\documentclass[final,5p,times,twocolumn]{elsarticle}

\usepackage{graphicx,amsmath,amssymb}

\newcommand{\myrm}{\relax}

\begin{document}


\begin{frontmatter}

\title{
\vspace*{-1cm}
\boldmath \bf
Adler Function, Sum Rules and Crewther Relation of Order  ${\cal O}(\alpha_s^4)$:
 the Singlet Case}

\author[a]{P.~A.~Baikov}
\author[b]{K. G. Chetyrkin}
\author[b]{J.~H.~K\"uhn}
\author[b]{J.~Rittinger}

\address[a]{
Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, 
1(2), Leninskie gory, Moscow 119234, Russian Federation
        }
\address[b]{Institut f\"ur Theoretische Teilchenphysik, Karlsruhe
  Institute of Technology (KIT), Wolfgang-Gaede-Stra\ss{}e 1, 726128 Karlsruhe, Germany}
\begin{abstract}
The analytic result for the singlet part of the Adler function of the
vector current in a general gauge theory is presented in five-loop
approximation.  Comparing this result with the corresponding singlet
part of the Gross-Llewellyn Smith sum rule \cite{Baikov:2010iw}, we successfully
demonstrate the validity of the generalized Crewther relation for the
singlet part. This provides  a  non-trivial test of both
our calculations and the generalized Crewther relation.
Combining the result    with the already available non-singlet part of the Adler function 
\cite{Baikov:2008jh,PhysRevLett.104.132004}
we  arrive at the complete  ${\cal O}(\alpha_s^4)$  expression for the Adler 
function and, as  a direct consequence, at  the complete  ${\cal O}(\alpha_s^4)$  
correction to the $e^+ e^-$ annihilation into hadrons in a general gauge theory. 
\end{abstract}
\begin{keyword}
QCD \sep   Adler function \sep     Gross-Llewellyn Smith  sum rule       \sep  Crewther relation 
\end{keyword}

\end{frontmatter}

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

The Crewther relation (CR) \cite{Crewther:1972kn,Broadhurst:1993ru}
connects in a non-trivial way two seemingly unrelated quantities,
namely the Adler function \cite{Adler:1974gd} and perturbative
corrections arising in the sum rules relevant for deep inelastic
scattering (DIS). Originally the CR had been formulated for the case
of a conformal-invariant gauge theory.  Subsequently, observing a
close relation between the ${\cal O}(\alpha_s^3)$ terms in the Adler
function and the corrections to the Bjorken sum rule, its
generalization for the case of QCD was suggested in
\cite{Broadhurst:1993ru}, introducing as modification additional terms
proportional to the $beta$-function. A more formal argument for
the validity of this ``generalized Crewther relation'' (GCR) was given
in \cite{Crewther:1997ux,Braun:2003rp}.  During the past years the
perturbative corrections both for Adler function and Bjorken sum rule
were extended from ${\cal O}(\alpha_s^3)$
\cite{Gorishnii:1990vf,Surguladze:1990tg,Chetyrkin:1996ez,Larin:1991tj}
to ${\cal O}(\alpha_s^4)$
\cite{Baikov:2008jh,PhysRevLett.104.132004}. 
However, these results were restricted to the respective non-singlet
parts. Nevertheless, they could be used to demonstrate the validity of
the GCR between non-singlet Adler function and Bjorken sum rule
\cite{Bjorken:1967px,Bjorken:1969mm}, thus providing at the same time
an important cross check of the underlying, demanding calculations.


The ${\cal O}(\alpha_s^4)$ singlet piece of the sum rule was published
in \cite{Baikov:2010iw}, thus completing the prediction for the
Gross-Llewellyn Smith (GLS) sum rule \cite{Gross:1969jf}. Below we
give the corresponding result for the Adler function. On the one hand
this leads to a prediction of the familiar $R$-ratio measured in
electron-positron annihilation, including the (small) up to now missing
singlet pieces of ${\cal O}(\alpha_s^4)$, on the other hand this
result allows to test the GCR also for the singlet case.



