We consider the $Z^* Q{\bar Q}$ vertex to order $\alpha_s^2$ in the QCD coupling for an on-shell massive quark-antiquark pair and for arbitrary momentum transfer of the $Z$ boson. We present closed analytic expressions for the two parity-violating form factors of that vertex at the two-loop level in QCD, excluding the contributions from triangle diagrams. These form factors are expressed in terms of 1-dimensional harmonic polylogarithms of maximum weight 4. \vskip .3cm \flushright{ \begin{minipage}{12.3cm} {Key words}: Feynman diagrams, Multi-loop calculations, Vertex diagrams, \hspace*{18.5mm} Heavy quarks.\\ {PACS}: 11.15.Bt, 12.38.Bx, 14.65.Fy, 14.65.Ha, 13.88.+e \end{minipage} } %{Key words}: Feynman diagrams, Multi-loop calculations, Vertex diagrams, %Heavy quarks. %{PACS}: 11.15.Bt, 12.38.Bx, 14.65.Fy, 14.65.Ha
QCD_AVff.tex
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%              : 17.12.04
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%\journal{Nuclear Physics B}

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\begin{titlepage}
\nopagebreak
{\flushright{
        \begin{minipage}{5cm}
        PITHA 04/11 \\%{\bf V3.2 11.06.04}\\
        Freiburg-THEP 04/10\\
        ZU-TH 11/04\\
        UCLA/04/TEP/25\\
%        CERN-TH/2004-XXX\\
        {\tt hep-ph/0412259}\\
        \end{minipage}        }
}
\vspace*{-1.5cm}                        
\vskip 1.5cm
\begin{center}
{\Large \bf Two-Loop QCD Corrections to the Heavy Quark \\[2mm]
Form Factors: Axial Vector Contributions}
\vskip 1.cm
{\large  W.~Bernreuther$\rm \, ^{a, \,}$\footnote{Email: 
{\tt breuther@physik.rwth-aachen.de}}},
{\large  R.~Bonciani$\rm \, ^{b, \,}$\footnote{Email: 
{\tt Roberto.Bonciani@physik.uni-freiburg.de}}},
{\large T.~Gehrmann$\rm \, ^{c, \,}$\footnote{Email: 
{\tt gehrt@physik.unizh.ch}}}, \\[2mm] 
{\large R.~Heinesch$\rm \, ^{a, \,}$\footnote{Email: 
{\tt heinesch@physik.rwth-aachen.de}}},
{\large T.~Leineweber$\rm \, ^{a, \,}$\footnote{Email: 
{\tt leineweber@physik.rwth-aachen.de}}}, 
{\large P.~Mastrolia$\rm \, ^{d, \,}$\footnote{Email: 
{\tt mastrolia@physics.ucla.edu}}}, \\[2mm] 
and {\large E.~Remiddi$\rm \, ^{e, \,}$\footnote{Email: 
{\tt Ettore.Remiddi@bo.infn.it}}}
%
\vskip .6cm
{\it $\rm ^a$ Institut f\"ur Theoretische Physik, RWTH Aachen,
D-52056 Aachen, Germany} 
\vskip .2cm
{\it $\rm ^b$ Fakult\"at f\"ur Mathematik und Physik, Albert-Ludwigs-Universit\"at
Freiburg, \\ D-79104 Freiburg, Germany} 
\vskip .2cm
{\it $\rm ^c$ Institut f\"ur Theoretische Physik, 
Universit\"at Z\"urich, CH-8057 Z\"urich, Switzerland}
\vskip .2cm
{\it $\rm ^d$ Department of Physics and Astronomy, UCLA,
Los Angeles, CA 90095-1547} 
\vskip .2cm
{\it $\rm ^e$ Dipartimento di Fisica dell'Universit\`a di Bologna, and
INFN, Sezione di Bologna, I-40126 Bologna, Italy} 
%{\it $\rm ^f$ Theory Division, CERN, CH-1211 Geneva 23, Switzerland} 
%\vskip .2cm
\end{center}
\vskip .4cm


\begin{abstract}
We consider the $Z^* Q{\bar Q}$ vertex to order $\alpha_s^2$ in the QCD
coupling for an on-shell massive quark-antiquark pair and for
arbitrary momentum transfer of the $Z$ boson. We present closed
analytic expressions for the two parity-violating
form factors of that vertex at the two-loop level in QCD, excluding the contributions from triangle
diagrams. These form factors are expressed 
in terms of 1-dimensional harmonic polylogarithms of maximum weight 4.

\vskip .3cm
\flushright{
        \begin{minipage}{12.3cm}
{\it Key words}:  Feynman diagrams, Multi-loop calculations,  Vertex diagrams,
\hspace*{18.5mm} Heavy quarks.\\
{\it PACS}: 11.15.Bt, 12.38.Bx, 14.65.Fy, 14.65.Ha, 13.88.+e
        \end{minipage}        }
%{\it Key words}: Feynman diagrams, Multi-loop calculations,  Vertex diagrams,
%Heavy quarks.
%{\it PACS}: 11.15.Bt, 12.38.Bx, 14.65.Fy, 14.65.Ha
\end{abstract}
\vfill
\end{titlepage}    


%%%%%%%%%%%%%%%%%%%%%% 72 characters %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\section{Introduction \label{sec_intro}}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

This paper is the second of a series that is devoted to the computation
of the  electromagnetic and neutral current form factors of
heavy quarks $Q$ at the two-loop level in QCD \cite{bbghlmr}.
These form factors are relevant for a number of applications. For
instance, they are part of the order $\alpha_s^2$ QCD corrections
to the differential electroproduction cross sections of heavy quarks
$e^+ e^- \to \gamma^*, Z^* \to Q {\bar Q} X$ and, in particular,
part of the second order QCD corrections to the forward-backward asymmetry
$A_{fb}^Q$. As far as $b$ quarks are concerned, an order $\alpha_s^2$ 
calculation
with a non-zero $b$ quark mass  is of interest in view of
the discrepancy between experimental measurement and theoretical
expectations -- see \cite{EWWorkingGroup} and the
references  given in 
\cite{bbghlmr}
concerning the state of the theoretical predictions.  
A review of the present status of electroweak corrections to
the forward-backward asymmetry is given in \cite{Freitas:2004mn}.
At a future  linear collider,
forward-backward asymmetries will play a  prominent role in
very precise measurements of the neutral current couplings of bottom
and of top quarks \cite{tesla}. Clearly, predictions
will be required taking the  mass of the heavy quark fully into account.
\par
In \cite{bbghlmr} we presented
closed analytic expressions to order $\alpha_s^2$ of the heavy-quark
electromagnetic vertex form factors for arbitrary momentum  transfer.
Up to an overall coupling factor these are identical to the
corresponding vector, i.e., parity-conserving form factors that appear
in the amplitude of the
decay of a virtual $Z$ boson into a heavy quark-antiquark pair.
In this paper, we compute the axial vector form factors
$G_1$ and $G_2$, excluding the anomalous triangle graph contributions,
Fig.~\ref{triangle-graph}~(a) and (b), which contribute only through the
mass splitting of a quark isospin doublet in the triangle loop. These
terms deserve a separate discussion and will be given elsewhere
\cite{triangle}. \par The paper is organized as follows. In Section 2
we fix our conventions and describe how the form factors can be
obtained from the vertex amplitude by appropriate projections. In
Section 3 the renormalization constants in the scheme that we use --
which is the same as the one chosen in \cite{bbghlmr} -- are collected
for the convenience of the reader. Sections 4 and 5 contain, for
spacelike momentum transfer, the unsubtracted and renormalized axial
vector form factors at one-loop and two-loop order, respectively, for
the cases of the renormalization scale being both equal and different
from the heavy quark mass. In Section 6 the form factors are
analytically continued above the ${\bar Q} Q$ threshold, and we give
there  also their threshold and asymptotic expansions. We
conclude in Section 7.
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\section{The Axial Vector Form Factors \label{sec_avff}}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We consider the amplitude $V^\mu_{c_1  c_2}(p_1,p_2)$ for the
decay of a virtual $Z$ boson of four-momentum $q=p_1+p_2$ into a massive quark-antiquark
pair of momenta $p_1$, $p_2$ and colors $c_1$, $c_2$. The quarks $Q,\,\bar{Q}$ are
on their mass shell, $p_1^2=p_2^2=m^2$, where $m$ denotes the pole
mass of $Q$. The squared center-of-mass energy is $S = (p_1+p_2)^2$.

A general decomposition of the vertex function $V^\mu(p_1,p_2)$
involves 6 form factors, two of which odd under a $CP$
transformation. As we consider here, besides QCD
interactions, Standard Model (SM) neutral current interactions to
lowest order, $CP$ invariance holds. This implies that $V^\mu$ depends
only on 4 form factors, and we use the decomposition
\bea
V^\mu_{c_1  c_2}(p_1,p_2) & = & \bar{u}_{c_1}(p_1)\Gamma^\mu_{c_1
  c_2}(q) v_{c_2}(p_2), \\
\Gamma^\mu_{c_1 c_2}(q) & = & (-i) \left( v_Q F_1(s)\gamma^\mu +
  v_Q\frac{1}{2m}F_2(s) i \sigma^{\mu \nu} q_\nu \right. \nn \\
& & \phantom{(-i)} + \left.a_Q G_1(s) \gamma^\mu \gamma_5 + a_Q
  \frac{1}{2m}G_2(s) \gamma_5 q^\mu \right) \delta_{c_1 c_2} \label{decomp},
\eea
where $s=S/m^2$, $\sigma^{\mu \nu} = \frac{i}{2}\left[\gamma^\mu,\gamma^\nu
\right]$, $\bar{u}_{c_1}(p_1)$, $v_{c_2}(p_2)$ are Dirac spinors and
$v_Q$, $a_Q$ are the SM vector and axial vector couplings of the massive
quark $Q$ to the Z boson:  
\be
v_Q = \frac{e}{s_w c_w} \left(
  \frac{T_3^Q}{2}-s_w^2 Q_Q\right), \quad a_Q = -
\frac{e}{s_w c_w}\frac{T_3^Q}{2},
\ee
where $s_w(c_w)$ is the sine (cosine) of the weak mixing angle,
$T_3^Q$ the third component of the weak isospin and $Q_Q$ is the charge
of the heavy quark in units of the positron charge $e>0$.

