~\cite{me84} with the help of so-called
$R^*$-operation. It was proven that any UV counterterm for any
(L+1)-loop Feynman integral can be expressed (in purely algebraic way)
in terms of pole and finite parts of some (L)-loop p-integrals.

Thus, to compute, say, $\als^3$ terms in $R(s)$ one should essentially
deal with two separate problems: (i) the reduction to 3-loop p-integrals and (ii) 
the calculation of the huge number of resulting p-integrals.

The second problem has been solved (for the 3-loop case only!) by
elaborating a dedicated program MINCER \cite{mincer2} written in FORM
\cite{Ver91}, which implements the generic solution \cite{IBP} of the
integration-by-parts recurrence relations to evaluate massless three
loop propagators. To generalize the algorithm to the  4-loop case
(needed for ${\cal O}(\als^4)$ calculations) seems to be  a formidable problem
at present.  We have used a different method based essentially on a
large $D$-expansion ($D$ is the space-time dimension). For more
details see Refs.~\cite{bai1,bai2,Baikov:2001aa,LL02:talk_by_Baikov}.


In original ${\cal O}(\als^3)$ calculations of $R(s)$
\cite{GorKatLar91SurSam91} the IRR and reduction to p-integrals was
performed manually --- an error-prone and time-consuming procedure
difficult to implement in computer algebra. The problem was completely
solved in Ref.~\cite{gvvq}. Here a general formula was found, which
explicitly expresses the result in terms of well-known renormalization
constants and three-loop massless propagators, thus resolving the
complicated combinatorics of ultraviolet and infrared subtractions.
This formula is valid for any number of loops and is, therefore, an
important ingredient of any ${\cal O}(\als^4)$ calculation.


The calculational effort for a full evaluation of $R(s)$ in ${\cal
O}(\als^4)$ is enormous.  As first step to this task we have computed 
a  gauge invariant subset, namely the terms of order
$\als^4 n_f^2$, where $n_f$ denotes the number of fermion flavours.
The terms of order $\als^4 n_f^3$ (and, in fact, all terms of order
$\als(\als n_f)^n$) have been obtained earlier by summing the
renormalon chains \cite{VV:renormalons}. These  are numerically
small and not directly sensitive to the nonabelian character of QCD,
in contrast to the terms of order $\als^4 n_f^2$.

