%Title: R(s), Bjorken sum rule and the Crewther Relation in order alpha_s^4
%Author: K.G. Chetyrkin

\documentclass{PoS}



\usepackage{dcolumn}
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\newcommand{\Tf}{ T_f }


\newcommand{\ice}[1]{\relax}

\newcommand{\re}[1]{(\ref{#1})}

\newcommand{\unl}{\underline}

\newcommand{\as}{a_s}
\newcommand{\api}{\frac{\alpha_s}{\pi}}
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\newcommand{\asthree}{{}^{2\times{\cal O}(\as^3)}_{{\rm no}\;{\cal O}(\as^3)}}
\newcommand{\asfour}{{}^{2\times{\cal O}(\as^4)}_{{\rm no}\;{\cal O}(\as^4)}}


%below from my SS-PLB paper of 97

\newcommand{\ds}{\displaystyle}
%\newcommand{\as}{\alpha_s}
\newcommand{\gm}{\gamma_m}
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\newcommand{\EQN}{\label}
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%below is from SS.tex

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\newcommand{\Break}{ \right. \nonumber \\ &{}& \left. }
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                     }


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%%% for qcd constraints
\newcommand{\nn}{\nonumber}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\newcommand{\dd}{{\rm d}}


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% this is version with T_f redifined as  T_f*tr !!! 

\ice{
Out[69]= {\cfC, \cfAtrBnlB, \cfBtrAnlA, \cfAtrAnlAcaA, \cfBcaA, \cfAcaB}

In[70]:= texclfs2["Rns"]

texclfs2["Rns"]

Out[70]= {\cfB, \cfAcaA, \cfAtrAnlA}
}

% as^1 

\newcommand{\sbz}{  }

% small space,

\newcommand{\smsp}{  }


\newcommand{\cfA}{C_F }

%2 as^2 

\newcommand{\cfB}{C_F^2 }
\newcommand{\cfAcaA}{C_F \smsp  C_A }
\newcommand{ \cfAtrAnlA}{C_F\smsp T_f}


%3 as^3

\newcommand{\cfC}{C_F^3 }

\newcommand{\cfAtrBnlB}{ C_F \smsp T_f^2  }

\newcommand{\cfBtrAnlA}{ C_F^2\smsp T_f }

\newcommand{\cfAtrAnlAcaA}{ C_F\smsp T_f\smsp   C_A}

\newcommand{\cfBcaA}{C_F^2\smsp  C_A}

\newcommand{\cfAcaB}{C_F\smsp  C_A^2}



%4 as^4

\newcommand{\cfD}{C_F^4 }
\newcommand{\dFFdRinvAnlA}{n_f\smsp \frac{d_F^{abcd}\smsp d_F^{abcd}}{d_R}}

\newcommand{\dFARinvA}{\frac{d_F^{abcd}\smsp d_A^{abcd}}{d_R}}

\newcommand{\cfAtrCnlC}{C_F  \smsp T_f^3}

\newcommand{\cfBtrBnlB}{C_F^2 \smsp   T_f^2}

\newcommand{\cfAtrBnlBcaA}{  C_F \smsp T_f^2\smsp C_A}

\newcommand{\cfCtrAnlA}{ C_F^3 \smsp  T_f }

\newcommand{\cfBtrAnlAcaA}{ C_F^2 \smsp T_f\smsp  C_A}

\newcommand{\cfAtrAnlAcaB}{ C_F \smsp T_f\smsp  C_A^2}

\newcommand{\cfCcaA}{C_F^3 \smsp C_A}

\newcommand{\cfBcaB}{C_F^2 \smsp C_A^2}

\newcommand{\cfAcaC}{C_F \smsp C_A^3}







\title{$R(s)$,  Bjorken sum rule and the Crewther Relation in order $\alpha_s^4$}

\ShortTitle{$R(s)$, Bjorken sum rule and the Crewther Relation in order $\alpha_s^4$}


\ice{
\author{\speaker{Konstantin Chetyrkin}
         \thanks{WWWWWWWWWWw.}\\
        Institut fuer Theoretische Teilchenphysik\\
        E-mail: \email{chet@particle.physik.uni-karlsruhe.de}}

%\author{Another Author\\
%        Affiliation\\
%        E-mail: \email{...}}

}


\author{\speaker{K.~G.~Chetyrkin}\thanks{
The talk is  based  on recent  results obtained in a collaboration to P.A.~Baikov and 
J.H.~K\"uhn \cite{Baikov:2010}. }\\
      Institut f\"ur Theoretische Teilchenphysik,
  Universit\"at Karlsruhe, D-76128 Karlsruhe, Germany
 \\
        E-mail: \email konstantin.chetyrkin@kit.edu}