The dimensionless form factors $F_i,\, G_i$ can be obtained from
$V^\mu_{c_1c_2}(p_1,p_2)$ by appropriate projections. We consider the spinor traces
\bea
&&\mathcal{P}_1=\mathrm{Tr}\left[ \gamma_\mu \left( \not\!
    p_1+m\right) \Gamma^\mu(q) \left( \not\!
    p_2-m\right)\right], \label{proj1}\\
&&\mathcal{P}_2=\mathrm{Tr}\left[ t_\mu \left( \not\!
    p_1+m\right) \Gamma^\mu(q) \left( \not\!
    p_2-m\right)\right],\label{proj2}\\
&&\mathcal{P}_3=\mathrm{Tr}\left[ \gamma_\mu \gamma_5\left( \not\!
    p_1+m\right) \Gamma^\mu(q) \left( \not\!
    p_2-m\right)\right], \label{proj3}\\
&&\mathcal{P}_4=\mathrm{Tr}\left[ \gamma_5 q_\mu \left( \not\!
    p_1+m\right) \Gamma^\mu(q) \left( \not\!
    p_2-m\right)\right],\label{proj4}
\eea
where $t^\mu = p_2^\mu - p_1^\mu$. Since we are working in $D=4 - 2 \epsilon$ dimensions we calculate
these traces in $D$ dimensions as well. Inserting Eq.~(\ref{decomp}) into Eqs.~(\ref{proj1}-\ref{proj4})
and performing the traces one obtains:
\bea
F_1 + F_2 & = & -\frac{i}{v_Q}\frac{2\mathcal{P}_2 +
  m(4-s)\mathcal{P}_1}{4sm^3(1-\epsilon)(4-s)} \label{formfactorf1}, \\ 
F_2 & = & -\frac{i}{v_Q}\frac{ [2+(1-\epsilon)s]\mathcal{P}_2 +
  m(4-s)\mathcal{P}_1}{sm^3(1-\epsilon)(4-s)^2}  \label{formfactorf2}, \\
G_1 & = & \phantom{-}
\frac{i}{a_Q}\frac{sm\mathcal{P}_3+2\mathcal{P}_4} {4sm^3(1-\epsilon)(4-s)},
\label{formfactorg1}
 \\
G_2 & = &
-\frac{i}{a_Q}\frac{ms\mathcal{P}_3+(2(3-2\epsilon)-(1-\epsilon)s)\mathcal{P}_4}
{s^2m^3(1-\epsilon)(4-s)}.
\label{formfactorg2}
\eea
The trace of the unit matrix is kept equal to four also in $D$
dimensions. In calculating the diagrams 
considered in this paper we use an anticommuting $\gamma_5$ in $D$
dimensions. (This prescription is also used in 
the derivation of the formulae (\ref{formfactorg1}),
(\ref{formfactorg2}).)  This prescription is appropriate
as the diagrams below correspond to the  order $\alpha_S^2$ 
``non-singlet'' contributions to the matrix
element of the axial vector current, and it is well known that for
these contributions a canonical, i.e., non-anomalous Ward identity
must hold to this order. Within dimensional regularisation this is most conveniently
implemented with an anticommuting $\gamma_5$. 
In a subsequent paper \cite{triangle}, the contributions involving closed triangle loops,
Fig. 1, will be computed for a mass-split doublet of fermions in the
triangle loop. In that context, the naive anticommuting  $\gamma_5$
used here is problematic in  $D\neq 4$.
However, it will be shown there that using a different
$\gamma_5$ prescription \cite{HV,Larin} does not affect the non-anomalous
contributions presented here. \\

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% TRIANGLE DIAGRAM
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bfig
\bc
\subfigure[]{
\begin{fmfgraph*}(40,25)
\fmfleft{i}
\fmfright{o1,o2}
\fmfright{o1,o2}
\fmf{dashes}{vz,i}
\fmf{phantom}{o1,v1,v2,v3,v4,vz,v5,v6,v7,v8,o2}
\fmffreeze
\fmf{plain}{vz,v3,v6,vz}
\fmf{dbl_plain}{o1,v1,v8,o2}
\fmf{gluon}{v1,v3}
\fmf{gluon}{v6,v8}
\end{fmfgraph*}
}
\subfigure[]{
\begin{fmfgraph*}(40,25)
\fmfleft{i}
\fmfright{o1,o2}
\fmfright{o1,o2}
\fmf{dashes}{vz,i}
\fmf{phantom}{o1,v1,v2,v3,v4,vz,v5,v6,v7,v8,o2}
\fmffreeze
\fmf{dbl_plain}{vz,v6,v3,vz}
\fmf{dbl_plain}{o1,v1,v8,o2}
\fmf{gluon}{v1,v3}
\fmf{gluon}{v6,v8}
\end{fmfgraph*}
}
\caption{\label{triangle-graph}Triangle diagram contributions to
  $V^\mu$. Crossed diagrams are not drawn.
The external dashed line refers to an incoming
  $Z$ boson, the curly lines to gluons, the double straight lines to the
  massive quark and the simple straight lines to massless quarks.}
\ec
\efig
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The formulae Eqs.~(\ref{formfactorf1}-\ref{formfactorg2}) show that with
the above projections the computation of the vector form factors
$F_{1,2}$ and the axial vector (i.e., parity-violating) form factors 
$G_{1,2}$ decouple from each other. The vector
form factors in Eqs.~(\ref{formfactorf1},\ref{formfactorf2}) were computed
in \cite{bbghlmr}. Here we determine the form factors in
Eqs.~(\ref{formfactorg1},\ref{formfactorg2}) to the second order in
the strong coupling constant $\alpha_S$, excluding the contributions
from the triangle diagrams shown in Fig.~\ref{triangle-graph}. Expanding the
renormalized form factors to the second order in
$\alpha_S$, we have:
\bea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
G_1\Bigl(s,\epsilon,\frac{\mu^2}{m^2}\Bigr) = &1 \; + & \! \! \! \! \left(
  \frac{\alpha_S}{2\pi}\right)G_{1,R}^{(1l)}\Bigl(s,\epsilon,\frac{\mu^2}{m^2}\Bigr)+\left(
  \frac{\alpha_S}{2\pi}\right)^2
G_{1,R}^{(2l)}\Bigl(s,\epsilon,\frac{\mu^2}{m^2}\Bigr) \nn \\
&& \! \! \! \! + \mathcal{O}
\left(\left(\frac{\alpha_S}{2\pi}\right)^3\right)\label{g1exp},\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
G_2\Bigl(s,\epsilon,\frac{\mu^2}{m^2}\Bigr) = && \! \! \! \! \left(
  \frac{\alpha_S}{2\pi}\right)G_{2,R}^{(1l)}\Bigl(s,\epsilon,\frac{\mu^2}{m^2}\Bigr)
+ \left(  \frac{\alpha_S}{2\pi}\right)^2
G_{2,R}^{(2l)}\Bigl(s,\epsilon,\frac{\mu^2}{m^2}\Bigr)\nn \\
&& \! \! \! \! + \mathcal{O}
\left(\left(\frac{\alpha_S}{2\pi}\right)^3\right)\label{g2exp},
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\eea
where the superscripts $(1l)$ and $(2l)$ denote the one- and two-loop
contributions. The subscript ``$R$'' labels the renormalization scheme
specified in the next Section. After having performed the
renormalization, the form factors still depend on the parameter
$\epsilon$, which regularizes the remaining
infrared divergencies. We keep $\alpha_S$ dimensionless also in
$D\not=4$ dimensions.

The form factors are represented as series in $\epsilon$ and expressed in terms of 1-dimensional harmonic
polylogarithms $\mbox{H}(\vec{a};x)$ up to weight 4 \cite{Polylog,Polylog3}, which are functions of the
dimensionless variable $x$ defined by
\be
x = \frac{\sqrt{-S+4m^2} - \sqrt{-S} }{\sqrt{-S+4m^2} + \sqrt{-S}} =
\frac{\sqrt{-s+4} - \sqrt{-s} }{\sqrt{-s+4} + \sqrt{-s}}.\label{xvar} 
\ee
%First we take $s$ to be space-like, $s<0$. In this region  $0\leq x <
%1$ and the form factors remain real. 
We give our results firstly in the kinematical region in which $s$ is
spacelike ($0\leq x \leq 1$), where the form factors are real. In Section \ref{sec_analytic} we
shall perform the analytical continuation to the physical region above
threshold, $s > 4$, $-1 < x \leq 0$, and explicitly decompose the form factors into real
and imaginary parts. 

In what follows  $N_f$ denotes the
number of light quarks (which we take to be massless) running in the
loops Fig.~\ref{2-loop-graph}~(g), and $C_F =(N_c^2-1)/(2N_c)$, $C_A =
N_c$, $T_R =1/2$, where  $N_c$ is the number of colors. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\section{Renormalization Scheme\label{sec_ren-scheme}}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


As in our previous paper \cite{bbghlmr}, we use renormalized
perturbation theory with $\alpha_S = g_s^2/(4\,\pi)$ being defined as
the standard $\overline{\mathrm{MS}}$ coupling in QCD with $N_f$
massless and one massive quark, while we define the mass $m$ and the
wave-function of the heavy quark $Q$ in the on-shell (OS) scheme. 