This talk presents  results for the (diagonal) vector correlator
at   order $\als^4 n_f^2$. We also compare  our
results with some of available predictions following from  various
``optimization'' schemes. 
%\section{Notations}
\section{Results}
To fix notation we start with considering  two-point correlator
of vector   quark currents and corresponding vacuum polarization
function  ($j_\mu^v= \ovl{Q}\gamma_\mu Q$;
$Q$ is a  quark field with  mass $m$, all other $n_f-1$ quarks  are
assumed to be massless)
\begin{eqnarray}
\nnb
\Pi_{\mu\nu}(q)  &=&
 i \int {\rm d} x e^{iqx}
\langle 0|T[ \;
\;j_{\mu}^{v}(x)j_{\nu}^{v}(0)\;]|0 \rangle
\\
&=&
\displaystyle
(-g_{\mu\nu}q^2  + q_{\mu}q_{\nu} )\Pi(q^2)
{}.
\label{PiV}
\end{eqnarray}
The  physical observable $R(s)$ is related to $\Pi(q^2)$
by
$
R(s) = 12 \pi \Im \, \Pi(q^2 + i\epsilon)
{}.
%\label{R(S):def}
$ 
It is convenient to decompose $R(s)$ into the  massless and the quadratic
in the quark mass contributions as follows:
\bea
\nnb
R(s) &=& 3 \left\{r^V_0 +  \frac{m^2}{s} r^{V}_2 \right\}+\dots 
\\
 &=& 3\left\{\sum_{i \ge 0} a_s^i\left(
 r^{V,i}_0 + \frac{m^2}{s} r^{V,i}_2 \right)
                                    \right\} +\dots
\nnb
{}.
\eea
Here we have set the normalization scale  $\mu^2= s$;
$a_s = \als(s)/\pi$ and dots stand for corrections proportional
to $m^4$ and higher powers of the quark mass.
Since the function $R(s)$ is  known to order $\als^3$ from works
\cite{GorKatLar91SurSam91,gvvq} (massless limit)
and \cite{ChetKuhn90} (${\mathcal{O}}(m^2)$ contribution)
we give below only the ($\msbar$-renormalized \cite{msbar}) 
${\cal O}(\als^4)$ results.
\begin{eqnarray}
\lefteqn{r^{V,4}_0 =  } 
\nonumber\\
&{+}& \, n_f^3
\left[
-\frac{6131}{5832} 
+\frac{11}{432}  \pi^2
+\frac{203}{324}  \,\zeta_{3}
\BreakI
\phantom{+ \, n_f^3}
-\frac{1}{54}  \pi^2 \,\zeta_{3}
+\frac{5}{18}  \,\zeta_{5}
%zero == 0
\right]
\nonumber\\
&{+}& \, n_f^2
\left[
\frac{1045381}{15552} 
-\frac{593}{432}  \pi^2
-\frac{40655}{864}  \,\zeta_{3}
\BreakI
\phantom{+ \, n_f^2}
+\frac{11}{12}  \pi^2 \,\zeta_{3}
+\frac{5}{6}  \,\zeta_3^2
-\frac{260}{27}  \,\zeta_{5}
%zero == 0
\right]
+\dots
{},
\label{r4^V_0:qcd}
\end{eqnarray}
%zero == 0
\begin{eqnarray}
\lefteqn{r^{V,4}_{2} =  } 
\nonumber\\
&{+}& \, n_f^3
\left[
-\frac{2705}{1944} 
+\frac{13}{108}  \pi^2
%zero == 0
\right]
\nonumber\\
&{+}& \, n_f^2
\left[
\frac{91943}{486} 
-\frac{1121}{108}  \pi^2
+\frac{13}{324}  \,\zeta_{3}
+\frac{53}{9}  \,\zeta_3^2
\BreakI
\phantom{+ \, n_f^2}
-\frac{3610}{81}  \,\zeta_{5}
%zero == 0
\right]
+\dots
{}
\label{r4^V_2:qcd}
\end{eqnarray}
%zero == 0
Here dots stand for
still unknown terms of order $\als^4 n_f$ and $\als^4 n_f^0$.
Numerically $r^V_0$ and $r^V_2$ read
\bea
\lefteqn{r^V_0 =
1 + a_s +   a_s^2 \left( 1.986 - 0.1153 \ n_f \right)
}
\nonumber
\\
&+& 
\nnb
a_s^3
\left(
-6.6369 - 1.2001 \ n_f - 0.00518 \ n_f^2
\right)
\\
\nonumber
&+& a_s^4
\left(
 0.02152 \ n_f^3 - 0.7974 \ n_f^2 
\nnb
\right.
\\
&{}& \phantom{MMMMMM} +
\left.
\ n_f \ r^{V,4}_{0,1} + r^{V,4}_{0,0}
\right)
{},
\label{RV:num}
\eea
%\input{/pchome/chet/MATH/problems/R5l2N/talk.lo}
\bea
\lefteqn{r^V_2 =
12  a_s +   a_s^2 \left( 126.5 - 4.33333 \ n_f \right)
}
\nonumber
\\
&+& 
\nnb
a_s^3
\left(
1032.14 - 104.167 n_f + 1.21819 n_f^2
\right)
\\
\nonumber
&+& a_s^4
\left(
49.0839 n_f^2  - 0.203453 n_f^3
\nnb
\right.
\\
&{}& \phantom{MMMMMM} +
\left.
\ n_f \ r^{V,4}_{2,1} + r^{V,4}_{2,0}
\right)
{}.
\label{RV_2:num}
\eea
The result for the $a_s^4$ terms in $r^{V}_0$ has been already
presented in \cite{Baikov:2001aa}, the coefficients of the $a_s^4$
terms in  $r^{V}_2$  are new.