\abstract{
We report on our calculation of  the order $\alpha_s^4$
contributions to the Bjorken sum rule for  polarized electron-nucleon  scattering
and to the (non-singlet) part of the cross section for electron-positron
annihilation into hadrons for the  case of a  generic colour gauge group.
We confirm at the same order a (generalized)
Crewther relation between the Bjorken sum rule and the corresponding Adler function  which provides a strong test
of the correctness of our previously obtained  results:  the ratio $R(s)$ in QCD
and the five-loop $\beta$-function in  quenched QED.
}

\FullConference{RADCOR 2009 - 9th International Symposium on Radiative Corrections
(Applications of Quantum Field Theory to Phenomenology)\\
October 25-30 2009\\
Ascona, Switzerland
}

\begin{document}

\section{Introduction}

At the previous RADCOR Symposium we presented \cite{Baikov:2008cp} two new results.
First, corrections of order $\alpha_s^4$ the (non-singlet) part of the
cross section for electron-positron annihilation into hadrons in QCD
with $n_f=3$. Since then the first result has been extended to  QCD with generic 
value of the quark flavours $n_f$  \cite{Baikov:2008jh}.  


Second,   
the five-loop contribution to the $\beta$ function of  quenched  QED
which  has revealed  an unexpected\footnote{
Unexpected, because there existed a wide-spread belief that the
rationality property is not accidental but holds also in higher orders
\cite{Bender:1976pw,Broadhurst:1999xk}  appearance of the irrational constant
$\zeta_3$ at five loops.} This  fact has raised   some  doubts on  the
correctness of the  QCD result for the order $\alpha_s^4$
and resulted in  a call for an independent testing and/or reevaluation of our results
 \cite{Kataev:2008sk,Kataev:2008nc}.



At the present  Symposium we are happy to report  about two new results  \cite{Baikov:2010} which 
are not only of interest by themselves but also provide us  with a highly non-trivial confirmation 
of the  validity of calculations  of \cite{Baikov:2008cp,Baikov:2008jh}.
Namely, we discuss calculations the order $\alpha_s^4$
contributions to the Bjorken sum rule for  polarized electron-nucleon  scattering 
and to the (non-singlet) Adler function for the  case of a  generic colour gauge group. 
%
These two seemingly  disconnected quantities
are related in  a non-obvious way by the (generalized)
Crewther relation 
%(gCR) 
\cite{Crewther:1972kn,Broadhurst:1993ru}.
%connects
The results are checked to be  in full agreement with 
the constraints imposed by the Crewther relation.

\section{ Adler Function and R(s) to order  $\alpha_s^4$ in a general gauge theory}

The  Adler function is defined through  the 
correlator of the vector  current $j_\mu$
\beq
3\smsp  Q^2\smsp  \Pi(Q^2) = 
i\int\dd^4 xe^{iq\cdot x}\langle 0|{\rm  T}j_\mu(x)j^\mu(0)|0\rangle
{},
\eeq
as follows
\beq
{D}(Q^2) =  -12\, \pi^2
Q^2\, \frac{\mathrm{d}}{\mathrm{d} Q^2} \Pi(Q^2)
=
 d_R\left(
1 
+ \frac{3}{4}\,C_F\, a_s +
\sum_{i=2}^{\infty} \  {d}_i \, a_s^i(Q^2)
\right),
\eeq
where  $Q^2 = -q^2$, $d_R$ is the dimension of the quark colour representation (for
QCD $d_R=3$), $a_s \equiv \alpha_s/\pi$ and  the normalization scale
$\mu^2$ is set  $\mu^2=Q^2$.
%
In fact, the Adler  function is the main theoretical object required to study
such important physical observables as the  cross section for electron-positron
annihilation into hadrons and  the hadronic  decay rates of both  the $Z$-boson and
the $\tau$-lepton (see, e.g.  \cite{Chetyrkin:1996ia}).
%
The relation between the Adler function and 
the ratio
$R(s)\equiv \sigma(e^+e^-\to {\rm hadrons}) / \sigma(e^+e^-\to\mu^+\mu^-)$
is (in the next expression  $\mu^2$ is set to $s$)
\beq
R(s) = D(s) - \pi^2 \beta_0^2 \{ \frac{ d_1}{3} a_s^3 + (d_2 + \frac{5}{6\,\beta_0 }d_1\,\beta_1)\,a_s^4\} + \dots
= d_R\left(
1 
+ \frac{3}{4}\,C_F\, a_s +
\sum_{i=2}^{\infty} \  {r}_i \, a_s^i(s)
\right)
{}.
\eeq
Here $\beta(a_s) = \mu^2\,\frac{\mathrm{d}}{\mathrm{d} \mu^2}
a_s(\mu) =-\sum_{i \ge 0} \beta_i a_s^{i+2}$  is the QCD
$\beta$-function with its first term $ \beta_0 = \frac{11}{12}\, C_A -\frac{T}{3}\,n_f$
being responsible for {asymptotic freedom} of QCD.
Note that we consider only 
the so-called ``non-singlet''
contribution to the Adler function and do not write explicitly a
common factor $\sum_i Q_i^2$ (with $Q_i$ being an electric charge of the $i$-th quark  flavour)
for $R(s)$. 