\section{Singlet ${\cal O}(\alpha_s^4)$ contributions to the Adler function and R(s)}

For the definition of the Adler function it is convenient to start with the polarization  function of the flavor singlet vector current:
\begin{equation}
3 \, Q^2 \, \Pi(Q^2) =
i\,\int{\rm d}^4 x \, e^{iq\cdot x}\langle 0|{\rm  T}j_\mu(x)j^\mu(0)|0\rangle~,
\end{equation}
with $j_{\mu} = \sum_i\overline{\psi}_i\gamma_{\mu}\psi_i$ and  $Q^2 = -q^2$. The  corresponding Adler function
\begin{equation}
D(Q^2) =  -12\, \pi^2 Q^2\, \frac{\rm{d}}{\rm{d} Q^2} \Pi(Q^2)
\end{equation}
is naturally decomposed into a sum of the non-singlet (NS) and singlet (SI) components (see Fig.~1):
\begin{equation}
D(Q^2) = n_f\, \,  D^{\myrm NS}(Q^2) + n_f^2 \, D^{\myrm SI}(Q^2)~.\label{D:decomp}
\end{equation}
Here $n_f$ stands for   the total number of quark flavours; all quarks  are  considered as massless.
%Similar decomposition holds for the very polarization operator.
\begin{figure*}
\begin{center}
\includegraphics[width=10cm]{DNS2_SI.eps}
\end{center}
\hspace{6cm} {\bf (a)}  \hspace{6cm} {\bf (b)}
\caption{
Lowest order non-singlet (a) and singlet (b) diagrams contributing to the Adler function.
}
\label{figure:1}
\end{figure*}
The Adler function $D^{\myrm EM}$ corresponding to the electromagnetic vector current $j^{\myrm EM}_{\mu} = \sum_i q_i\, \overline{\psi}_i\gamma_{\mu}\psi_i$ ($q_i$ stands for the electric charge of the quark field $\psi_i$)  is thus  given by the following combination:
\begin{equation}
D^{\myrm EM} = \bigg(  \sum_i q_i^2 \bigg) D^{\myrm NS} + \bigg(  \sum_i q_i \bigg)^2 \, D^{\myrm SI}~.
\end{equation}
Similar decompositions hold for the  corresponding polarization functions  $\Pi$ and $\Pi^{\myrm EM}$.



The  physical observable $ R(s) ={\sigma(e^+e^-\to {\rm hadrons})\over \sigma(e^+e^-\to \mu^+\mu^-)}$ is related to $\Pi^{\myrm EM}(Q^2)$ by the optical theorem
\begin{equation}
R(s) = 12 \pi \,\Im \, \Pi^{\myrm EM}(-s- i\epsilon)~.
\label{R(S):def}
\end{equation}
The result for the perturbative expansions of  the non-singlet part ($a_s \equiv \frac{\alpha_s}{\pi}$)
\begin{equation}
{D}^{\myrm NS}(Q^2) = d_R\,\bigg( 1 + \sum_{i=1}^{\infty} \ d^{\myrm NS}_i \, a_s^i(Q^2) \bigg)
\end{equation}
has been presented in \cite{PhysRevLett.104.132004}. For the  singlet 
%Adler functions 
part it reads:
\begin{equation}
D^{\myrm SI}(Q^2) =  d_R\,\bigg( \sum_{i=3}^{\infty} \  d^{\myrm SI}_i \, a_s^i(Q^2) \bigg)~,
\end{equation}
where the parameter $d_R$ (the dimension of the quark color representation, $d_R= 3$ in QCD) is  factorized in {\em both} non-singlet and singlet components. 