For calculation of $r^{V,4}_0$ we have had to deal with divergent
parts of five-loop diagrams and finite parts of the four-loop ones. In
fact, as was first discovered in Ref.~\cite{ChetKuhn90},  the
quadratic mass correction  $r^{V,4}_2$ can be obtained with the help of
renormalization group methods {\em exclusively} from 
the four-loop function $\Pi_2$ defined as
($ a_s = \als(\mu^2),\mu = -q^2$ ) 
\[
\nnb
\Pi   = 
\frac{3}{16 \pi^2} \left(
\Pi_0  + \frac{m^2}{-q^2} \Pi_2
\right)
{},
\ \ \ 
\Pi_2 = \sum_{i \ge 0}\   a_s^i k^{V,i}_2
{}.
\]
The coefficients  $k^{V,i}_2$ for $i=1,2$ are known from \cite{GorKatLar86}
while our result for  $k^{V,3}_2$ reads
\begin{eqnarray}
\lefteqn{k^{V,3}_{2} =  } 
\nonumber\\
&{+}& \, n_f^2
\left[
-\frac{5161}{1458} 
-\frac{8}{27}  \,\zeta_{3}
%zero == 0
\right]
\nonumber\\
&{+}& \,n_f 
\left[
\frac{62893}{162} 
-\frac{4150}{243}  \,\zeta_{3}
+\frac{424}{27}  \,\zeta_3^2
+\frac{20}{3}  \,\zeta_{4}
\BreakI
\phantom{+ \,n_f }
-\frac{28880}{243}  \,\zeta_{5}
%zero == 0
\right]
+ \dots
{}
\label{kV4_2:qcd}
\end{eqnarray}
or, numerically,
\bea
\nnb
\lefteqn{\Pi_2 =
-  8 - 21.333 \, a_s 
+ ( 10.56 \, n_f -224.80 ) \, a_s^2
 }
\\
&+& a_s^3\, ( 274.37\, n_f - 3.896 \, n_f^2  + k^{V,3}_{2,0} )
{}.
\label{Pi_2:num}
\eea



\section{Discussion}

In this section we will compare our results with predictions of
FAC/PMS optimization methods. FAC (Fastest Apparent Convergence) and
PMS (Principle of Minimal Sensitivity) methods are both based
eventually on the concept of scheme-invariant properties and the idea
of the choice of an "optimal" scheme to provide better convergence of
the resulting perturbative series \cite{ste,gru,krasn,KS}.


 For both methods the optimal scheme depends on the physical
 observable we are dealing with. With FAC it should be a scheme which
 minimizes (set to zero by construction) all the terms of order
 $\als^2$ and higher, while the PMS scheme is fixed by the requirement
 that the perturbative expansion for the observable is as insensitive
 as possible to a change in the scheme. The assumption that a
 renormalization scheme is in a sense optimal sets certain constraints
 on not yet computed higher order corrections in any other
 scheme. 



Since the massless case  have been already discussed in  Ref.~\cite{Baikov:2001aa}
we concentrate here on the mass correction.  
In Table 1 we list the FAC/PMS predictions for the the
function $\Pi_2$. The predictions have been obtained
following the work  \cite{CheKniSir97}.
{}From the three entries corresponding to $n_f = 3,4$ and $5$ one
easily restores the FAC/PMS prediction for the $n_f$ dependence of the
$\as^3$ term in $\Pi_{2}$ 
\bea 
k^{V,3}_{2}(FAC) &=& -2886 + 277 \, n_f  -4.2 \, n_f^2  
{},
\\
k^{V,3}_{2}(PMS) &=& -2916 + 282 \, n_f  -4.2 \, n_f^2  
\label{K4:nf}
{}.
\eea
A comparison to (\ref{Pi_2:num}) shows a very good agreement with exactly known
terms of order $n_f$ and $n_f^2$.

At last, encouraged  by the agreement, we use the approach of
\cite{ChetKuhn90} to deduce from Table 1 the full result for the
quadratic mass correction to $R(s)$.  We will do it separately for
$n_f =2,3,4,5$ and 6
respectively (as FAC and PMS predictions are so close one each other
we will consider for definiteness the latter case).

\bea
\nnb
\lefteqn{r^V_2 (n_f =2)=} \\  &{}& 12 \,
(\, a_s + 9.819 \, a_s^2  + 69.056 \, a_s^3  +371 \, a_s^4)
\\
\nnb
\lefteqn{r^V_2 (n_f =3)=} \\  &{}& 12 \,
(\, a_s + 9.459 \, a_s^2  + 60.883 \, a_s^3  + 271 \, a_s^4)
\\
\nnb
\lefteqn{r^V_2 (n_f =4)=} \\  &{}&  12\, 
(\, a_s + 9.097 \, a_s^2  +  52.913 \, a_s^3  +  179 \, a_s^4)
\\
\nnb
\lefteqn{r^V_2 (n_f =5) =}  \\  &{}&  12\, 
(\, a_s + 8.736 \, a_s^2  +   45.146 \, a_s^3  +  96\, a_s^4)
\\
\nnb
\lefteqn{r^V_2 (n_f =6) =}  \\  &{}&  12\, 
(\, a_s + 8.375 \, a_s^2  +   37.582 \, a_s^3  +  21\, a_s^4)
\label{fin}
\eea