Our results for the  coefficients $d_4$  and $r_4$
for  a general gauge group read\footnote{The full expressions
for lower order terms  can be found in the  original
publications \cite{Gorishnii:1990vf,Surguladze:1990tg}.} 

%Adler function and and $R(s)$
\ice{
\bea
d_2 &=&
-\frac{3}{32}\cfB
{+}\cfAtrAnlA
\!
\left[
 \sbz \zeta_{3}
-\frac{11}{8}
%zero == 0
\right]
{+}\cfAcaA
\!
\left[
\frac{123}{32}
-\frac{11\zeta_{3} }{4  } 
%\sbz \zeta_{3}
%zero == 0
\right]
{},
\label{d2}
\\
d_3 &=&  
-\frac{69}{128}
{}\cfC
%\nonumber\\
{+}\cfBtrAnlA
\left[
-\frac{29}{64}
+\frac{19}{4}  \sbz \zeta_{3}
-5  \sbz \zeta_{5}
%zero == 0
\right]
\nonumber
{+}
{
\cfAtrBnlB
\left[
\frac{151}{54}
-\frac{19}{9}  \sbz \zeta_{3}
%zero == 0
\right]
\nn
%\\
%&{+}&
+
\cfBcaA
\left[
-\frac{127}{64}
-\frac{143}{16}  \sbz \zeta_{3}
+\frac{55}{4}  \sbz \zeta_{5}
%zero == 0
\right]
}
\\
{}
%\hspace{1.6cm}
&+&
\cfAtrAnlAcaA
\left[
-\frac{485}{27}
+\frac{112}{9}  \sbz \zeta_{3}
+\frac{5}{6}  \sbz \zeta_{5}
%zero == 0
\right]
\nonumber
{}
+
\cfAcaB
\left[
\frac{90445}{3456}
-\frac{2737}{144}  \sbz \zeta_{3}
-\frac{55}{24}  \sbz \zeta_{5}
%zero == 0
\right]
{},
\label{d3}
\end{eqnarray}
}
\begin{eqnarray}
%\lefteqn{d_4 =  } \nonumber\\
d_4 & = &\dFARinvA
\left[
\frac{3}{16} 
-\frac{1}{4}  \sbz \zeta_{3}
-\frac{5}{4}  \sbz \zeta_{5}
%zero == 0
\right]
%\nonumber\\
{+}\dFFdRinvAnlA
\left[
-\frac{13}{16} 
-  \sbz \zeta_{3}
+\frac{5}{2}  \sbz \zeta_{5}
%zero == 0
\right]
%\nonumber\\
{+}\cfD
\left[
\frac{4157}{2048} 
+\frac{3}{8}  \sbz \zeta_{3}
%zero == 0
\right]
\nonumber\\
&{+}&\cfCtrAnlA
\left[
\frac{1001}{384} 
+\frac{99}{32}  \sbz \zeta_{3}
-\frac{125}{4}  \sbz \zeta_{5}
+\frac{105}{4}  \sbz \zeta_{7}
%zero == 0
\right]
%\nonumber\\
{+}\cfBtrBnlB
\left[
\frac{5713}{1728} 
-\frac{581}{24}  \sbz \zeta_{3}
+\frac{125}{6}  \sbz \zeta_{5}
+3  \sbz \zeta_3^2
%zero == 0
\right]
\nonumber\\
&{+}&\cfAtrCnlC
\left[
-\frac{6131}{972} 
+\frac{203}{54}  \sbz \zeta_{3}
+\frac{5}{3}  \sbz \zeta_{5}
%zero == 0
\right]
{+}\cfCcaA
\left[
-\frac{253}{32} 
-\frac{139}{128}  \sbz \zeta_{3}
+\frac{2255}{32}  \sbz \zeta_{5}
-\frac{1155}{16}  \sbz \zeta_{7}
%zero == 0
\right]
\nonumber\\
&{+}&\cfBtrAnlAcaA
\left[
\frac{32357}{13824} 
+\frac{10661}{96}  \sbz \zeta_{3}
-\frac{5155}{48}  \sbz \zeta_{5}
-\frac{33}{4}  \sbz \zeta_3^2
-\frac{105}{8}  \sbz \zeta_{7}
%zero == 0
\right]
\nonumber\\
&{+}&\cfAtrBnlBcaA
\left[
\frac{340843}{5184} 
-\frac{10453}{288}  \sbz \zeta_{3}
-\frac{170}{9}  \sbz \zeta_{5}
-\frac{1}{2}  \sbz \zeta_3^2
%zero == 0
\right]
\nonumber\\
&{+}&\cfBcaB
\left[
-\frac{592141}{18432} 
-\frac{43925}{384}  \sbz \zeta_{3}
+\frac{6505}{48}  \sbz \zeta_{5}
+\frac{1155}{32}  \sbz \zeta_{7}
%zero == 0
\right]
\nonumber\\
&{+}&\cfAtrAnlAcaB
\left[
-\frac{4379861}{20736} 
+\frac{8609}{72}  \sbz \zeta_{3}
+\frac{18805}{288}  \sbz \zeta_{5}
-\frac{11}{2}  \sbz \zeta_3^2
+\frac{35}{16}  \sbz \zeta_{7}
%zero == 0
\right]
\nonumber\\
&{+}&\cfAcaC
\left[
\frac{52207039}{248832} 
-\frac{456223}{3456}  \sbz \zeta_{3}
-\frac{77995}{1152}  \sbz \zeta_{5}
+\frac{605}{32}  \sbz \zeta_3^2
-\frac{385}{64}  \sbz \zeta_{7}
%zero == 0
\right]
{},
\label{d4}
\end{eqnarray}