For the renormalization procedure we need the coupling
renormalization and the gluon wave function to one-loop:
\bea
Z_g^{\overline{\mathrm{MS}}} =  1 -
C(\epsilon)\; \frac{\alpha_S}{2\pi}\;
\frac{1}{4\epsilon}\; \left(\frac{11}{3}C_A-\frac{4}{3}T_R(N_f+1)\right)\,+\,\mathcal{O}\left(\left(\frac{\alpha_S}{2\,\pi}\right)^2\right),\label{zg}
\eea
where
\bea
C(\epsilon) = (4 \pi)^{\epsilon} \, \Gamma \left( 1 + \epsilon \right) 
\,  \label{cofd}
\eea
and, in the Feynman gauge,
\bea
Z_3^{\overline{\mathrm{MS}}} = 1 +
C(\epsilon)\; \frac{\alpha_S}{2\pi}\;
\frac{1}{2\epsilon}\;
\left(\frac{5}{3}C_A-\frac{4}{3}T_R(N_f+1)\right)\, + 
\,\mathcal{O}\left(\left(\frac{\alpha_S}{2\,\pi}\right)^2\right)
\label{z3msb}.
\eea
The renormalization constants concerning the heavy quark are
defined in the on-shell scheme. Here we need
\bea
Z_m^{\mathrm{OS}} = 1 +
\frac{\alpha_S}{2\pi}\;C(\epsilon)\;C_F\;\frac{1}{2\epsilon}\;\frac{2\epsilon-3}{1-2\epsilon}\left(\frac{\mu^2}{m^2}\right)^{\epsilon}+
\mathcal{O}\left(\left(\frac{\alpha_S}{2\,\pi}\right)^2\right),
\label{zmos}
\eea
to one-loop order. Here and in the following $\mu$
denotes the mass scale of dimensional regularization/renormalization.
The wave function $Z_2^{\rm{OS}}$ is needed to two-loop
order. The latter was computed in \cite{Broadhurst,Melnikov}. Using the result of
\cite{Melnikov} and expressing it in terms of the renormalized
$\overline{\mathrm{MS}}$ coupling $\alpha_S$ we have 
\bea
Z_2^{\mathrm{OS}} = &&\nr 1 +
\frac{\alpha_S}{2\pi}\;C(\epsilon)\;C_F\;\frac{1}{2\epsilon}\;\frac{2\epsilon-3}{1-2\epsilon}\left(\frac{\mu^2}{m^2}\right)^{\epsilon}\nn
\\
&&\nr \phantom{1} + \,C^2(\epsilon) \left(\frac{\alpha_S}{2\pi}\right)^2 \;
Z_2^{(2)}+\mathcal{O}\left(\left(\frac{\alpha_S}{2\,\pi}\right)^2\right)\label{z2os}
\eea
where $Z_2^{(2)}$ is: 
\bea
%%%%%%%%%%%%%%%
Z_{2}^{(2)} = &&\nr C_{F}^2
\left(\frac{\mu^2}{m^2}\right)^{2\epsilon}\Bigg\{{\frac
  {9}{8\,\epsilon^2}} + {\frac {51}{16\,\epsilon}}+{\frac {433}{32}}-{\frac {39}{2}}\,\zeta \left( 2 \right) +24\,\zeta
 \left( 2 \right) \ln  \left( 2 \right)\nn \\
%%%%%%%%%%%%%%%
&&\nr  -6\,\zeta \left( 3 \right)\Bigg\} + C_{F} C_{A}  \Bigg\{\left(\frac{\mu^2}{m^2}\right)^{\epsilon}\bigg[{\frac {11}{12}}\,\bigg(\frac{3}{\epsilon^2}+\frac{4}{\epsilon}+8\bigg)
\bigg]\nn \\
%%%%%%%%%%%%%%%
&&\nr +\left(\frac{\mu^2}{m^2}\right)^{2\epsilon}\bigg[-{\frac
  {11}{8\,\epsilon^2}}-{\frac {101}{16\,\epsilon}} + \frac{15}{2}\,\zeta \left( 2 \right) -{\frac {803}{32}}-12\,\zeta \left( 2
 \right) \ln
  \left( 2 \right) \nn \\
%%%%%%%%%%%%%%%
&&\nr  +3\,\zeta \left( 3 \right)\bigg]\Bigg\} + C_{F} T_{R} N_f  \Bigg\{\left(\frac{\mu^2}{m^2}\right)^{\epsilon}\bigg[-\frac{1}{3}\,\bigg(\frac{3}{\epsilon^2}+\frac{4}{\epsilon}+8\bigg)
\bigg]\nn \\
%%%%%%%%%%%%%%%
&&\nr
+\left(\frac{\mu^2}{m^2}\right)^{2\epsilon}\bigg[\frac{1}{2\,\epsilon^{2}}+\frac{9}{4\,\epsilon}+2\,\zeta
\left( 2 \right) +{\frac {59}{8}}\bigg]\Bigg\} \nn\\
%%%%%%%%%%%%%%%
&&\nr + C_{F} T_{R}
\Bigg\{\left(\frac{\mu^2}{m^2}\right)^{\epsilon}\bigg[-\frac{1}{3} \,\bigg(\frac{3}{\epsilon^2}+\frac{4}{\epsilon}+8\bigg)
\bigg]\nn \\
%%%%%%%%%%%%%%%
&&\nr +\left(\frac{\mu^2}{m^2}\right)^{2\epsilon}\bigg[\frac{1}{{\epsilon}^{2}}+{
\frac {19}{12\,\epsilon}}+{\frac
  {1139}{72}}-8\,\zeta \left( 2 \right)\bigg]\Bigg\} .
%%%%%%%%%%%%%%%
\eea
Further we need the renormalization constant $Z_{\mathrm{1F}}$ for the $Q\bar{Q}$
gluon vertex to one loop. Using a Slavnov-Taylor
identitiy and Eqs.~(\ref{zg},\ref{z3msb},\ref{z2os}) we get:
\bea
Z_{\mathrm{1F}}
= &&\nr Z_g^{\overline{\mathrm{MS}}} \; Z_2^{\mathrm{OS}} \;
\sqrt{Z_3^{\overline{\mathrm{MS}}}} \nn \\
= &&\nr 1 - C(\epsilon)\; \frac{\alpha_S}{2\pi} \;
\frac{1}{2\epsilon} \; \left(C_A - C_F
  \;\frac{2\epsilon-3}{1-2\epsilon}\left(\frac{\mu^2}{m^2}\right)^{\epsilon}\right)\nn \\
&&\nr \!\!+\,\mathcal{O}\left(\left(\frac{\alpha_S}{2\,\pi}\right)^2\right)
\label{z1f}.
\eea
For the counterterm contributions to the renormalized form factors it
is convenient to define (here our notation differs slightly
  from that in \cite{bbghlmr})
\bea
\delta_{\mathrm{1F}} & = & Z_{\mathrm{1F}} \, - \, 1,\\
\delta_2 & = & Z_2^{\mathrm{OS}} \, - \, 1 = \delta_2^{(1l)} \, + \, \delta_2^{(2l)},\\
\delta_3 & = & Z_3^{\overline{\mathrm{MS}}} \, - \, 1, \\
\delta_m & = & \left(Z_2^{\mathrm{OS}}\,Z_m^{\mathrm{OS}} - 1\right) \; m,
\eea
where $\delta_2^{(1l)}$, $\delta_2^{(2l)}$  can be read off from
Eq.~(\ref{z2os}). 

In this renormalization scheme the renormalized vertex function
$\Gamma^\mu$ to order $\alpha_S^2$ is given by the contributions from
the 1-particle irreducible diagrams, Figs.~\ref{1-loop-graph}
and \ref{2-loop-graph}, which we call the unsubtracted contributions,
and the counterterms given below.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\begin{boldmath}
%\section{Results for $G_{i,R}^{\,(1l)},\,i=1,2$ \label{sec_1l_results}}
\section{One-loop Unsubtracted and UV-Renormalized Form Factors}\label{sec_1l_results}
\end{boldmath}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{fmfgroup}
\bfig
\bc
\begin{fmfgraph*}(40,25)
\fmfleft{i}
\fmfright{o1,o2}
\fmf{dashes}{vz,i}
\fmf{dbl_plain}{o1,v1,v2,v3,v4,vz,v5,v6,v7,v8,o2}
\fmffreeze
\fmf{gluon}{v1,v8}
\fmflabel{$q$}{i}
\fmflabel{$p_1$}{o2}
\fmflabel{$p_2$}{o1}
\end{fmfgraph*}
\caption{\label{1-loop-graph}One-loop QCD contribution to $V^\mu_{c_1
    c_2}(q)$.} 
\ec
\efig
\end{fmfgroup}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this Section we give the results for $G_{1,R}^{(1l)},\,
  G_{2,R}^{(1l)}$. First we write down the contribution from the diagram
Fig.~\ref{1-loop-graph} including the terms of order $\epsilon$. They
are needed for computing the order $\alpha_S^2$ counterterms
shown in Figs.~\ref{ct-graph}~(c), (d) and (f) of Section \ref{subsec_sub_cont}
below. We obtain:
\bea
G_1^{(\mathrm{Fig.\ref{1-loop-graph}})}\Bigl(s,\epsilon,\frac{\mu^2}{m^2}\Bigr)
= && \nr \frac{\alpha_S}{2\,\pi}\;C_F \,
C(\epsilon)\,\left(\frac{\mu^2}{m^2}\right)^{\epsilon}\left( \frac{1}{\epsilon} \; a_1 +
  a_2 + \epsilon \; a_3 \right) + \mathcal{O}(\epsilon^2), \label{g11l} \\
G_2^{(\mathrm{Fig.\ref{1-loop-graph}})}\Bigl(s,\epsilon,\frac{\mu^2}{m^2}\Bigr)
= && \nr \frac{\alpha_S}{2\,\pi}\;C_F \,
C(\epsilon)\,\left(\frac{\mu^2}{m^2}\right)^{\epsilon}\left( \frac{1}{\epsilon} \; b_1 +
  b_2 + \epsilon \; b_3 \right) + \mathcal{O}(\epsilon^2), \label{g21l}  
\eea
where 
$C(\epsilon)$ is defined in Eq.~(\ref{cofd}).

The coefficients $a_i$ and $b_i$ ($i=1\dots 3$) are:
\bea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
a_1 = && \nr (x^2+1)/[(x-1)(x+1)]\,H(0;x)+1/2,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
a_2 = && \nr 1/2\,(3\,x^2-2\,x+3)/[(x-1)(x+1)]\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(x^2+1)/[(x-1)(x+1)]\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+(x^2+1)/[(x-1)(x+1)]\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-\zeta(2)\,(x^2+1)/[(x-1)(x+1)],\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
a_3 = && \nr 2\,\zeta(2)\,(x^2+1)/[(x-1)(x+1)]\,H(-1;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-(x^2+1)\,(-4+\zeta(2))/[(x-1)(x+1)]\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-(3\,x^2-2\,x+3)/[(x-1)(x+1)]\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4\,(x^2+1)/[(x-1)(x+1)]\,H(-1,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(x^2+1)/[(x-1)(x+1)]\,H(0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/2\,(3\,x^2-2\,x+3)/[(x-1)(x+1)]\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(x^2+1)/[(x-1)(x+1)]\,H(-1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+(x^2+1)/[(x-1)(x+1)]\,H(0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/2\,(3\,\zeta(2)\,x^2+4\,\zeta(3)\,x^2-2\,\zeta(2)\,x\nn \\
%%%%%%%%%%%%%%%
&& \nr+3\,\zeta(2)+4\,\zeta(3))/[(x-1)(x+1)],
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
\eea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
respectively,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
b_1 = && \nr 0,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
b_2 = && \nr 2\,(3\,x^2-2\,x+3)/[(x+1)(x-1)^3]\,x\,H(0;x)-4/(x-1)^2\,x,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
b_3 = && \nr 4\,(3\,x^2-2\,x+3)/[(x+1)(x-1)^3]\,x\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4\,(3\,x^2-2\,x+3)/[(x+1)(x-1)^3]\,x\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(3\,x^2-2\,x+3)/[(x+1)(x-1)^3]\,x\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(3\,\zeta(2)\,x^2+4\,x^2-2\,\zeta(2)\,x-4+3\,\zeta(2))/[(x+1)(x-1)^3]\,x.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\eea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bfig
\bc
\begin{fmfgraph*}(40,25)
\fmfleft{i}
\fmfright{o1,o2}
\fmf{dashes}{vz,i}
\fmf{dbl_plain}{o1,v1,v2,v3,v4,vz,v5,v6,v7,v8,o2}
\fmffreeze
\fmfv{decor.shape=cross,label=$\delta_2^{(1l)}$,label.angle=90.
  ,label.dist=0.1w}{vz} 
\end{fmfgraph*}
\caption{\label{1-loop-ct-graph}Subtraction term for the one-loop
  renormalization.} 
\ec
\efig
The counterterm of Fig.~\ref{1-loop-ct-graph} contributes to $G_1$:
\bea
G_1^{\mathrm{(Fig.~\ref{1-loop-ct-graph})}}\Bigl(s,\epsilon,\frac{\mu^2}{m^2}\Bigr) = \delta_2^{(1l)}\label{g11lct}
\eea
From Eqs.~(\ref{g11l}), (\ref{g21l}) and (\ref{g11lct}) we obtain the
renormalized one-loop form factors
\bea
G_{1,R}^{(1l)}\Bigl(s,\epsilon,\frac{\mu^2}{m^2}\Bigl) = && \nr C_F \,
C(\epsilon)\,\left(\frac{\mu^2}{m^2}\right)^{\epsilon}\,\left(\frac{1}{\epsilon} \;
  \tilde{a}_1 + \tilde{a}_2  + \epsilon \, \tilde{a}_3 \right) +
\mathcal{O}\left(\epsilon^2\right), \label{g1_1l_ren_decomp}\\
G_{2,R}^{(1l)}\Bigl(s,\epsilon,\frac{\mu^2}{m^2}\Bigr) = && \nr C_F \,
C(\epsilon)\,\left(\frac{\mu^2}{m^2}\right)^{\epsilon}\,\left(\frac{1}{\epsilon} \;
  \tilde{b}_1 + \tilde{b}_2  + \epsilon \, \tilde{b}_3 \right) +
\mathcal{O}\left(\epsilon^2\right), \label{g2_1l_ren_decomp}
\eea
with
\bea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\tilde{a}_1 = && \nr a_1 - 3/2,\\  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\tilde{a}_2 = && \nr a_2 - 2,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\tilde{a}_3 = && \nr a_3 - 4
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\eea
and $\tilde{b}_i = b_i$ for $i=1\ldots3$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% ENDE 1-loop
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace{.5cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\begin{boldmath}
%\section{Results for $G_{i,R}^{\,(2l)},\,i=1,2$ \label{sec_2l_results}}
\section{Two-Loop Unsubtracted and Renormalized\\
Form Factors}\label{sec_2l_results}
\end{boldmath}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Two-Loop Unsubtracted Form Factors \label{subsec_2l_cont}}