\begin{table}[hbt]
%\setlength{\tabcolsep}{1.5pc}
% -----------------------------------------------------
% adapted from TeX book, p. 241
%%%\newlength{\digitwidth} \settowidth{\digitwidth}{\rm 0}
%%%\catcode`?=\active \def?{\kern\digitwidth}
% -----------------------------------------------------
\caption{\label{t1}
Estimates for  the  coefficient $k^{V,3}_2$
based on  FAC and PMS optimizations.
}
\renewcommand{\arraystretch}{1.2}
%\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
Method& $n_f=3$ & $n_f=4$ & $n_f=5$  \\
\hline\hline
FAC & $-2092  $  & $ - 1844 $ & $- 1604 $ \\
\hline
PMS & $ -2109 $ & $-1856$ & $-1612 $\\
\hline
\end{tabular}
%\end{center}
\end{table}






\section{Conclusions}


We have presented the results of first genuine QCD five-loop
calculations viz. analytical contributions of order $\alpha_s^4 n_f^2$
and $m_q^2 \alpha_s^4 n_f^2$ to the vacuum polarization function of
vector currents. The results are tested against predictions
of FAC/PMS optimization methods.  A good agreement is found.


Our results have direct relevance for the theoretical description of
the $\tau$ decay rate, including its strange quark mas dependence.
For a detailed discussion of the issue see Ref.~\cite{preparations}.




The present calculation does not include all possible topological
types of diagrams appearing in order $\als^4$.  Our experience shows
that these new topologies can be solved in the same way, given
sufficient computer resources. Work in this direction is under way.
  
The authors are grateful to M.~Steinhauser  for the carefull reading 
of the manuscript and   very useful comments.  This work was supported by the
DFG-Forschergruppe {\it ``Quantenfeldtheorie, Computeralgebra und
Monte-Carlo-Simulation''} (contract FOR 264/2-1), by INTAS (grant
00-00313), by RFBR (grant 01-02-16171), by Volkswagen Foundation and
by the European Union under contract HPRN-CT-2000-00149.



\begin{thebibliography}{9}

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\def\cmp#1#2#3{  {Comm.\ Math.\ Phys.\ }{\ice{\bf}#1} (#2) #3}
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\def\sovnp#1#2#3{{Sov.\ J.\ Nucl.\ Phys.\ }{\ice{\bf} #1} (#2) #3}
\def\tmf#1#2#3{  {Teor.\ Mat.\ Fiz.\ }{\ice{\bf} #1} (#2) #3}
\def\yadfiz#1#2#3{{Yad.\ Fiz.\ }{\ice{\bf} #1} (#2) #3}
\def\zpc#1#2#3{  {Z.\ Phys.\ C }{\ice{\bf} #1} (#2) #3}
\def\ibid#1#2#3{ {ibid.\ }{\ice{\bf} #1} (#2) #3}


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\end{thebibliography}
\end{document}