\ice{
\beq
r_2 =   
{+}\cfB
\left[
-\frac{3}{32}\right]
{+}\cfAcaA
\left[
\frac{123}{32} 
-\frac{11}{4}  \sbz \zeta_{3}
%zero == 0
\right]
{+}\cfAtrAnlA
\left[
-\frac{11}{8} 
+  \sbz \zeta_{3}
%zero == 0
\right]
{},
\label{r2}
\eeq
\begin{eqnarray}
r_3 =   
&{}&\cfC
\left[
-\frac{69}{128}\right]
{+}\cfAtrBnlB
\left[
\frac{151}{54} 
-\frac{1}{36}  \pi^2
-\frac{19}{9}  \sbz \zeta_{3}
%zero == 0
\right]
\nonumber
{+}\cfBtrAnlA
\left[
-\frac{29}{64} 
+\frac{19}{4}  \sbz \zeta_{3}
-5  \sbz \zeta_{5}
%zero == 0
\right]
\nonumber\\
&{+}&\cfAtrAnlAcaA
\left[
-\frac{485}{27} 
+\frac{11}{72}  \pi^2
+\frac{112}{9}  \sbz \zeta_{3}
+\frac{5}{6}  \sbz \zeta_{5}
%zero == 0
\right]
{+}\cfBcaA
\left[
-\frac{127}{64} 
-\frac{143}{16}  \sbz \zeta_{3}
+\frac{55}{4}  \sbz \zeta_{5}
%zero == 0
\right]
\nonumber
\\
&{+}&\cfAcaB
\left[
\frac{90445}{3456} 
-\frac{121}{576}  \pi^2
-\frac{2737}{144}  \sbz \zeta_{3}
-\frac{55}{24}  \sbz \zeta_{5}
%zero == 0
\right]
{},
\label{r3}
\end{eqnarray}
%zero == 0
}
\begin{eqnarray}
&{}&r_4 =   
\nonumber
\cfD
\left[
\frac{4157}{2048} 
+\frac{3}{8}  \sbz \zeta_{3}
%zero == 0
\right]
{+}\dFFdRinvAnlA
\left[
-\frac{13}{16} 
-  \sbz \zeta_{3}
+\frac{5}{2}  \sbz \zeta_{5}
%zero == 0
\right]
{+}\dFARinvA
\left[
\frac{3}{16} 
-\frac{1}{4}  \sbz \zeta_{3}
-\frac{5}{4}  \sbz \zeta_{5}
%zero == 0
\right]
\nonumber
\\
&{+}&\cfAtrCnlC
\left[
-\frac{6131}{972} 
+\frac{11}{72}  \pi^2
+\frac{203}{54}  \sbz \zeta_{3}
-\frac{1}{9}  \pi^2 \sbz \zeta_{3}
+\frac{5}{3}  \sbz \zeta_{5}
%zero == 0
\right]
\nonumber
{+}\cfBtrBnlB
\left[
\frac{5713}{1728} 
-\frac{1}{24}  \pi^2
-\frac{581}{24}  \sbz \zeta_{3}
+3  \,\zeta_3^2
+\frac{125}{6}  \sbz \zeta_{5}
%zero == 0
\right]
\nonumber\\
&{+}&\cfAtrBnlBcaA
\left[
\frac{340843}{5184} 
-\frac{65}{48}  \pi^2
-\frac{10453}{288}  \sbz \zeta_{3}
+\frac{11}{12}  \pi^2 \sbz \zeta_{3}
-\frac{1}{2}  \,\zeta_3^2
-\frac{170}{9}  \sbz \zeta_{5}
%zero == 0
\right]
\nonumber\\
&{+}&\cfCtrAnlA
\left[
\frac{1001}{384} 
+\frac{99}{32}  \sbz \zeta_{3}
-\frac{125}{4}  \sbz \zeta_{5}
+\frac{105}{4}  \sbz \zeta_{7}
%zero == 0
\right]
\nonumber
\\
&{+}&\cfBtrAnlAcaA
\left[
\frac{32357}{13824} 
+\frac{11}{128}  \pi^2
+\frac{10661}{96}  \sbz \zeta_{3}
-\frac{33}{4}  \,\zeta_3^2
-\frac{5155}{48}  \sbz \zeta_{5}
-\frac{105}{8}  \sbz \zeta_{7}
%zero == 0
\right]
\nonumber\\
&{+}&\cfAtrAnlAcaB
\left[
-\frac{4379861}{20736} 
+\frac{249}{64}  \pi^2
+\frac{8609}{72}  \sbz \zeta_{3}
-\frac{121}{48}  \pi^2 \sbz \zeta_{3}
-\frac{11}{2}  \,\zeta_3^2
+\frac{18805}{288}  \sbz \zeta_{5}
+\frac{35}{16}  \sbz \zeta_{7}
%zero == 0
\right]
\nonumber
\\
&{+}&\cfCcaA
\left[
-\frac{253}{32} 
-\frac{139}{128}  \sbz \zeta_{3}
+\frac{2255}{32}  \sbz \zeta_{5}
-\frac{1155}{16}  \sbz \zeta_{7}
%zero == 0
\right]
\nonumber\\
&{+}&\cfBcaB
\left[
-\frac{592141}{18432} 
+\frac{121}{1536}  \pi^2
-\frac{43925}{384}  \sbz \zeta_{3}
+\frac{6505}{48}  \sbz \zeta_{5}
+\frac{1155}{32}  \sbz \zeta_{7}
%zero == 0
\right]
\nonumber\\
&{+}&\cfAcaC
\left[
\frac{52207039}{248832} 
-\frac{16753}{4608}  \pi^2
-\frac{456223}{3456}  \sbz \zeta_{3}
+\frac{1331}{576}  \pi^2 \sbz \zeta_{3}
+\frac{605}{32}  \,\zeta_3^2
-\frac{77995}{1152}  \sbz \zeta_{5}
-\frac{385}{64}  \sbz \zeta_{7}
%zero == 0
\right]
{}.
\label{r4}
\end{eqnarray}
%zero == 0
Here
 $C_F$ and
 $ C_A$ are the quadratic Casimir
operators of the
fundamental and the adjoint representation of the Lie algebra, 
$T$ is the trace normalization of the
fundamental representation, $\Tf \equiv  T\,n_f $,  with $n_f$ being  the number of
quark flavors. The exact definitions of
${d_F^{a b c d} d_A^{a b c d}}$ and ${d_F^{a b c d}d_F^{a b c d}}$
are given in \cite{Vermaseren:1997fq}.
For  QCD (colour gauge group SU(3)):
\beq
C_F =4/3\,,\, C_A=3\,,\, T=1/2\,,\, d_R = 3\,,\,
{d_F^{a b c d} d_A^{a b c d}} = \frac{15}{2}\,,\, {d_F^{a b c d}d_F^{a b c d}} = \frac{5}{12}
{}.
\label{SU3}
\eeq