%%%%%%%%%%%%%%%%%%%%%%% 2-Loop graphs %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bfig
\bc
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\subfigure[]{
\begin{fmfgraph*}(40,25)
\fmfleft{i}
\fmfright{o1,o2}
\fmf{dashes}{vz,i}
\fmf{dbl_plain}{o1,v1,v2,v3,v4,vz,v5,v6,v7,v8,o2}
\fmffreeze
\fmf{gluon,left}{v6,v8}
\fmf{gluon}{v2,v7}
%\fmflabel{$Q$}{i}
%\fmflabel{$p_1$}{o2}
%\fmflabel{$p_2$}{o1}
\end{fmfgraph*}
\label{2-loop-graph-1}
}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\subfigure[]{
\begin{fmfgraph*}(40,25)
\fmfleft{i}
\fmfright{o1,o2}
\fmf{dashes}{vz,i}
\fmf{dbl_plain}{o1,v1,v2,v3,v4,vz,v5,v6,v7,v8,o2}
\fmffreeze
\fmf{gluon}{v1,v8}
\fmf{gluon,left}{v5,v7}
%\fmflabel{$Q$}{i}
%\fmflabel{$p_1$}{o2}
%\fmflabel{$p_2$}{o1}
\end{fmfgraph*}
\label{2-loop-graph-2}
}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\subfigure[]{
\begin{fmfgraph*}(40,25)
\fmfleft{i}
\fmfright{o1,o2}
\fmf{dashes}{vz,i}
\fmf{dbl_plain}{o1,v1,v2,v3,v4,vz,v5,v6,v7,v8,o2}
\fmfforce{.76w,.5h}{vtg}
\fmffreeze
\fmf{gluon,tension=0.}{vtg,v8}
\fmf{gluon,tension=0.}{v6,vtg}
\fmf{gluon,tension=0.}{v1,vtg}
%\fmflabel{$Q$}{i}
%\fmflabel{$p_1$}{o2}
%\fmflabel{$p_2$}{o1}
\end{fmfgraph*}
\label{2-loop-graph-3}
}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\subfigure[]{
\begin{fmfgraph*}(40,25)
\fmfleft{i}
\fmfright{o1,o2}
\fmf{dashes}{vz,i}
\fmf{dbl_plain}{o1,v1,v2,v3,v4,vz,v5,v6,v7,v8,o2}
\fmffreeze
\fmf{gluon}{v1,v8}
\fmf{gluon}{v2,v7}
%\fmflabel{$Q$}{i}
%\fmflabel{$p_1$}{o2}
%\fmflabel{$p_2$}{o1}
\end{fmfgraph*}
\label{2-loop-graph-4}
}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\subfigure[]{
\begin{fmfgraph*}(40,25)
\fmfleft{i}
\fmfright{o1,o2}
\fmf{dashes}{vz,i}
\fmf{dbl_plain}{o1,v1,v2,v3,v4,vz,v5,v6,v7,v8,o2}
\fmffreeze
\fmf{gluon}{v1,v7}
\fmf{gluon,rubout}{v2,v8}
%\fmflabel{$Q$}{i}
%\fmflabel{$p_1$}{o2}
%\fmflabel{$p_2$}{o1}
\end{fmfgraph*}
\label{2-loop-graph-5}
}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\subfigure[]{
\begin{fmfgraph*}(40,25)
\fmfleft{i}
\fmfright{o1,o2}
\fmf{dashes}{vz,i}
\fmf{dbl_plain}{o1,v1,v2,v3,v4,vz,v5,v6,v7,v8,o2}
\fmffreeze
\fmf{gluon}{v1,vn1}
\fmf{gluon}{vn2,v8}
\fmf{dbl_plain,left,tension=0.4}{vn1,vn2,vn1}
%\fmflabel{$Q$}{i}
%\fmflabel{$p_1$}{o2}
%\fmflabel{$p_2$}{o1}
\end{fmfgraph*}
\label{2-loop-graph-6}
}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\subfigure[]{
\begin{fmfgraph*}(40,25)
\fmfleft{i}
\fmfright{o1,o2}
\fmf{dashes}{vz,i}
\fmf{dbl_plain}{o1,v1,v2,v3,v4,vz,v5,v6,v7,v8,o2}
\fmffreeze
\fmf{gluon}{v1,vn1}
\fmf{gluon}{vn2,v8}
\fmf{plain,left,tension=0.4}{vn1,vn2,vn1}
%\fmflabel{$Q$}{i}
%\fmflabel{$p_1$}{o2}
%\fmflabel{$p_2$}{o1}
\end{fmfgraph*}
\label{2-loop-graph-7}
}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\subfigure[]{
\begin{fmfgraph*}(40,25)
\fmfleft{i}
\fmfright{o1,o2}
\fmf{dashes}{vz,i}
\fmf{dbl_plain}{o1,v1,v2,v3,v4,vz,v5,v6,v7,v8,o2}
\fmffreeze
\fmf{gluon}{v1,vn1}
\fmf{gluon}{vn2,v8}
\fmf{gluon,left,tension=0.4}{vn1,vn2,vn1}
%\fmflabel{$Q$}{i}
%\fmflabel{$p_1$}{o2}
%\fmflabel{$p_2$}{o1}
\end{fmfgraph*}
\label{2-loop-graph-8}
}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\subfigure[]{
\begin{fmfgraph*}(40,25)
\fmfleft{i}
\fmfright{o1,o2}
\fmf{dashes}{vz,i}
\fmf{dbl_plain}{o1,v1,v2,v3,v4,vz,v5,v6,v7,v8,o2}
\fmffreeze
\fmf{gluon}{v1,vn1}
\fmf{gluon}{vn2,v8}
\fmf{dashes,left,tension=0.4}{vn1,vn2,vn1}
%\fmflabel{$Q$}{i}
%\fmflabel{$p_1$}{o2}
%\fmflabel{$p_2$}{o1}
\end{fmfgraph*}
\label{2-loop-graph-9}
}
\caption{\label{2-loop-graph}Two-loop QCD contributions to $V^\mu_{c_1
    c_2}(p_1,p_2)$. The
  inner dashed line refers to ghosts. Straight lines represent massless
  quarks and the double straight line the massive quark (c.f. Fig.~\ref{triangle-graph}).}
\ec
\efig

First we compute the two-loop unsubtracted contributions represented
by the diagrams of Figs.~\ref{2-loop-graph}~(a)-(i) and the ``mirror''
diagrams to Figs.~\ref{2-loop-graph}~(a)-(c) to the
form factors $G_1,\, G_2$. 

Performing the $\gamma$ algebra we obtain, as explained in
\cite{RoPieRem1,RoPieRem2,RoPieRem3}, the form factors in
Eqs.~(\ref{formfactorg1},\ref{formfactorg2}) expressed in terms of
hundreds of different scalar integrals. These integrals are reduced to
a small set of master integrals by means of the so-called Laporta
algorithm \cite{Lap} with the help of integration-by-parts identities
\cite{Chet} and Lorentz-invariance identities \cite{Rem3}. Symmetry
relations which one can establish between different integrals are also
used during the reduction. The master integrals themselves were
evaluated with the method of differential equations
\cite{Rem3,Kot,Rem1,Rem2} in \cite{RoPieRem1,RoPieRem3}. The master
integrals, and thus the form factors are represented as series in the
regularization parameter $\epsilon$ and expressed in terms of
1-dimensional harmonic polylogarithms up to weight 4
\cite{Polylog,Polylog3}, which are functions of the dimensionless
variable $x$ defined in Eq.~(\ref{xvar}). As in the one-loop case we
take $s$ to be space-like. 