\ice{
and
\beq
\EQN{ss}
 \widetilde{\Pi}(Q^2)
=  i\int dx e^{iqx}\langle 0|\;T[\;j^{s}_f(x)
j^{s}_{f'}(0)\,]\;|0\rangle
\label{PiS}
{}.
\eeq
}
\ice{
Both correlators are divergent even after standard renormalization of
the QCD coupling constant $\alpha_s = g_s^2/(4 \pi)$ (and of the scalar
current for (\ref{ss})) has been performed. The divergency is related
to the integration region $x \approx 0$, does not appear in the
absorptive part of the polarization functions and is traditionally
avoided by defining "Adler functions" as follows:
\beq
D(Q^2) \equiv \frac{1}{4}Q^2 \frac{ {\rm d}}{{\rm d}\  Q^2} \Pi(Q^2)
= Q^2\int_0^\infty \frac{R(s) d s }{(s+Q^2)^2}
\eeq
and
\beq
\ovl{D}(Q^2) \equiv \frac{1}{4} Q^2
\left(
\frac{ {\rm d}}{ {\rm d} \ Q^2}
\right)^2
\widetilde{\Pi}(Q^2) = 2 Q^2\int_0^\infty \frac{s \widetilde{R}(s) d s }{(s+Q^2)^3}
\label{DSbar}
\eeq
Here, $R(s) = 1+ \alpha_s/\pi + \dots$ is the famous $R$-ratio in the massless
limit and
$\widetilde{R}(s) = 1+ \frac{17}{3} \alpha_s/\pi + \dots $ is  properly normalized
imaginary part of the scalar polarization function.
Our results for $D, \ovl{D}, R$ and  , $\tilde{R}$ are presented below. First, we  define  
\beq
D(Q^2) = 1+ \sum_{i=1}^{4} d_i\  a_s^i, \ \ 
R(s)   = 1 + \sum_{i=1}^{4} r_i \ a_s^i
\label{V:expan:def}
\eeq
and 
\beq
\ovl{D}(Q^2) = 1+ \sum_{i=1}^{4} \  \ovl{d}_i a_s^i,  \ \ 
\tilde{R}(s) = 1 + \sum_{i=1}^{4} \  \tilde{r}_i a_s^i
\label{S:expan:def}
{},
\eeq
where we have set the normalization scale $\mu^2$ to $Q^2$  for the  Adler functions
and to  $s$ for the  "$R$"-functions.

{\it another version of the phrase above:}

where we have set the normalization scale $\mu^2=Q^2$ or to $\mu^2= s$  for the  
euclidean and minkowskian  functions respectively. 
}


\ice{
The results for generic values of $\mu$  can  be  easily recovered with the standard
RG tecniques. Since the functions (\ref{V:expan:def}) and  (\ref{S:expan:def})
are known from Refs.~\cite{} to order $\als^3$ we cite below only 
the ${\cal O}(\als^4)$ results. 
\begin{eqnarray}
d_4 &=&   
 \,C_F  T^3 \, N_f^3
\left[
-\frac{6131}{972} 
+\frac{203}{54}  \,\zeta_{3}
+\frac{5}{3}  \,\zeta_{5}
%zero == 0
\right]
{+} C_F^2 T^2 \, N_f^2
\left[
\frac{5713}{1728} 
-\frac{581}{24}  \,\zeta_{3}
+3  \,\zeta(3)^2
+\frac{121}{6}  \,\zeta_{5}
%zero == 0
\right]
\nonumber\\
&{+}& \,C_F \,C_A  T^2 \, N_f^2
\left[
\frac{340843}{5184} 
-\frac{10453}{288}  \,\zeta_{3}
-\frac{1}{2}  \,\zeta(3)^2
-\frac{170}{9}  \,\zeta_{5}
%zero == 0
\right]
+\dots
{},
\label{d4}
\end{eqnarray}
%
\begin{eqnarray}
r_4 &=&   
{+} \,C_F  T^3 \, N_f^3
\left[
-\frac{6131}{972} 
+\frac{11}{72}  \pi^2
+\frac{203}{54}  \,\zeta_{3}
-\frac{1}{9}  \pi^2 \,\zeta_{3}
+\frac{5}{3}  \,\zeta_{5}
%zero == 0
\right]
\nonumber\\
&{+}& C_F^2 T^2 \, N_f^2
\left[
\frac{5713}{1728} 
-\frac{1}{24}  \pi^2
-\frac{581}{24}  \,\zeta_{3}
+3  \,\zeta(3)^2
+\frac{121}{6}  \,\zeta_{5}
%zero == 0
\right]
\\
\nonumber
&{+}& \,C_F \,C_A  T^2 \, N_f^2
\left[
\frac{340843}{5184} 
-\frac{65}{48}  \pi^2
-\frac{10453}{288}  \,\zeta_{3}
+\frac{11}{12}  \pi^2 \,\zeta_{3}
-\frac{1}{2}  \,\zeta(3)^2
-\frac{170}{9}  \,\zeta_{5}
%zero == 0
\right]
+\dots
{},
\label{r4}
\end{eqnarray}