It is of interest to  note that the coefficient $d_4$ is free from 
$\zeta_4$, $\zeta_3^2$ and $\zeta_6$. Separate diagrams do contain such contributions,
however, they {\em all cancel  each other out} in a   non-trivial way  in the final result.  The reasons for such  mysterious
cancellations are unclear (see for some more details \cite{Baikov:2009zz}). 


\section{ Bjorken sum rule to order  $\alpha_s^4$ in a general gauge theory}

The Bjorken sum rule  
%(BSR) 
expresses
the integral over the  spin distributions of quarks inside of the nucleon in terms of 
its axial charge times a  coefficient function
${C}^{Bjp}$:
\beq
%\ba{c}
\Gamma_1^{p-n}(Q^2) = 
\int_0^1 [g_1^{ep}(x,Q^2)-g_1^{en}(x,Q^2)]dx
=\frac{g_A}{6}
C^{Bjp}(a_s) +
%\dots
\sum_{i=2}^{\infty}\frac{\mu_{2i}^{p-n}(Q^2)}{Q^{2i-2}}
{},
\label{gBSR}
\eeq
where $g_1^{ep}$ and $g_1^{en}$ are the spin-dependent proton and neutron
structure functions, $g_A$ is the nucleon axial charge as measured in 
neutron $\beta$-decay. The coefficient function
$C^{Bjp}(a_s)=1+{\cal O}(a_s)$  is proportional to the flavour-nonsinglet axial vector
current $\bar{\psi}\gamma^{\mu}\gamma_5\psi$ in the corresponding
short distance Wilson expansion. The sum  in the second line of \re{gBSR} describes for
the nonperturbative  power corrections (higher twist) which are  inaccessible for pQCD.
The function   $C^{Bjp}(a_s)$ is known to order 
$\alpha_s^3$  from \cite{Larin:1991tj}.
It can be conveniently represented as 
\beq
{C}^{Bjp}(Q^2) = 
1  - \frac{3}{4}\,C_F\, a_s +
\sum_{i=2}^{\infty} \  {c}_i\, a_s^i(Q^2),  
\ \ 
1/{C}^{Bjp}(Q^2) = 
1  + \frac{3}{4}\,C_F\, a_s +
\sum_{i=2}^{\infty} \  {b}_i\, a_s^i(Q^2)
{}.
\eeq
We have computed the next 
$\alpha_s^4$ term in  $C^{Bjp}(a_s)$ using the same methods as in  \cite{Baikov:2008jh}
as well as the  ``method of projectors'' \cite{Gorishnii:1986gn}.
The result  for   $b_4$ reads:  
\ice{
\beq
b_2 &=&  
-\frac{3}{32}
\cfB
%\nonumber\\
{+}\cfAtrAnlA
\!
\left[
-\frac{1}{2}\right]
{+}\cfAcaA
\!
\left[
 \frac{23}{16}\right]
{},
\label{b2}
\\
b_3 &=& 
{-}
\frac{69}{128} \cfC
{+}\cfBtrAnlA
\left[
-\frac{299}{576}
+\frac{5}{12}  \sbz \zeta_{3}
%zero == 0
\right]
+
\cfAtrBnlB
\left[
 \frac{115}{216}\right]