The singlet component has the following structure at orders $\alpha_s^3$ and $\alpha_s^4$:
\begin{eqnarray}
d_3^{\myrm SI} &=& d^{abc}\,d^{abc}/d_R\,\Big( \tfrac{11}{192} - \tfrac{1}{8}\,\zeta_{3}\Big)~,
\\ 
d_4^{\myrm SI} &=& d^{abc}\,d^{abc}/d_R\,\Big( C_F \, d_{4,1}^{\myrm SI} +C_A \, d_{4,2}^{\myrm SI} + T\, n_f\, d_{4,3}^{\myrm SI}\Big)~.
\label{SI:d3:d4}
\end{eqnarray}
Here  $C_F$ and $ C_A$ are the quadratic Casimir operators of the fundamental and the adjoint representation of the Lie algebra, $d^{abc} = 2\, \mathrm{Tr}(\{\frac{\lambda^a}{2},\frac{\lambda^b}{2}\},  \frac{\lambda^c}{2}\})$, $T$ is the trace normalization of the fundamental representation. 
% \ice{
% The exact definitions of
% ${d_F^{a b c } d_F^{a b c }}$
% are given in \cite{Vermaseren:1997fq}.
% }
For  QCD (colour gauge group SU(3)):
\begin{equation*}
C_F =4/3~,~~~ C_A=3~,~~~ T=1/2~,~~~ {d^{a b c }\,d^{a b c }} = 40/3~.
\end{equation*}

% \ice{
% The evaluation of the NLO terms of $r_V^{\myrm S}$ requires the
% calculation of the absorptive parts of five-loop diagrams with
% massless propagators which, with the help of some complicated
% combinatorics based on the $R^*$-operation \cite{Chetyrkin:1984xa}, can be boiled down
% to the calculation of  four-loop propagator
% diagrams. The  latter have been computed  
% via reduction to 28  master integrals, based on evaluating  
% sufficiently many terms of the $1/D$ expansion \cite{Baikov:2005nv} of
% the corresponding coefficient functions \cite{Baikov:1996rk}. 
% This direct procedure required huge computing resources and
% was performed using a parallel version \cite{Tentyukov:2004hz} of FORM
% \cite{Vermaseren:2000nd}. The master integrals are reliably  known from
% \cite{Baikov:2010hf,Smirnov:2010hd,Lee:2011jt}. 
% The details of the calculation, the results in analytic form and their 
% relation to the Gross-Llewellyn Smith sum rule will be given in  \cite{Baikov:2012}.
% }
Using the methods described  in  \cite{Chetyrkin:1984xa,Baikov:2005nv,Baikov:1996rk,Baikov:2008jh,Baikov:2010hf,PhysRevLett.104.132004} we obtain
\begin{eqnarray}
d_{4,1}^{\myrm SI} &=& -\tfrac{13}{64} - \tfrac{1}{4}\,\zeta_3  + \tfrac{5}{8}\,\zeta_5~, 
\label{d41}
\\
d_{4,2}^{\myrm SI} &=& \tfrac{ 3893}{4608} - \tfrac{169}{128}\,\zeta_3 + \tfrac{45}{64}\,\zeta_5  -\tfrac{11}{32}\,\zeta_3^2~,
\label{d42}
\\
d_{4,3}^{\myrm SI} &=& -\tfrac{149}{576} + \tfrac{13}{32}\,\zeta_3 - \tfrac{5}{16}\,\zeta_5 + \tfrac{1}{8}\,\zeta_3^2~.
\label{d43}
\end{eqnarray}