\begin{eqnarray}
\bar{d}_4 &=&   
{+} \,C_F  T^3 \, N_f^3
\left[
-\frac{142531}{93312} 
-\frac{5}{24}  \,\zeta_{3}
+\frac{1}{24}  \,\zeta_{4}
+\frac{5}{3}  \,\zeta_{5}
%zero == 0
\right]
\nonumber\\
&{+}& C_F^2 T^2 \, N_f^2
\left[
\frac{2186821}{124416} 
-\frac{2161}{96}  \,\zeta_{3}
+3  \,\zeta(3)^2
-\frac{15}{32}  \,\zeta_{4}
+\frac{55}{6}  \,\zeta_{5}
%zero == 0
\right]
\\
\nonumber
&{+}& \,C_F \,C_A  T^2 \, N_f^2
\left[
\frac{5977453}{248832} 
+\frac{149}{144}  \,\zeta_{3}
-\frac{1}{2}  \,\zeta(3)^2
+\frac{15}{32}  \,\zeta_{4}
-\frac{2501}{144}  \,\zeta_{5}
%zero == 0
\right]
+ \dots
{},
\label{ds4}
\end{eqnarray}
%
\begin{eqnarray}
&{}&\tilde{r}_4 =   
{+} \,C_F  T^3 \, N_f^3
\left[
-\frac{520771}{93312} 
+\frac{275}{648}  \pi^2
-\frac{1}{360}  \pi^4
+\frac{65}{72}  \,\zeta_{3}
-\frac{1}{9}  \pi^2 \,\zeta_{3}
+\frac{1}{24}  \,\zeta_{4}
+\frac{5}{3}  \,\zeta_{5}
%zero == 0
\right]
\nonumber\\
&{+}& C_F^2 T^2 \, N_f^2
\left[
\frac{7009861}{124416} 
-\frac{6605}{1728}  \pi^2
+\frac{11}{480}  \pi^4
-\frac{4073}{96}  \,\zeta_{3}
+\frac{11}{12}  \pi^2 \,\zeta_{3}
+3  \,\zeta(3)^2
-\frac{15}{32}  \,\zeta_{4}
+\frac{85}{6}  \,\zeta_{5}
%zero == 0
\right]
\\
\nonumber
&{+}& \,C_F \,C_A  T^2 \, N_f^2
\left[
\frac{18248293}{248832} 
-\frac{16031}{3456}  \pi^2
+\frac{11}{480}  \pi^4
-\frac{1243}{144}  \,\zeta_{3}
+\frac{19}{24}  \pi^2 \,\zeta_{3}
-\frac{1}{2}  \,\zeta(3)^2
+\frac{15}{32}  \,\zeta_{4}
-\frac{2621}{144}  \,\zeta_{5}
%zero == 0
\right]
+\dots
{},
\label{rs4}
\end{eqnarray}
Here 
we have distinguished between contributions proportional to different 
color factors (for SU(3) $C_F = 4/3$, $C_A = 3$, and
$T{}=1/2$, $N_f$ represents the number of fermions);
$\dots$ stand for still unknown terms of order $\als^4 N_f$ and  $\als^4 N_f^0$. 

Numerically  the $R$-rations read ($\as = \als(s)/\pi$):
\bea
R(s) = 
%\left\{
1 &+& a_s +
 a_s^2 \left( 1.98571 - 0.115295 \ N_f
\right)
%\right.
\nonumber
+ a_s^3
\left(
-6.63694 - 1.20013 \ N_f - 0.00518 \ N_f^2   
\right)
\nonumber
\\
&+& a_s^4
%\left.
\left(
 0.0215161 \ N_f^3  -1.10467 \ N_f^2  + \ N_f \ r_{(4,1)} + r_{(4,1)} 
\right)
%\right\}
{}.
\label{RV:num}
\eea
%
\bea
\widetilde{R}(s) = 
%\left\{
1 &+& 5.66667 a_s +
 a_s^2 \left(1.98571 - 0.11529 \ N_f
\right)
%\right.
\nonumber
+ a_s^3
\left(164.139 - 25.7712 \ N_f + 0.25897 \ N_f^2
\right)
\nonumber
\\
&+& a_s^4
%\left.
\left(
- 0.0204599 \ N_f^3 + 9.37757 \ N_f^2  + \ \tilde{r}_{(4,1)} + \tilde{r}_{(4,1)} 
\right)
%\right\}
{}.
\label{RS:num}
\eea
}