%\nonumber\\
{+}\cfBcaA
\left[
\frac{1}{576}
+\frac{11}{12}  \sbz \zeta_{3}
%zero == 0
\right]
\nonumber\\
&{+}&
\cfAtrAnlAcaA
\left[
-\frac{3535}{864}
-\frac{3}{4}  \sbz \zeta_{3}
+\frac{5}{6}  \sbz \zeta_{5}
%zero == 0
\right]
{+}\cfAcaB
\left[
\frac{5437}{864}
-\frac{55}{24}  \sbz \zeta_{5}
%zero == 0
\right]
{},
\label{b3}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
}
\begin{eqnarray}
%\lefteqn{b_4 =  } \nonumber\\
b_4 & = &\dFARinvA
\left[
\frac{3}{16} 
-\frac{1}{4}  \sbz \zeta_{3}
-\frac{5}{4}  \sbz \zeta_{5}
%zero == 0
\right]
%\nonumber\\
{+}\dFFdRinvAnlA
\left[
-\frac{13}{16} 
-  \sbz \zeta_{3}
+\frac{5}{2}  \sbz \zeta_{5}
%zero == 0
\right]
%\nonumber\\
{+}\cfD
\left[
\frac{4157}{2048} 
+\frac{3}{8}  \sbz \zeta_{3}
%zero == 0
\right]
\nonumber\\
&{+}&\cfCtrAnlA
\left[
-\frac{473}{2304} 
-\frac{391}{96}  \sbz \zeta_{3}
+\frac{145}{24}  \sbz \zeta_{5}
%zero == 0
\right]
%\nonumber\\
{+}\cfBtrBnlB
\left[
\frac{869}{576} 
-\frac{29}{24}  \sbz \zeta_{3}
%zero == 0
\right]
%\nonumber\\
{+}\cfAtrCnlC
\left[
-\frac{605}{972}\right]
\nonumber\\
&{+}&\cfCcaA
\left[
-\frac{8701}{4608} 
+\frac{1103}{96}  \sbz \zeta_{3}
-\frac{1045}{48}  \sbz \zeta_{5}
%zero == 0
\right]
%\nonumber\\
{+}\cfBtrAnlAcaA
\left[
-\frac{17309}{13824} 
+\frac{1127}{144}  \sbz \zeta_{3}
-\frac{95}{144}  \sbz \zeta_{5}
-\frac{35}{4}  \sbz \zeta_{7}
%zero == 0
\right]
\nonumber\\
&{+}&\cfAtrBnlBcaA
\left[
\frac{165283}{20736} 
+\frac{43}{144}  \sbz \zeta_{3}
-\frac{5}{12}  \sbz \zeta_{5}
+\frac{1}{6}  \sbz \zeta_3^2
%zero == 0
\right] 
%eqs.~(\ref{d4},\ref{b4}) 
{+}\cfBcaB
\left[
-\frac{435425}{55296} 
-\frac{1591}{144}  \sbz \zeta_{3}
+\frac{55}{9}  \sbz \zeta_{5}
+\frac{385}{16}  \sbz \zeta_{7}
%zero == 0
\right]
\nonumber
\\
%\hspace{-15mm}
\nonumber
&{+}&\cfAtrAnlAcaB
\left[
-\frac{1238827}{41472} 
-\frac{59}{64}  \sbz \zeta_{3}
+\frac{1855}{288}  \sbz \zeta_{5}
-\frac{11}{12}  \sbz \zeta_3^2
+\frac{35}{16}  \sbz \zeta_{7}
%zero == 0
\right]
\nonumber\\
&{+}&
+\cfAcaC
\left[
\frac{8004277}{248832} 
-\frac{1069}{576}  \sbz \zeta_{3}
-\frac{12545}{1152}  \sbz \zeta_{5}
+\frac{121}{96}  \sbz \zeta_3^2
-\frac{385}{64}  \sbz \zeta_{7}
%zero == 0
\right]
{}.
\label{b4}
\end{eqnarray}