The two-loop unsubtracted form factors are found to be:
\bea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
G_1^{\mathrm{(Fig.~\ref{2-loop-graph})}}\Bigl(s,\epsilon,\frac{\mu^2}{m^2}\Bigr) = && \nr  \left(\frac{\alpha_S}{2\,\pi}\right)^2\;C^2(\epsilon) \,
\left(\frac{\mu^2}{m^2}\right)^{2\epsilon}\;\Bigg\{\nn \\
&& \nr  \phantom{+}\frac{1}{\epsilon^2}\; \left(C_F\; T_R\; N_f
\;  c_1\; + \;C_F \;T_R\; c_2 \;+ \;C_F\; C_A\; c_3\; + \;C_F^2\;
c_4\right)\nn \\
&& \nr + \,\frac{1}{\epsilon}\; \left(C_F\; T_R\; N_f
\;  c_5\; + \;C_F \;T_R\; c_6 \;+ \;C_F\; C_A\; c_7\; + \;C_F^2\;
c_8\right)\nn \\
&& \nr + \, C_F\; T_R\; N_f
\;  c_9\; + \;C_F \;T_R\; c_{10} \;+ \;C_F\; C_A\; c_{11}\; + \;C_F^2\;
c_{12} \nn \\ 
&& \nr +\,\mathcal{O}(\epsilon)\Bigg\} \label{g12lu}, \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
G_2^{\mathrm{(Fig.~\ref{2-loop-graph})}}\Bigl(s,\epsilon,\frac{\mu^2}{m^2}\Bigr) = && \nr
\left(\frac{\alpha_S}{2\,\pi}\right)^2\; 
C^2(\epsilon)\,\left(\frac{\mu^2}{m^2}\right)^{2\epsilon}\;\Bigg\{\nn \\ 
&& \nr \phantom{+}\frac{1}{\epsilon^2}\;\left(C_F\; T_R\; N_f
\;  d_1\; + \;C_F \;T_R\; d_2 \;+ \;C_F\; C_A\; d_3\; + \;C_F^2\;
d_4\right)\nn \\
&& \nr +\,\frac{1}{\epsilon}\;\left(C_F\; T_R\; N_f
\;  d_5\; + \;C_F \;T_R\; d_6 \;+ \;C_F\; C_A\; d_7\; + \;C_F^2\;
d_8\right)\nn \\
&& \nr +\,C_F\; T_R\; N_f
\;  d_9\; + \;C_F \;T_R\; d_{10} \;+ \;C_F\; C_A\; d_{11}\; + \;C_F^2\;
d_{12} \nn \\
&& \nr + \,\mathcal{O}(\epsilon)\Bigg\}\label{g22lu},
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
\eea
where $C(\epsilon)$ is defined in Eq.~(\ref{cofd}) and 
 $N_f$,  $C_F$, $C_A$, and $T_R$ are given at the end of Section 2.
 One finds for
the $c_i$ ($i=1\dots 12$):
\bea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
c_1 = && \nr \frac{1}{2}\,c_2\, = -\frac{4}{11}\,c_3\, \nn \\
%%%%%%%%%%%%%%%
 = && \nr -1/3\,(x^2+1) / [(x-1)(x+1)]\,H(0;x)-1/6,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
c_4 = && \nr 1/2\,(x^4+20\,x^3+14\,x^2+20\,x+1)/[(x+1)^3(x-1)]\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+(x^2+1)^2/[(x-1)^2(x+1)^2]\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/8\,(25\,x^2+2\,x+25)/(x+1)^2,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
c_5 = && \nr -2/9\,(7\,x^2-3\,x+7)/[(x-1)(x+1)]\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4/3\,(x^2+1)/[(x-1)(x+1)]\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2/3\,(x^2+1)/[(x-1)(x+1)]\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/36\,(-13\,x^2+24\,\zeta(2)\,x^2+13+24\,\zeta(2))/[(x-1)(x+1)],\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
c_6 = && \nr -1/3\,(3\,x^2-2\,x+3)/[(x-1)(x+1)]\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4/3\,(x^2+1)/[(x-1)(x+1)]\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2/3\,(x^2+1)/[(x-1)(x+1)]\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/12\,(-3\,x^2+8\,\zeta(2)\,x^2+3+8\,\zeta(2))/[(x-1)(x+1)],\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
c_7 = && \nr-1/18\,(27\,\zeta(2)\,x^4-83\,x^4+33\,x^3+18\,\zeta(2)\,x^2-33\,x-9\,\zeta(2)+83)\nn\\
%%%%%%%%%%%%%%%
&& \nr \hspace{1.2cm}/[(x+1)^2(x-1)^2]\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-14/3\,(x^2+1)/[(x-1)(x+1)]\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+(x^2+1)^2/[(x+1)^2(x-1)^2]\,H(0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/6\,(23\,x^2+11)/[(x-1)(x+1)]\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(x^2+1)/[(x+1)^2(x-1)^2]\,x^2\,H(0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+(x^2+1)/[(x-1)(x+1)]\,H(1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-(x^2+1)^2/[(x+1)^2(x-1)^2]\,H(0,1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/144\,(48\,\zeta(2)\,x^4-185\,x^4+72\,\zeta(3)\,x^4+370\,x^2\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.6cm}+288\,\zeta(2)\,x^2+144\,\zeta(3)\,x^2-185-336\,\zeta(2)+72\,\zeta(3))\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.6cm}/[(x+1)^2(x-1)^2],\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
c_8 = && \nr-1/4\,(4\,\zeta(2)\,x^5-3\,x^5+4\,\zeta(2)\,x^4-49\,x^4+18\,x^3\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.1cm}+8\,\zeta(2)\,x^3+8\,\zeta(2)\,x^2-18\,x^2+4\,\zeta(2)\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.1cm}+49\,x+3+4\,\zeta(2))/[(x+1)^3(x-1)^2]\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-(x^4+20\,x^3+14\,x^2+20\,x+1)/[(x+1)^3(x-1)]\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(x^2+1)^2/[(x+1)^2(x-1)^2]\,H(0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/2\,(7\,x^5+21\,x^4+2\,x^3+14\,x^2-17\,x +5)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}/[(x+1)^3(x-1)^2]\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4\,(x^2+1)^2/[(x-1)^2(x+1)^2]\,H(-1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+3\,(x^2+1)^2/[(x-1)^2(x+1)^2]\,H(0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/16\,(35\,x^4+8\,\zeta(2)\,x^4+6\,x^3+160\,\zeta(2)\,x^3+112\,\zeta(2)\,x^2-6\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.4cm}+160\,\zeta(2)\,x-35+8\,\zeta(2))/[(x+1)^3(x-1)],\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
c_9 = && \nr -8/3\,\zeta(2)\,(x^2+1)/[(x-1)(x+1)]\,H(-1;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/54\,(353\,x^2-78\,x+353)/[(x-1)(x+1)]\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+8/9\,(7\,x^2-3\,x+7)/[(x-1)(x+1)]\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-16/3\,(x^2+1)/[(x-1)(x+1)]\,H(-1,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+8/3\,(x^2+1)/[(x-1)(x+1)]\,H(0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4/9\,(7\,x^2-3\,x+7)/[(x-1)(x+1)]\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+8/3\,(x^2+1)/[(x-1)(x+1)]\,H(-1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4/3\,(x^2+1)/[(x-1)(x+1)]\,H(0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/216\,(-169\,x^2+576\,\zeta(3)\,x^2+528\,\zeta(2)\,x^2-288\,\zeta(2)\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.6cm}+576\,\zeta(3)+169+816\,\zeta(2))/[(x-1)(x+1)],\\ 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
c_{10} = && \nr -4/3\,\zeta(2)\,(x^2+1)/[(x-1)(x+1)]\,H(-1;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/54\,(409\,x^4+80\,x^3+302\,x^2-216\,\zeta(2)\,x^2+80\,x+409)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.2cm}/[(x-1)(x+1)^3]\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2/3\,(3\,x^2-2\,x+3)/[(x-1)(x+1)]\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-8/3\,(x^2+1)/[(x-1)(x+1)]\,H(-1,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4/3\,(x^2+1)/[(x-1)(x+1)]\,H(0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2/9\,(5\,x^5-27\,x^4+5\,x^3-23\,x^2+6\,x-14)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}/[(x+1)^4(x-1)]\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4/3\,(x^2+1)/[(x-1)(x+1)]\,H(-1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4/3\,(x^2-x+1)\,(x^2+3\,x+1)/[(x-1)(x+1)^3]\,H(0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/216\,(288\,\zeta(3)\,x^5+223\,x^5+1728\,\zeta(2)\,x^5-291\,x^4-2304\,\zeta(2)\,x^4\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.6cm}+864\,\zeta(3)\,x^4+1152\,\zeta(3)\,x^3+1872\,\zeta(2)\,x^3-514\,x^3\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.6cm}+1152\,\zeta(3)\,x^2+514\,x^2-1008\,\zeta(2)\,x^2+291\,x+864\,\zeta(3)\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.6cm}+3312\,\zeta(2)\,x-1296\,\zeta(2)+288\,\zeta(3)-223)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.6cm}/[(x+1)^4(x-1)],\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
c_{11} = && \nr1/3\,(\zeta(2)\,x^4+6\,\zeta(3)\,x^4+81\,\zeta(2)\,x^3+12\,\zeta(3)\,x^2-108\,\zeta(2)\,x^2\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.8cm}+81\,\zeta(2)\,x+6\,\zeta(3)-37\,\zeta(2))/[(x+1)^2(x-1)^2]\,H(-1;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+\zeta(2)\,(7\,x^6+6\,x^5-29\,x^4+24\,x^3-31\,x^2+6\,x+5)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.1cm}/[(x+1)^3(x-1)^3]\,H(0,-1;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/216\,(648\,\zeta(2)\,x^6+4129\,x^6-1620\,\zeta(3)\,x^6+3024\,\zeta(3)\,x^5\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.6cm}-12528\,\zeta(2)\,x^5-570\,x^5-4129\,x^4+18792\,\zeta(2)\,x^4\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.6cm}-13716\,\zeta(3)\,x^4+25920\,\zeta(3)\,x^3-12744\,\zeta(2)\,x^3+1140\,x^3\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.6cm}-13500\,\zeta(3)\,x^2+3456\,\zeta(2)\,x^2-4129\,x^2-2808\,\zeta(2)\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.6cm}-570\,x+3024\,\zeta(3)\,x-1404\,\zeta(3)+4129)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.6cm}/[(x+1)^3(x-1)^3]\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/18\,(161\,x^4-276\,x^3-72\,\zeta(2)\,x^2+276\,x-161-72\,\zeta(2))\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.2cm}/[(x-1)^2(x+1)^2]\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-10\,(x^2+1)^2/[(x-1)^2(x+1)^2]\,H(0,-1,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/3\,(79\,x^4+6\,x^3-66\,x^2+6\,x-13)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}/[(x-1)^2(x+1)^2]\,H(0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4\,(x^2+1)^2/[(x-1)^2(x+1)^2]\,H(-1,0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(11\,x^6+5\,x^5-15\,x^4+26\,x^3-27\,x^2+5\,x-1)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}/[(x+1)^3(x-1)^3]\,H(0,0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4\,(x^2+1)^2/[(x-1)^2(x+1)^2]\,H(1,0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-6\,(x^2+1)/[(x-1)(x+1)]\,H(1,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+6\,(x^2+1)^2/[(x-1)^2(x+1)^2]\,H(0,1,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/18\,(18\,\zeta(2)\,x^6-59\,x^6-72\,\zeta(2)\,x^5-174\,x^5+176\,x^4\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.4cm}+306\,\zeta(2)\,x^4-252\,\zeta(2)\,x^3+240\,x^3+270\,\zeta(2)\,x^2\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.4cm}-283\,x^2-72\,\zeta(2)\,x-66\,x+166-18\,\zeta(2))\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.4cm}/[(x+1)^3(x-1)^3]\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/3\,(64\,x^4-141\,x^3+228\,x^2-141\,x-28)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}/[(x-1)^2(x+1)^2]\,H(-1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(7\,x^6-7\,x^5+33\,x^4-56\,x^3+25\,x^2-7\,x-1)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}/[(x+1)^3(x-1)^3]\,H(0,-1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/3\,(50\,x^6-201\,x^5+244\,x^4-177\,x^3+13\,x^2-12\,x+11)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}/[(x+1)^3(x-1)^3]\,H(0