With the use of eqs.~(\ref{d41})-(\ref{d43}) and the result for $D^{\myrm NS}$ from \cite{PhysRevLett.104.132004} we arrive at  the complete  result for the ratio $R(s)$ at  order $\alpha_s^4$ in (massless)  QCD:
\begin{align}
 R(s) =&~ 3\sum_f q_f^2 \bigg\{
1 + a_s + a_s^2 \Big( \tfrac{365}{24} - 11\,\zeta_3 - \tfrac{11}{12}\,n_f + \tfrac{2}{3}\,\zeta_3 \,n_f \Big)
\nonumber\\[-1mm]
&~+ a_s^3\,\Big[n_f^2\,\Big( \tfrac{151}{162} - \tfrac{1}{108}  \pi^2 - \tfrac{19}{27}\,\zeta_{3}\Big)
\nonumber \\
&\qquad  + n_f\,\Big( - \tfrac{7847}{216} + \tfrac{11}{36}\,\pi^2 + \tfrac{262}{9}\,\zeta_{3} - \tfrac{25}{9}\,\zeta_{5}\Big)
\nonumber \\
&\qquad + \tfrac{87029}{288} - \tfrac{121}{48}\,\pi^2 - \tfrac{1103}{4}\,\zeta_{3} + \tfrac{275}{6}\,\zeta_{5} \Big]
\nonumber \\
&~+ a_s^4\,\Big[ n_f^3\,\Big( - \tfrac{6131}{5832} + \tfrac{11}{432}\,\pi^2 + \tfrac{203}{324}\,\zeta_{3} - \tfrac{1}{54}\,\pi^2\,\zeta_{3}
  + \tfrac{5}{18}\,\zeta_{5}\Big)
\nonumber\\
&\qquad + n_f^2\,\Big( \tfrac{1045381}{15552} - \tfrac{593}{432}\,\pi^2 - \tfrac{40655}{864}\,\zeta_{3}
\nonumber\\
&\qquad\qquad\qquad + \tfrac{11}{12}\,\pi^2\,\zeta_{3}  + \tfrac{5}{6}\,\zeta_3^2 - \tfrac{260}{27}\,\zeta_{5}\Big)
\nonumber\\
&\qquad + n_f\,\Big( - \tfrac{13044007}{10368} + \tfrac{2263}{96}\,\pi^2 + \tfrac{12205}{12}\,\zeta_{3} - \tfrac{121}{8}\,\pi^2\,\zeta_{3}
\nonumber\\
&\qquad\qquad\qquad - 55\,\zeta_3^2 + \tfrac{29675}{432}\,\zeta_{5} + \tfrac{665}{72}\,\zeta_{7}\Big)
\nonumber\\
&\qquad + \tfrac{144939499}{20736} - \tfrac{49775}{384}\,\pi^2 - \tfrac{5693495}{864}\,\zeta_{3} + \tfrac{1331}{16}\,\pi^2\,\zeta_{3}
\nonumber\\
&\qquad\qquad\qquad  +\tfrac{5445}{8}\,\zeta_3^2 + \tfrac{65945}{288}\,\zeta_{5} - \tfrac{7315}{48}\,\zeta_{7}\Big]
\bigg\}
\nonumber
\end{align}
\begin{align}
&~ + \bigg(\sum_f q_f \bigg)^2 \bigg\{ a_s^3 \Big( \tfrac{55}{72} - \tfrac{5}{3}\,\zeta_3 \Big)
\nonumber\\[-2mm]
&\qquad\qquad + a_s^4\,\Big[ n_f\,\Big( - \tfrac{745}{432} + \tfrac{65}{24}\,\zeta_{3} + \tfrac{5}{6}\,\zeta_3^2 - \tfrac{25}{12}\,\zeta_{5}\Big)
\nonumber\\
&\qquad\qquad\qquad + \Big( \tfrac{5795}{192} - \tfrac{8245}{144}\,\zeta_{3} - \tfrac{55}{4}\,\zeta_3^2
  + \tfrac{2825}{72}\,\zeta_{5}\Big)\Big]\bigg\}~,
\label{Rqcd}
\end{align}
where we set $\mu=Q$. The full results for Adler function and $R(s)$  for generic color factors
and  generic value of $\mu$ are  rather lenghty and can be found  
(in computer-readable form) in 
{\tt http://www-ttp.physik.uni-karlsruhe.de/Progdata\\/ttp12/ttp12-017.} Numerically, it reads:
\begin{align}
R(s) =&~ 3\sum_f q_f^2\,\bigg\{
1 + a_s + a_s^2\,\Big( 1.986 - 0.1153 \,n_f \Big)
\nonumber\\[-2mm]
&\qquad+ a_s^3\,\Big( -6.637 - 1.200 \,n_f - 0.00518 \,n_f^2 \Big)
\nonumber\\
&\qquad +a_s^4\,\Big( -156.608 + 18.7748\,n_f  - 0.797434 \,n_f^2
\nonumber\\
&\qquad\qquad + 0.0215161 \,n_f^3 \Big)\bigg\}
\nonumber
\\
&~- \bigg(\sum_f q_f\bigg)^2 \Big( 1.2395\, a_s^3   + \Big(17.8277 - 0.57489 \,n_f \Big)\, a_s^4 \Big)~.
\nonumber
\end{align}
% \ice{
% \mbox{for $\,n_f$=5}  
% \[
% \frac{11}{3}
% \left[
% 1 + a_s  + a_s^2\, 1.409 
% -12.767\, a_s^3  - 79.98\, a_s^4 
% \right]
% + \frac{1}{9}
% \left[
% -1.240\, a_s^3 {- 14.95\, a_s^4 }
% \right]
% \]
% }