\section{ Crewther relation  to order  $\alpha_s^4$ }

The 
%generalized 
Crewther relation has the form\footnote{
Very recently an  extension of the (generalized)  Crewther relation  \re{gCrewther} 
has been suggested in \cite{Kataev:2010dm}.}
\beq  {D}(a_s)\,{C}^{Bjp}(a_s)
= d_R \left[ 1 + \frac{\beta(a_s)}{a_s}\, K(a_s) \right], 
\ \ 
K(a_s) = 
K_0 +
a_s\,K_1 +a_s^2\,K_2 +a_s^3\,K_3 
+ \dots 
\label{gCrewther} 
\eeq
 The term proportional the $\beta$-function describes the deviation
 from the limit of exact conformal invariance, with the deviations
 starting in order $\alpha_s^2$ (hence, $K_0\equiv 0$), and was suggested
 \cite{Broadhurst:1993ru} on the basis of ${\cal O}(\alpha_s^3)$
 calculations of $D(a_s)$ \cite{Gorishnii:1990vf,Surguladze:1990tg}
 and ${C}^{Bjp}(a_s)$ \cite{Larin:1991tj}.  A formal proof was carried
 out in \cite{Crewther:1997ux,Braun:2003rp}.  The original relation
 without this term was first proposed in
\cite{Crewther:1972kn} (see, also, \cite{Adler:1973kz}).



The Crewther relation is trivially met at order $\alpha_s$.
It was  demonstrated in \cite{Broadhurst:1993ru} that fulfillment
of \re{gCrewther} at orders $\alpha_s^2$ and  $\alpha_s^3 $
puts as many as 2 and 3 constraints  on the differences $d_2- b_2$ and
$d_3 -b_3$ respectively. At order  $\alpha_s^4$ the number of constraints 
on the difference  $d_4 -b_4$ is increased to 6 and 
our results do meet all of them   (see \cite{Baikov:2010} for details).

Here we mention only three  most simple relations, namely those between coefficients
in front of colour structures  ($C_F^4, \dFFdRinvAnlA$ and $\dFARinvA$) appearing in $d_4$ and $b_4$.
As it follows directly from eq. ~\re{gCrewther} the coefficients must be (pairwise) identical.
The prediction is  indeed in agreement  with our results as expressed by eqs. (\ref{d4}) and (\ref{b4})!

In particular the equality of the coefficients  of $C_F^4$ in  eqs. (\ref{d4}) and (\ref{b4})
provides us with a strong check of the correctness of 
the result for the five-loop  $\beta$ function of  quenched  QED published 
in \cite{Baikov:2008cp}. Note that the test was first suggested in
 \cite{Kataev:2008sk,Kataev:2008nc}.



\section{Conclusion}






The  calculations have been
performed on a SGI ALTIX 24-node IB-interconnected cluster of 8-cores
Xeon computers and on the HP XC4000 supercomputer of the federal state
Baden-W\"urttemberg
%(claster of Dual Core AMD Opteron 2.6 GH),
using  parallel MPI-based  \cite{Tentyukov:2004hz,Tentyukov:2006pr} as well as thread-based 
\cite{Vermaseren:2006ag,Tentyukov:2007mu} versions  of FORM
\cite{Vermaseren:2000nd}.  For evaluation of color factors  the FORM program {\em COLOR}
\cite{vanRitbergen:1998pn} has been used. 
The diagrams have been generated with QGRAF \cite{Nogueira:1991ex}.