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+74/3\,(x^2+1)/[(x-1)(x+1)]\,H(-1,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(x^2+3)\,(x^2+1)/[(x-1)^2(x+1)^2]\,H(-1,0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(x^5-5\,x^4+18\,x^3-26\,x^2+x-5)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}/[(x-1)^2(x+1)^3]\,H(1,0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(2\,x^4+5\,x^3-14\,x^2+5\,x+6)/[(x-1)^2(x+1)^2]\,H(1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-(12\,x^5-x^4+10\,x^3-28\,x^2-2\,x-1)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.4cm}/[(x+1)^3(x-1)^3]\,x\,H(0,0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(x^6-5\,x^5+21\,x^4-34\,x^3+17\,x^2-5\,x-3)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}/[(x+1)^3(x-1)^3]\,H(0,1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+(-3\,x^5+4\,\zeta(2)\,x^5-x^4-8\,\zeta(2)\,x^4+40\,\zeta(2)\,x^3+2\,x^3\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.5cm}-48\,\zeta(2)\,x^2-2\,x^2+x+4\,\zeta(2)\,x+3-8\,\zeta(2))\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(x-1)^2(x+1)^3]\,H(1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-6\,(x^2+1)/[(x-1)(x+1)]\,H(-1,1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+6\,(x^2+1)^2/[(x-1)^2(x+1)^2]\,H(0,-1,1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4\,(x^2+1)^2/[(x-1)^2(x+1)^2]\,H(-1,0,1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(7\,x^5+7\,x^4+12\,x^3+4\,x^2+x+1)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}/[(x-1)^2(x+1)^3]\,H(0,0,1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4\,(x^2+1)^2/[(x-1)^2(x+1)^2]\,H(1,0,1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(x^2+1)/[(x-1)(x+1)]\,H(1,1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(x^2+1)^2/[(x-1)^2(x+1)^2]\,H(0,1,1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+8\,(x^2+1)/[(x-1)(x+1)^2]\,x\,H(0,1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-(x^2+1)\,(\zeta(2)\,x^2+2\,\zeta(3)\,x^2-\zeta(2)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,\zeta(3))/[(x-1)^2(x+1)^2]\,H(1;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+\zeta(2)\,(x^2+1)^2/[(x-1)^2(x+1)^2]\,H(0,1;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/4320\,(85920\,\zeta(2)\,x^6-77760\,\zeta(2)\,\ln(2)\,x^6+25200\,\zeta(3)\,x^6\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}-12925\,x^6+15336\,\zeta(2)^2\,x^6+155520\,\zeta(2)\,\ln(2)\,x^5+28512\,\zeta(2)^2\,x^5\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}+181440\,\zeta(3)\,x^5-279360\,\zeta(2)\,x^5+38775\,x^4-99576\,\zeta(2)^2\,x^4\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}-344880\,\zeta(3)\,x^4+230160\,\zeta(2)\,x^4-77760\,\zeta(2)\,\ln(2)\,x^4\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}+83520\,\zeta(2)\,x^3+109728\,\zeta(2)^2\,x^3+268560\,\zeta(3)\,x^2-38775\,x^2\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}-294720\,\zeta(2)\,x^2+77760\,\zeta(2)\,\ln(2)\,x^2-115560\,\zeta(2)^2\,x^2\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}+28512\,\zeta(2)^2\,x-181440\,\zeta(3)\,x+195840\,\zeta(2)\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}-155520\,\zeta(2)\,\ln(2)\,x-648\,\zeta(2)^2+77760\,\zeta(2)\,\ln(2)+12925\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}-21360\,\zeta(2)+51120\,\zeta(3))/[(x+1)^3(x-1)^3],\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
c_{12} = && \nr (13\,x^5-23\,x^4+12\,x^3+24\,x^2-61\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr+11)\,\zeta(2)/[(x+1)^3(x-1)^2]\,H(-1;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(7\,x^6+10\,x^5-65\,x^4+104\,x^3-75\,x^2\nn\\
%%%%%%%%%%%%%%%
&& \nr+10\,x-3)/[(x+1)^3(x-1)^3]\,H(0,-1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/2\,(17\,x^6-58\,x^5-213\,x^4+224\,x^3-37\,x^2\nn\\
%%%%%%%%%%%%%%%
&& \nr-46\,x+17)/[(x+1)^3(x-1)^3]\,H(0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-12\,(x^2+1)^2/[(x-1)^2(x+1)^2]\,H(-1,0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+(27\,x^6-2\,x^5+27\,x^4-56\,x^3-7\,x^2-2\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr-7)/[(x+1)^3(x-1)^3]\,H(0,0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+8\,(x^5-2\,x^4+18\,x^3-14\,x^2+4\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr+1)/[(x+1)^3(x-1)^2]\,H(1,0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,\zeta(2)\,(5\,x^6+6\,x^5-31\,x^4+24\,x^3-29\,x^2\nn\\
%%%%%%%%%%%%%%%
&& \nr+6\,x+7)/[(x+1)^3(x-1)^3]\,H(0,-1;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(8\,x^4-19\,x^3+46\,x^2-19\,x+8)/[(x-1)^2(x+1)^2]\,H(1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4\,(x^6-x^5+14\,x^4-20\,x^3+14\,x^2-x\nn\\
%%%%%%%%%%%%%%%
&& \nr+1)/[(x+1)^3(x-1)^3]\,H(0,1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4\,(4\,x^5+\zeta(2)\,x^5-5\,\zeta(2)\,x^4+2\,x^4+34\,\zeta(2)\,x^3-2\,x^3+2\,x^2\nn\\
%%%%%%%%%%%%%%%
&& \nr-30\,\zeta(2)\,x^2+7\,\zeta(2)\,x-2\,x-4+\zeta(2))\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(x+1)^3(x-1)^2]\,H(1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(5\,x^4-10\,x^3+18\,x^2-10\,x+5)/[(x-1)^2(x+1)^2]\,H(0,1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4\,(3\,x^5+3\,x^4+6\,x^3-2\,x^2-x\nn\\
%%%%%%%%%%%%%%%
&& \nr-1)/[(x+1)^3(x-1)^2]\,H(0,0,1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4\,\zeta(3)\,(x^2+1)^2/[(x-1)^2(x+1)^2]\,H(1;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/160\,(1915\,x^6+4640\,\zeta(2)\,x^6+2896\,\zeta(2)^2\,x^6-640\,\zeta(3)\,x^6\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}-5760\,\zeta(2)\,\ln(2)\,x^6+11520\,\zeta(2)\,\ln(2)\,x^5-7680\,\zeta(3)\,x^5\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}-3520\,\zeta(2)\,x^5-1152\,\zeta(2)^2\,x^5-3840\,x^5+1935\,x^4+9584\,\zeta(2)^2\,x^4\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}-8160\,\zeta(2)\,x^4+25280\,\zeta(3)\,x^4-5760\,\zeta(2)\,\ln(2)\,x^4\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}-21248\,\zeta(2)^2\,x^3-1920\,\zeta(3)\,x^3-5760\,\zeta(2)\,x^3-17280\,\zeta(3)\,x^2\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}+5760\,\zeta(2)\,\ln(2)\,x^2+13920\,\zeta(2)\,x^2+7632\,\zeta(2)^2\,x^2-1935\,x^2\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}-1152\,\zeta(2)^2\,x+1920\,\zeta(3)\,x+4160\,\zeta(2)\,x-11520\,\zeta(2)\,\ln(2)\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}+3840\,x+5760\,\zeta(2)\,\ln(2)-1915-5280\,\zeta(2)+320\,\zeta(3)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}+944\,\zeta(2)^2)/[(x+1)^3(x-1)^3]\nn\\
%%%%%%%%%%%%%%%
&& \nr+4\,(x^2+1)^2/[(x-1)^2(x+1)^2]\,H(0,-1,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+8\,(x^2+1)^2/[(x-1)^2(x+1)^2]\,H(-1,0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-8\,(x^2+1)^2/[(x-1)^2(x+1)^2]\,H(1,0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+8\,(x^2+1)^2/[(x-1)^2(x+1)^2]\,H(1,0,1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+(13\,x^5-59\,x^4+72\,x^3+60\,x^2-21\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr+15)/[(x+1)^3(x-1)^2]\,H(0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(11\,x^6-2\,x^5+33\,x^4-56\,x^3+23\,x^2\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,x+1)/[(x+1)^3(x-1)^3]\,H(0,0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/8\,(148\,\zeta(2)\,x^6-64\,\zeta(3)\,x^6-19\,x^6-160\,\zeta(3)\,x^5-224\,\zeta(2)\,x^5\nn\\
%%%%%%%%%%%%%%%
&& \nr-326\,x^5+1056\,\zeta(3)\,x^4-676\,\zeta(2)\,x^4+275\,x^4-1536\,\zeta(3)\,x^3\nn\\
%%%%%%%%%%%%%%%
&& \nr+992\,\zeta(2)\,x^3+275\,x^2-724\,\zeta(2)\,x^2+1056\,\zeta(3)\,x^2-160\,\zeta(3)\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr+140\,x^3-326\,x+96\,\zeta(2)\,x-19+4\,\zeta(2)\nn\\
%%%%%%%%%%%%%%%
&& \nr-64\,\zeta(3))/[(x+1)^3(x-1)^3]\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4\,(-9\,x^5+\zeta(2)\,x^5-4\,x^4+\zeta(2)\,x^4+2\,\zeta(2)\,x^3+13\,x^3-13\,x^2\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,\zeta(2)\,x^2+4\,x+\zeta(2)\,x+9+\zeta(2))/[(x+1)^3(x-1)^2]\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/2\,(26\,\zeta(2)\,x^6+123\,x^6-8\,\zeta(2)\,x^5-112\,x^5-33\,x^4+62\,\zeta(2)\,x^4\nn\\
%%%%%%%%%%%%%%%
&& \nr+120\,x^3-160\,\zeta(2)\,x^3+42\,\zeta(2)\,x^2-135\,x^2-8\,\zeta(2)\,x+56\,x-19\nn\\
%%%%%%%%%%%%%%%
&& \nr+6\,\zeta(2))/[(x+1)^3(x-1)^3]\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(x^4+20\,x^3+14\,x^2+20\,x+1)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(x+1)^3(x-1)]\,H(-1,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-(9\,x^5-3\,x^4+116\,x^3+128\,x^2-41\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr+7)/[(x+1)^3(x-1)^2]\,H(-1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+16\,(x^2+1)^2/[(x-1)^2(x+1)^2]\,H(-1,-1,0,0;x).
%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\eea
The $d_i$ ($i=1\dots 12$) are:
\bea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
d_1 = && \nr d_2 = d_3 = 0,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
d_4 = && \nr -6\,(x^2+1)/[(x+1)^3(x-1)]\,x\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+6/(x+1)^2\,x,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
d_5 = && \nr d_6 = -\frac{4}{11}\,d_7 = -4/3\,(3\,x^2-2\,x+3)/[(x+1)(x-1)^3]\,x\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+8/3/(x-1)^2\,x,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
d_8 = && \nr -(9\,x^4-60\,x^3+22\,x^2-60\,x+9)/[(x+1)^3(x-1)^3]\,x\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+12\,(x^2+1)/[(x+1)^3(x-1)]\,x\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(x^2+1)\,(3\,x^3+11\,x^2-7\,x+9)/[(x+1)^3(x-1)^4]\,x\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(-3\,x^3+3\,\zeta(2)\,x^3-25\,x^2-3\,\zeta(2)\,x^2\nn\\
%%%%%%%%%%%%%%%
&& \nr+3\,\zeta(2)\,x-25\,x-3-3\,\zeta(2))/[(x+1)^3(x-1)^2]\,x,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
d_9 = && \nr -2/9\,(87\,x^2-50\,x+87)/[(x+1)(x-1)^3]\,x\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+16/3\,(3\,x^2-2\,x+3)/[(x+1)(x-1)^3]\,x\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-8/3\,(3\,x^2-2\,x+3)/[(x+1)(x-1)^3]\,x\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4/9\,(31\,x^2+18\,\zeta(2)\,x^2-12\,\zeta(2)\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr-31+18\,\zeta(2))/[(x+1)(x-1)^3]\,x,\\ 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
d_{10} = && \nr -2/9\,(87\,x^4-128\,x^3-46\,x^2-72\,\zeta(2)\,x^2\nn\\
%%%%%%%%%%%%%%%
&& \nr-128\,x+87)/[(x+1)^3(x-1)^3]\,x\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+8/3\,(3\,x^2-2\,x+3)/[(x+1)(x-1)^3]\,x\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-8/3\,(12\,x^4-3\,x^3+9\,x^2-5\,x+3)/[(x+1)^4(x-1)^3]\,x\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+16/[(x+1)^3(x-1)^3]\,x^3\,H(0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+8/9\,(2\,x^5+27\,\zeta(2)\,x^5-18\,x^4-57\,\zeta(2)\,x^4-20\,x^3+63\,\zeta(2)\,x^3\nn\\