Specifically, for the  particular values of $n_f = 4 $ and 5 one  obtains (for the terms of order $\alpha_s^3$ and  $\alpha_s^4$ we have explicitly decomposed the coefficient into  non-singlet and singlet contributions):
\begin{align}
R^{n_f=4}(s)  =&~ \frac{10}{3} \bigg[ 1 + a_s + 1.5245 a_s^2
\nonumber \\
&~+ a_s^3\, \Big( -11.686 =-11.52 - 0.16527^{\myrm SI}\Big)
\nonumber \\
&~+ a_s^4\, \Big(  - 94.961 = -92.891 - 2.0703^{\myrm SI}\Big) \bigg]~,
\\
R^{n_f=5}(s)  =&~ \frac{11}{3} \bigg[ 1 + a_s + 1.40902 a_s^2
\nonumber \\
&~+ a_s^3\, \Big( - 12.80=-12.767 - 0.037562^{\myrm SI}\Big)
\nonumber \\
&~+ a_s^4\, \Big(  - 80.434 = -79.981 - 0.4531^{\myrm SI}\Big) 
\bigg]
{}.
\end{align}
Note that for $n_f=3$ the singlet contributions vanish in every order in $\alpha_s$ as the corresponding
global coefficient $(\sum_i q_i)^2$ happens to be zero. 
Implications of this result for the determination of $\alpha_s$ in 
electron-positron annihilation and in $Z$-boson decays are discussed in
\cite{Baikov:2012er}.



\section{GLS sum rule at order ${\cal O}(\alpha_s^4)$ and the Crewther relation}
The second quantity of interest, the GLS sum rule,
\begin{equation}
\frac{1}{2}\int_0^1 F_3(x,Q^2) dx = 3\,  C^{CLS}(a_s) 
{},
\end{equation}
 relates the lowest  moment of 
the  isospin singlet structure function $F_3^{\nu p + \bar\nu p}(x,Q^2)$ to a coefficient 
$C^{CLS}(a_s)$, 
which appears in the operator product expansion of the axial and vector
non-singlet  currents
\begin{equation}
i\int T{A_{\mu}^{a}(x)V_{\nu}^{b}(0)}e^{iqx}dx
|_{\normalsize q^2\rightarrow -\infty}
\approx C_{\mu\nu\alpha}^{V,ab}\,V_{\alpha}(0)+ \dots
\label{OPE}
\end{equation}
where 
\[C_{\mu\nu\alpha}^{V,ab}
=
\delta^{ab}\epsilon_{\mu\nu\alpha\beta}
\frac{q^{\beta}}{Q^2}C^{GLS}(a_s)
{}
\]
and
$V_{\alpha} =\overline{\psi}\gamma_{\alpha} \psi$ is a flavour singlet  quark current.
At last 
$A_{\mu}^{a} = \overline{\psi}\gamma_{\mu} \gamma_5 t^a \psi$,  
$V_{\nu}^{b} = \overline{\psi}\gamma_{\nu} t^b \psi$ are axial vector    and  vector 
non-singlet quark currents, with  $t^a, \, t^b$  being the generators of the flavour group
$SU(n_f)$.