This work was supported by the Deutsche Forschungsgemeinschaft in the
Sonderforschungsbereich/Transregio SFB/TR-9 ``Computational Particle
Physics'' and  by RFBR grant 08-02-01451. We thank V.M.~~Braun for useful discussions.



\begin{thebibliography}{10}

\bibitem{Baikov:2010}
P.A. Baikov, K.G. Chetyrkin and J.H. K\"uhn,
\newblock (2010), 1001.3606v1.

\bibitem{Baikov:2008cp}
P.A. Baikov, K.G. Chetyrkin and J.H. K{\"u}hn,
\newblock PoS RADCOR2007 (2007) 023, 0810.4048.

\bibitem{Baikov:2008jh}
P.A. Baikov, K.G. Chetyrkin and J.H. K\"uhn,
\newblock Phys. Rev. Lett. 101 (2008) 012002, 0801.1821.

\bibitem{Bender:1976pw}
C.M. Bender, R.W. Keener and R.E. Zippel,
\newblock Phys. Rev. D15 (1977) 1572.

\bibitem{Broadhurst:1999xk}
D.J. Broadhurst,
\newblock (1999), hep-th/9909185.

\bibitem{Kataev:2008sk}
A.L. Kataev,
\newblock Phys. Lett. B668 (2008) 350, 0808.3121.

\bibitem{Kataev:2008nc}
A.L. Kataev,
\newblock (2008), 0805.0621.

\bibitem{Crewther:1972kn}
R.J. Crewther,
\newblock Phys. Rev. Lett. 28 (1972) 1421.

\bibitem{Broadhurst:1993ru}
D.J. Broadhurst and A.L. Kataev,
\newblock Phys. Lett. B315 (1993) 179, hep-ph/9308274.

\bibitem{Chetyrkin:1996ia}
K.G. Chetyrkin, J.H. K\"uhn and A. Kwiatkowski,
\newblock Phys. Rept. 277 (1996) 189.

\bibitem{Gorishnii:1990vf}
S.G. Gorishnii, A.L. Kataev and S.A. Larin,
\newblock Phys. Lett. B259 (1991) 144.

\bibitem{Surguladze:1990tg}
L.R. Surguladze and M.A. Samuel,
\newblock Phys. Rev. Lett. 66 (1991) 560.

\bibitem{Vermaseren:1997fq}
J.A.M. Vermaseren, S.A. Larin and T. van Ritbergen,
\newblock Phys. Lett. B405 (1997) 327, hep-ph/9703284.

\bibitem{Baikov:2009zz}
P.A. Baikov, K.G. Chetyrkin and J.H. K\"uhn,
\newblock Nucl. Phys. Proc. Suppl. 189 (2009) 49.

\bibitem{Larin:1991tj}
S.A. Larin and J.A.M. Vermaseren,
\newblock Phys. Lett. B259 (1991) 345.

\bibitem{Gorishnii:1986gn}
S.G. Gorishnii and S.A. Larin,
\newblock Nucl. Phys. B283 (1987) 452.

\bibitem{Kataev:2010dm}
A.L. Kataev and S.V. Mikhailov,
\newblock (2010), 1001.0728.

\bibitem{Crewther:1997ux}
R.J. Crewther,
\newblock Phys. Lett. B397 (1997) 137, hep-ph/9701321.

\bibitem{Braun:2003rp}
V.M. Braun, G.P. Korchemsky and D. M\"uller,
\newblock Prog. Part. Nucl. Phys. 51 (2003) 311, hep-ph/0306057.

\bibitem{Adler:1973kz}
S.L. Adler et~al.,
\newblock Phys. Rev. D6 (1972) 2982.

\bibitem{Tentyukov:2004hz}
M. Tentyukov et~al.,
\newblock (2004), cs/0407066.

\bibitem{Tentyukov:2006pr}
M. Tentyukov, H.M. Staudenmaier and J.A.M. Vermaseren,
\newblock Nucl. Instrum. Meth. A559 (2006) 224.

\bibitem{Vermaseren:2006ag}
J.A.M. Vermaseren and M. Tentyukov,
\newblock Nucl. Phys. Proc. Suppl. 160 (2006) 38.

\bibitem{Tentyukov:2007mu}
M. Tentyukov and J.A.M. Vermaseren,
\newblock (2007), hep-ph/0702279.

\bibitem{Vermaseren:2000nd}
J.A.M. Vermaseren,
\newblock (2000), math-ph/0010025.

\bibitem{vanRitbergen:1998pn}
T. van Ritbergen, A.N. Schellekens and J.A.M. Vermaseren,
\newblock Int. J. Mod. Phys. A14 (1999) 41, hep-ph/9802376.

\bibitem{Nogueira:1991ex}
P. Nogueira,
\newblock J. Comput. Phys. 105 (1993) 279.

\end{thebibliography}



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