%%%%%%%%%%%%%%%
&& \nr+20\,x^2-45\,\zeta(2)\,x^2+18\,x+78\,\zeta(2)\,x-2\nn\\
%%%%%%%%%%%%%%%
&& \nr-18\,\zeta(2))/[(x+1)^4(x-1)^3]\,x,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
d_{11} = && \nr -18\,\zeta(2)\,(x^2+1)\,(x^2-4\,x+1)/[(x+1)^2(x-1)^4]\,x\,H(-1;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+24\,\zeta(2)\,(x^2+1)\,(x^2-5\,x+1)/[(x+1)^3(x-1)^5]\,x^2\,H(0,-1;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/18\,(1083\,x^6+324\,\zeta(2)\,x^6-238\,x^5+1008\,\zeta(3)\,x^5-3384\,\zeta(2)\,x^5\nn\\
%%%%%%%%%%%%%%%
&& \nr+3960\,\zeta(2)\,x^4-1083\,x^4-4752\,\zeta(3)\,x^4+476\,x^3+9216\,\zeta(3)\,x^3\nn\\
%%%%%%%%%%%%%%%
&& \nr-7056\,\zeta(2)\,x^3+1620\,\zeta(2)\,x^2-1083\,x^2-4752\,\zeta(3)\,x^2-238\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr+1008\,\zeta(3)\,x-648\,\zeta(2)\,x+1083)/[(x+1)^3(x-1)^5]\,x\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4/3\,(18\,x^2-55\,x+18)/[(x+1)(x-1)^3]\,x\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-48\,(x^2-x+1)/[(x+1)^2(x-1)^4]\,x^2\,H(0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+8\,(5\,x^4-14\,x^3+30\,x^2-14\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr+5)/[(x+1)^3(x-1)^5]\,x^2\,H(0,0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2/3\,(3\,x^6+24\,\zeta(2)\,x^5+103\,x^5-126\,\zeta(2)\,x^4-27\,x^4-98\,x^3\nn\\
%%%%%%%%%%%%%%%
&& \nr+24\,\zeta(2)\,x^3+57\,x^2-126\,\zeta(2)\,x^2+24\,\zeta(2)\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr-5\,x-33)/[(x+1)^3(x-1)^5]\,x\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(9\,x^4-92\,x^3+130\,x^2-92\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr+9)/[(x+1)^2(x-1)^4]\,x\,H(-1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-8\,(7\,x^4-29\,x^3+62\,x^2-29\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr+7)/[(x+1)^3(x-1)^5]\,x^2\,H(0,-1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(9\,x^4-112\,x^3+146\,x^2-196\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr+9)/[(x+1)^3(x-1)^5]\,x^3\,H(0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4\,(x^4-5\,x^3+38\,x^2-5\,x+1)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(x+1)^3(x-1)^5]\,x^2\,H(0,0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-16\,(3\,x^2-8\,x+3)/[(x+1)^3(x-1)^3]\,x^2\,H(1,0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-24\,(3\,x^2-2\,x+3)/[(x+1)^2(x-1)^4]\,x^2\,H(1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+8\,(5\,x^4-14\,x^3+42\,x^2-14\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr+5)/[(x+1)^3(x-1)^5]\,x^2\,H(0,1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4\,(x^4+12\,\zeta(2)\,x^3-2\,x^2-32\,\zeta(2)\,x^2+12\,\zeta(2)\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr+1)/[(x+1)^3(x-1)^3]\,x\,H(1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-16/[(x+1)^2(x-1)^2]\,x^2\,H(0,1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-32/[(x+1)^3(x-1)^3]\,x^3\,H(0,0,1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2/45\,(270\,\zeta(3)\,x^6-945\,\zeta(2)\,x^6-935\,x^6+540\,\zeta(2)\,\ln(2)\,x^6\nn\\
%%%%%%%%%%%%%%%
&& \nr-3690\,\zeta(3)\,x^5-594\,\zeta(2)^2\,x^5-2160\,\zeta(2)\,\ln(2)\,x^5+5415\,\zeta(2)\,x^5\nn\\
%%%%%%%%%%%%%%%
&& \nr+2805\,x^4+2439\,\zeta(2)^2\,x^4+2700\,\zeta(2)\,\ln(2)\,x^4+5490\,\zeta(3)\,x^4\nn\\
%%%%%%%%%%%%%%%
&& \nr-3465\,\zeta(2)\,x^4-2010\,\zeta(2)\,x^3-1584\,\zeta(2)^2\,x^3-2700\,\zeta(2)\,\ln(2)\,x^2\nn\\
%%%%%%%%%%%%%%%
&& \nr+2439\,\zeta(2)^2\,x^2-2805\,x^2-5490\,\zeta(3)\,x^2+4185\,\zeta(2)\,x^2\nn\\
%%%%%%%%%%%%%%%
&& \nr-594\,\zeta(2)^2\,x+3690\,\zeta(3)\,x-3405\,\zeta(2)\,x+2160\,\zeta(2)\,\ln(2)\,x+935\nn\\
%%%%%%%%%%%%%%%
&& \nr-540\,\zeta(2)\,\ln(2)+225\,\zeta(2)-270\,\zeta(3))/[(x+1)^3(x-1)^5]\,x,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
d_{12} = && \nr 24\,(x^2+1)\,(x^3-3\,x^2-6\,x+2)\,\zeta(2)/[(x+1)^3(x-1)^4]\,x\,H(-1;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-48\,\zeta(2)\,(x^2+1)\,(x^2-5\,x+1)/[(x+1)^3(x-1)^5]\,x^2\,H(0,-1;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/2\,(84\,\zeta(2)\,x^6+45\,x^6-310\,x^5-32\,\zeta(3)\,x^5-168\,\zeta(2)\,x^5\nn\\
%%%%%%%%%%%%%%%
&& \nr+467\,x^4-1116\,\zeta(2)\,x^4+1056\,\zeta(3)\,x^4-1664\,\zeta(3)\,x^3-404\,x^3\nn\\
%%%%%%%%%%%%%%%
&& \nr+544\,\zeta(2)\,x^3+467\,x^2-564\,\zeta(2)\,x^2+1056\,\zeta(3)\,x^2+104\,\zeta(2)\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr-32\,\zeta(3)\,x-310\,x+45-36\,\zeta(2))/[(x+1)^3(x-1)^5]\,x\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4\,(13\,x^2+46\,x+13)/[(x+1)^3(x-1)]\,x\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-24\,(x^2+1)/[(x+1)^3(x-1)]\,x\,H(-1,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4\,(9\,x^5-29\,x^4+114\,x^3+90\,x^2-11\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr+3)/[(x+1)^3(x-1)^4]\,x\,H(0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-16\,(x^4+19\,x^3-28\,x^2+19\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr+1)/[(x+1)^3(x-1)^5]\,x^2\,H(0,0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(63\,x^6-78\,x^5+48\,\zeta(2)\,x^4+13\,x^4-216\,\zeta(2)\,x^3+188\,x^3\nn\\
%%%%%%%%%%%%%%%
&& \nr+48\,\zeta(2)\,x^2-215\,x^2+50\,x-21)/[(x+1)^3(x-1)^5]\,x\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-8\,(6\,x^5+4\,x^4+85\,x^3+97\,x^2-5\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr+9)/[(x+1)^3(x-1)^4]\,x\,H(-1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-16\,(x^4-37\,x^3+54\,x^2-37\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr+1)/[(x+1)^3(x-1)^5]\,x^2\,H(0,-1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(3\,x^6-40\,x^5-329\,x^4+88\,x^3-7\,x^2-24\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr+21)/[(x+1)^3(x-1)^5]\,x\,H(0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-8\,(x^4-5\,x^3+38\,x^2-5\,x+1)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(x+1)^3(x-1)^5]\,x^2\,H(0,0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-32\,(x^2-17\,x+1)/[(x+1)^3(x-1)^3]\,x^2\,H(1,0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+8\,(3\,x^4-7\,x^3+64\,x^2-7\,x+3)/[(x+1)^2(x-1)^4]\,x\,H(1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-16\,(x^4+19\,x^3-16\,x^2+19\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr+1)/[(x+1)^3(x-1)^5]\,x^2\,H(0,1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+32\,(x^4-\zeta(2)\,x^3+x^3+17\,\zeta(2)\,x^2-\zeta(2)\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr+x+1)/[(x+1)^3(x-1)^3]\,x\,H(1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-8\,(3\,x^4-6\,x^3+14\,x^2-6\,x+3)/[(x+1)^2(x-1)^4]\,x\,H(0,1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+64/[(x+1)^3(x-1)^3]\,x^3\,H(0,0,1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/5\,(75\,x^6+240\,\zeta(2)\,\ln(2)\,x^6-230\,\zeta(2)\,x^6-120\,\zeta(3)\,x^6\nn\\
%%%%%%%%%%%%%%%
&& \nr+560\,\zeta(3)\,x^5-960\,\zeta(2)\,\ln(2)\,x^5+320\,x^5+8\,\zeta(2)^2\,x^5\nn\\
%%%%%%%%%%%%%%%
&& \nr+340\,\zeta(2)\,x^5-865\,x^4-3760\,\zeta(3)\,x^4+1200\,\zeta(2)\,\ln(2)\,x^4\nn\\
%%%%%%%%%%%%%%%
&& \nr-1160\,\zeta(2)^2\,x^4+790\,\zeta(2)\,x^4+480\,\zeta(3)\,x^3+1400\,\zeta(2)\,x^3\nn\\
%%%%%%%%%%%%%%%
&& \nr+3240\,\zeta(2)^2\,x^3-1200\,\zeta(2)\,\ln(2)\,x^2+2920\,\zeta(3)\,x^2+865\,x^2\nn\\
%%%%%%%%%%%%%%%
&& \nr-1160\,\zeta(2)^2\,x^2-2330\,\zeta(2)\,x^2-140\,\zeta(2)\,x-320\,x+8\,\zeta(2)^2\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr-80\,\zeta(3)\,x+960\,\zeta(2)\,\ln(2)\,x-75+170\,\zeta(2)\nn\\
%%%%%%%%%%%%%%%
&& \nr -240\,\zeta(2)\,\ln(2))/[(x+1)^3(x-1)^5]\,x, 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\eea
with $\zeta(n)$ being Riemann's $\zeta$-function. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\subsection{Contributions from the Counterterms}\label{subsec_sub_cont}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bfig
\bc
\subfigure[]{
\begin{fmfgraph*}(40,25)
\fmfleft{i}
\fmfright{o1,o2}
\fmf{dashes}{vz,i}
\fmf{dbl_plain}{o1,v1,v2,v3,v4,vz,v5,v6,v7,v8,o2}
\fmffreeze
\fmf{gluon}{v1,v8}
\fmfv{decor.shape=cross,label=$\delta_2^{(1l)}\not\!p-\delta_m$,label.angle=90.,label.dist=0.15w}{v6}
%\fmflabel{$Q$}{i}
%\fmflabel{$p_1$}{o2}
%\fmflabel{$p_2$}{o1}
\end{fmfgraph*}
\label{ct-graph-1}
}
\subfigure[]{
\begin{fmfgraph*}(40,25)
\fmfleft{i}
\fmfright{o1,o2}
\fmf{dashes}{vz,i}
\fmf{dbl_plain}{o1,v1,v2,v3,v4,vz,v5,v6,v7,v8,o2}
\fmffreeze
\fmf{gluon}{v1,v8}
\fmfv{decor.shape=cross,label=$\delta_2^{(1l)}\not\!p-\delta_m$,label.angle=-90,label.dist=0.15w}{v3}
%\fmflabel{$Q$}{i}
%\fmflabel{$p_1$}{o2}
%\fmflabel{$p_2$}{o1}
\end{fmfgraph*}
\label{ct-graph-2}
}
\subfigure[]{
\begin{fmfgraph*}(40,25)
\fmfleft{i}
\fmfright{o1,o2}
\fmf{dashes}{vz,i}
\fmf{dbl_plain}{o1,v1,v2,v3,v4,vz,v5,v6,v7,v8,o2}
\fmffreeze
\fmfv{decor.shape=cross}{v8}
\fmfv{label=$\delta_{1{\mathrm F}}$,label.angle=135,label.dist=0.1w}{v8}
\fmf{gluon}{v1,v8}
%\fmflabel{$Q$}{i}
%\fmflabel{$p_1$}{o2}
%\fmflabel{$p_2$}{o1}
\end{fmfgraph*}
\label{ct-graph-3}
}
\subfigure[]{
\begin{fmfgraph*}(40,25)
\fmfleft{i}
\fmfright{o1,o2}
\fmf{dashes}{vz,i}
\fmf{dbl_plain}{o1,v1,v2,v3,v4,vz,v5,v6,v7,v8,o2}
\fmffreeze
\fmfv{decor.shape=cross}{v1}
\fmfv{label=$\delta_{1{\mathrm F}}$,label.angle=-135,label.dist=0.1w}{v1}
\fmf{gluon}{v1,v8}
%\fmflabel{$Q$}{i}
%\fmflabel{$p_1$}{o2}
%\fmflabel{$p_2$}{o1}
\end{fmfgraph*}
\label{ct-graph-4}
}
\subfigure[]{
\begin{fmfgraph*}(40,25)
\fmfleft{i}
\fmfright{o1,o2}
\fmf{dashes}{vz,i}
\fmf{dbl_plain}{o1,v1,v2,v3,v4,vz,v5,v6,v7,v8,o2}
\fmfforce{.76w,.5h}{vn}
\fmffreeze
\fmfv{decor.shape=cross,label=$\delta_3$,label.dist=.1w}{vn}
\fmf{gluon}{v1,vn}
\fmf{gluon}{vn,v8}
%\fmflabel{$Q$}{i}
%\fmflabel{$p_1$}{o2}
%\fmflabel{$p_2$}{o1}
\end{fmfgraph*}
\label{ct-graph-5}
}
\subfigure[]{
\begin{fmfgraph*}(40,25)
\fmfleft{i}
\fmfright{o1,o2}
\fmf{dashes}{vz,i}
\fmf{dbl_plain}{o1,v1,v2,v3,v4,vz,v5,v6,v7,v8,o2}
\fmffreeze
\fmfv{decor.shape=cross,label=$\delta_2^{(1l)}$,label.angle=90.,label.dist=0.1w}{vz}
\fmf{gluon}{v1,v8}
%\fmflabel{$Q$}{i}
%\fmflabel{$p_1$}{o2}
%\fmflabel{$p_2$}{o1}
\end{fmfgraph*}
\label{ct-graph-6}
}
\subfigure[]{
\begin{fmfgraph*}(40,25)
\fmfleft{i}
\fmfright{o1,o2}
\fmf{dashes}{vz,i}
\fmf{dbl_plain}{o1,v1,v2,v3,v4,vz,v5,v6,v7,v8,o2}
\fmffreeze
\fmfv{decor.shape=cross,label=$\delta_2^{(2l)}$,label.angle=90.,label.dist=0.1w}{vz}
%\fmflabel{$Q$}{i}
%\fmflabel{$p_1$}{o2}
%\fmflabel{$p_2$}{o1}
\end{fmfgraph*}
\label{ct-graph-7}
}
\caption{\label{ct-graph}Subtraction terms for the two-loop
  renormalization. Note that the diagrams (a)-(b) and (c)-(d)
  yield the same contribution, respectively.}
\ec
\efig