\begin{figure*}
\begin{center}
\includegraphics[width=12cm]{BjGLS.eps}
\end{center}
\hspace{3.6cm} {\bf (a)}  \hspace{6.2cm} {\bf (b)}
\caption{
(a),(b): ${\cal O}(\alpha_s^3)$ non-singlet and singlet diagrams contributing to the Gross-Llewellyn Smith sum rule;
note that the  coefficient function  $C^{Bjp}$ is contributed by only non-singlet diagrams.
}
\label{table:1}
\end{figure*}


Again diagrams contributing to $C^{GLS}(a_s)$ can be separated in two groups:
non-singlet and singlet ones (see Fig. 2):
\begin{align}
C^{GLS} =&~ C^{\myrm NS} + C^{\myrm SI}~,
\label{GLSdecomp}
\\
C^{\myrm NS}(Q^2) =&~ 
1+
\sum_{i=1}^{\infty} \  c^{\myrm NS}_i \, a_s^i(Q^2)~,
\\
{C}^{\myrm SI}(Q^2) =&~
\sum_{i=3}^{\infty} \  c^{\myrm SI}_i \, a_s^i(Q^2)~.
\end{align}


The results for both functions  $C^{NS}$ and $C^{SI}$ at order 
$\alpha_s^3$ are known since early 90-ties \cite{Larin:1991tj}. 
Note that as a consequence of  chiral invariance the closely related
Bjorken sum rule receives contributions  from   the non-singlet piece only  \cite{Larin:1991tj}: 
\begin{equation}
C^{Bjp} \equiv C^{\myrm NS}
\label{chiral:inv}
{}.
\end{equation}


The ${\cal O}(\alpha_s^4)$ contribution to $C^{Bjp}$  has been  computed  some  time ago
\cite{PhysRevLett.104.132004}. 
The calculation of the  ${\cal O}(\alpha_s^4)$ contribution to  ${C}^{\myrm SI}$
has been  published in  \cite{Baikov:2010iw} for
 a  generic gauge group and is repeated below:
\begin{align}
c_3^{\myrm SI} =&~ n_f\, \frac{d^{abc}\,d^{abc}}{d_R} \left(
c_{3,1}^{\myrm SI} \equiv
-\frac{11}{192}
+\frac{1}{8}  \, \zeta_{3}
\right)~,
\label{c31SI}
\\
c_4^{\myrm SI} =&~
n_f\,\frac{d^{abc}\,d^{abc}}{d_R} \left(
C_F \,c_{4,1}^{\myrm SI} +C_A \, c_{4,2}^{\myrm SI} + T\, n_f\, c_{4,3}^{\myrm SI}
\right)~,
\label{c4SI}
\\
 c_{4,1}^{\myrm SI} =&~ {
\frac{37}{128}
+\frac{1}{16}  \, \zeta_{3}
-\frac{5}{8}  \, \zeta_{5}
}~,
\label{c41SI}
\\
c_{4,2}^{\myrm SI} =&~  { -\frac{481}{1152}
+\frac{971}{1152}  \, \zeta_{3}
-\frac{295}{576}  \, \zeta_{5}
+\frac{11}{32}  \, \zeta_3^2
}~,
\label{c42SI}
\\
c_{4,3}^{\myrm SI} =&~ {
\frac{119}{1152}
-\frac{67}{288}  \, \zeta_{3}
+\frac{35}{144}  \, \zeta_{5}
-\frac{1}{8}  \, \zeta_3^2
}~.
\label{c43SI}
\end{align}

% \ice{
% The polarized Bjorken sum rule  reads
% \begin{equation}
% Bjp(Q^2)=\int_0^1[g_1^{ep}(x,Q^2)-g_1^{en}(x,Q^2)]dx=\frac{1}{6}
% |\frac{g_A}{g_V}|C^{Bjp}(a_s)
% \end{equation}
% Coefficient function $C^{Bjp}(a_s)$ is  fixed by OPE
% of two non-singlet vector
% currents (up to power suppressed corrections)
% \begin{equation}
% i\int T{V_{\alpha}^{a}(x)V_{\beta}^{b}(0)}e^{iqx}dx|_{q^2\rightarrow{\infty}}
% \approx C_{\alpha\beta\rho}^{Q,abc}A_{\rho}^{c}(0)+\dots
% \label{7}
% \end{equation}
% where
% \begin{equation}
% C_{\alpha\beta\rho}^{Q,abc}\sim
% id^{abc}\epsilon_{\alpha\beta\rho\sigma}
% \frac{q^{\sigma}}{Q^2}C^{Bjp}(a_s)
% \nonumberb
% \end{equation}
% and $Q^2=-q^2$
% }

Using  the input from eqs. (\ref{d41}-\ref{d43}) and (\ref{c31SI}-\ref{c43SI}), 
the validity of the GCR can now be investigated. 
In fact, there exist two of them \cite{Broadhurst:1993ru}, one  involving  the non-singlet parts only
 and one involving also a singlet piece:
\begin{eqnarray}
D^{\myrm NS}(a_s)\, C^{Bjp}(a_s) &=& d_R \left[ 1+ \frac{\beta(a_s)}{a_s}\, K^{\myrm NS}(a_S)
\right]~,
\label{gCrewtherNS}
\\
  K^{\myrm NS}(a_s) &=&
a_s\,K^{\myrm NS}_1 
+ a_s^2\,K^{\myrm NS}_2 +a_s^3\,K^{\myrm NS}_3
+ \dots
\nonumber
\end{eqnarray}
and 
\begin{eqnarray}
{ \displaystyle D(a_s)\,  C^{GLS}(a_s)}
\label{gCrewtherFull}
&=& 
{d_R \, n_f} \left[ 1 + 
\frac{\beta(a_s)}{a_s}\, K(a_s)
\right]
{},
\\
K(a_s) &=&
a_s\,K_1 
+ a_s^2\,K_2 +a_s^3\,K_3
+ \dots
\nonumber
\end{eqnarray}
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$. 
The term proportional to the $\beta$-function describes the deviation from
the limit of exact conformal invariance, with the deviations starting in order $\alpha_s^2$.

Relation (\ref{gCrewtherNS}) has been studied in detail in \cite{PhysRevLett.104.132004},
where its  validity at order $\alpha_s^4$ was demonstrated  
(a  detailed  discussion at orders $\alpha_s^2 $ and $\alpha_s^3$
can be found in  \cite{Broadhurst:1993ru}).


Let us consider now eq. (\ref{gCrewtherFull}).
Combining  
eqs.~(\ref{D:decomp},\ref{GLSdecomp},\ref{chiral:inv}) and (\ref{gCrewtherNS}) leads  
to the following relations between coefficients $K_i^{\myrm NS}$ and $K_i$:
\begin{eqnarray}
K_1 &=& K_1^{\myrm NS}, \ \ K_2 = K_2^{\myrm NS}~,
\\
 K_3 &=& K^{\myrm NS}_3 + K_3^{\myrm SI}~,
\\
K_3^{\myrm SI} &=& k^{\myrm SI}_{3,1} \,\, n_f\,\frac{d^{abc}\,d^{abc}}{d_R}~,
\end{eqnarray}
with  $k^{\myrm SI}_{3,1}$ being a numerical parameter.
 
Thus, we conclude that eq. (\ref{gCrewtherFull}) puts $3-1 = 2$
constraints between two triplets of (purely numerical) parameters $\{
d^{\myrm SI}_{4,1}, d^{\myrm SI}_{4,2}, d^{\myrm SI}_{4,3}\}$ and $\{ c^{\myrm SI}_{4,1},
c^{\myrm SI}_{4,2}, c^{\myrm SI}_{4,3}\}$ appearing in eqs. (\ref{SI:d3:d4}) and (\ref{c4SI}) and
completely describing the order $\alpha_s^4$ singlet contributions to
the Adler function and the Gross-Llewellyn Smith sum rule
respectively.


The solution of the constraints and eqs.~(\ref{c41SI}-\ref{c43SI}) 
produces  the following relations for  $d_4^{\myrm SI}$:
\begin{eqnarray} d_{4,1}^{\myrm SI} &=& -\frac{3}{2}c_{3,1}^{\myrm SI} - c_{4,1}^{\myrm SI}
 =
 -\frac{13}{64} - \frac{\zeta_3}{4}  + \frac{5\zeta_5}{8} ~,
\\
d_{4,2}^{\myrm SI} &=& - c_{4,2}^{\myrm SI}  -  \frac{11}{12}\, k_{3,1}^{\myrm SI}~,
\\
 d_{4,3}^{\myrm SI} &=& - c_{4,3}^{\myrm SI}  +  \frac{1}{3} \,k_{3,1}^{\myrm SI} ~,
\end{eqnarray}
whose validity is indeed confirmed by the explicit calculations. As a
result the remaining unknown $k^{\myrm SI}_{3,1}$ is fixed as:
% \ice{
% K3SI = -179/384 + (25*z3)/48 - (5*z5)/24
% 
% \begin{equation}
% k^{\myrm SI}_{3,1} = -\frac{179}{1152} + \frac{254}{1443} \, \zeta_3 - \frac{10}{72}\, \zeta_5
% {}????.
% \end{equation}
% }
\begin{equation}
k^{\myrm SI}_{3,1} = -\frac{179}{384} + \frac{25}{48} \, \zeta_3 - \frac{5}{24}\, \zeta_5~.
\end{equation}


% \ice{
% \begin{eqnarray}
% &{}& \left(D^{\myrm NS} + \frac{n_f}{d_R}\,D^{\myrm SI}\right) C^{GLS} 
% =
% \\
% &{}&
%  1+ \frac{\beta(a_s)}{a_s}\, K(a_s),
% \\
% &{}& K^{\myrm NS}(a_s) =
% K^{\myrm NS}_0 +
% a_s\,K^{\myrm NS}_1 
% \nonumber
% \\
% && \hspace{1.7cm} + a_s^2\,K^{\myrm NS}_2 +a_s^3\,K^{\myrm NS}_3
% + \dots
% \label{gCrewtherFull}
% \end{eqnarray}
% }



\section{Conclusion}

We have analytically computed coeffcients of  all three colour 
structures  contributing to the singlet part of the Adler function in massless QCD at ${\cal O}(\alpha_s^4)$. 
We have  checked that all constraints on these coefficients derived previously in \cite{Baikov:2010iw} 
on the base of the GCR are  really fulfilled. This is  an important cross-check of 
our calculations of $D^{\myrm SI}$, $C^{\myrm SI}$ and  the very GCR. 




The calculations  has been performed
on a SGI ALTIX 24-node IB-interconnected cluster of 8-cores Xeon
computers 
using  parallel  MPI-based \cite{Tentyukov:2004hz} as well as thread-based 
\cite{Tentyukov:2007mu} versions  of FORM
\cite{Vermaseren:2000nd}.  For the evaluation of color factors we have used the FORM program {\em COLOR}
\cite{vanRitbergen:1998pn}. The diagrams have been generated with QGRAF \cite{Nogueira:1991ex}.
The figures have been drawn with the  the help of
Axodraw \cite{Vermaseren:1994je} and JaxoDraw  \cite{Binosi:2003yf}.


This work was supported by the Deutsche Forschungsgemeinschaft in the
Sonderforschungsbereich/Transregio SFB/TR-9 ``Computational Particle
Physics'' and  by RFBR grants  11-02-01196, 10-02-00525.


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