%%%%%%%%%%%%%%%% explanation of the ct-graphs %%%%%%%%%%%%%%%%%%%%%%%%%%

The terms in Fig.~\ref{ct-graph}~(a)-(f) are
defined as one-loop vertex amplitudes in $D$ dimensions including the
corresponding one-loop
renormalization insertions. They read explicitly:

\begin{fmfgroup}
\fmfset{curly_len}{2mm}

$\bullet$ Figs.~\ref{ct-graph}~(a) and \ref{ct-graph}~(b):
%
%
\bea
\parbox{25mm}{
\begin{fmfgraph*}(24,15)
\fmfleft{i}
\fmfright{o1,o2}
\fmf{dashes}{vz,i}
\fmf{dbl_plain}{o1,v1,v2,v3,v4,vz,v5,v6,v7,v8,o2}
\fmffreeze
\fmf{gluon}{v1,v8}
\fmfv{decor.shape=cross,decor.size=4mm}{v6}
\end{fmfgraph*}
}
& = &
\parbox{25mm}{
\begin{fmfgraph*}(24,15)
\fmfleft{i}
\fmfright{o1,o2}
\fmf{dashes}{vz,i}
\fmf{dbl_plain}{o1,v1,v2,v3,v4,vz,v5,v6,v7,v8,o2}
\fmffreeze
\fmf{gluon}{v1,v8}
\fmfv{decor.shape=cross,decor.size=4mm}{v3}
\end{fmfgraph*}
} \nn \\
& \stackrel{\mathrm{def.}}{=} & \frac{\alpha_S}{2\pi}\,C_F \,
C(\epsilon)\,\left(\frac{\mu^2}{m^2}\right)^{\epsilon}\nn \\
&&\times\int \mathfrak{D}^Dk \frac{\mathcal{N}_1}{\left[\left(p_1+k\right)^2 -
    m^2\right]^2\left[\left(p_2-k\right)^2 -m^2\right]k^2},
\eea
%
%
with
%
%
\bea
\mathcal{N}_1 & = &\gamma^\rho\left(\not\!k_{\phantom{1}} + \not\!p_1 + m\right)\left(\delta_2^{\left(1l\right)}
\left(\not\!k_{\phantom{1}} + \not\!p_1\right) - \delta_m\right)\left(\not\!k_{\phantom{1}} + \not\!p_1 + m\right)\nn \\
&&\times\left(v^Q\gamma^\mu + a^Q\gamma^\mu\gamma_5\right) \left(\not\!k_{\phantom{1}} - \not\!p_2 +
m\right)\gamma_\rho \label{ct_n1}  
\eea
%
%
and
%
%
\be
\mathfrak{D}^Dk = \frac{1}{C(\epsilon)} \left( \frac{m^2}{\mu^2}  \right)^\epsilon \frac{d^Dk}{(2\pi)^{D-2}}.
\ee
%
%
The contributions from these two diagrams to the axial vector form factors are:
\bea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
G_1^{(a,b)}\Bigl(s,\epsilon,\frac{\mu^2}{m^2}\Bigr) = && \nr  \left(\frac{\alpha_S}{2\,\pi}\right)^2\;C_F^2 \,
C^2(\epsilon)\,\left(\frac{\mu^2}{m^2}\right)^{2\,\epsilon}\,\Bigg\{\,
\nn \\
%%%%%%%%%%%%%%%
&& \nr \phantom{+}\frac{1}{\epsilon^2}\, \bigg[3/2\,(x-1)\,(x^2+x+1)/(x+1)^3\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} -3/4\,(x-1)^2/(x+1)^2\bigg]\nn \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
&& \nr + \, \frac{1}{\epsilon} \, \bigg[
1/4\,(17\,x^4+4\,x^3+6\,x^2+4\,x+17)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.5cm}/[(x-1)(x+1)^3]\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-3\,(x-1)\,(x^2+x+1)/(x+1)^3\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+3/2\,(x-1)\,(x^2+x+1)/(x+1)^3\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/2\,(x^2+x+1)\,(3\,\zeta(2)\,x-4\,x-3\,\zeta(2)-4)/(x+1)^3\bigg]\nn \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
&& \nr +3\,(x-1)\,(x^2+x+1)\,\zeta(2)/(x+1)^3\,H(-1;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/2\,(-26\,x^4+3\,\zeta(2)\,x^4-6\,x^3-3\,\zeta(2)\,x^3+8\,x^2-3\,\zeta(2)\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr-6\,x+3\,\zeta(2)-26)/[(x-1)(x+1)^3]\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/2\,(17\,x^4+4\,x^3+6\,x^2+4\,x+17)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}/[(x-1)(x+1)^3]\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+6\,(x-1)\,(x^2+x+1)/(x+1)^3\,H(-1,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-3\,(x-1)\,(x^2+x+1)/(x+1)^3\,H(0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/4\,(17\,x^4+4\,x^3+6\,x^2+4\,x+17)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}/[(x-1)(x+1)^3]\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-3\,(x-1)\,(x^2+x+1)/(x+1)^3\,H(-1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+3/2\,(x-1)\,(x^2+x+1)/(x+1)^3\,H(0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/4\,(17\,\zeta(2)\,x^4+12\,\zeta(3)\,x^4+16\,x^4-16\,x^3-12\,\zeta(3)\,x^3\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}+4\,\zeta(2)\,x^3+6\,\zeta(2)\,x^2+4\,\zeta(2)\,x+16\,x-12\,\zeta(3)\,x+17\,\zeta(2)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}+12\,\zeta(3)-16)/[(x-1)(x+1)^3]\Bigg\},\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
G_2^{(a,b)}\Bigl(s,\epsilon,\frac{\mu^2}{m^2}\Bigr) = && \nr\left(\frac{\alpha_S}{2\,\pi}\right)^2\;
C_F^2\,C^2(\epsilon)\,\left(\frac{\mu^2}{m^2}\right)^{2\,\epsilon}\,\Bigg\{
\nn \\
%%%%%%%%%%%%%%%
&& \nr \phantom{+}\frac{1}{\epsilon^2}\bigg[ 3\,(x^2+1)/[(x-1)(x+1)^3]\,x\,H(0;x)-3/(x+1)^2\,x\bigg] \nn \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
&& \nr +\frac{1}{\epsilon}\bigg[ (13\,x^2+6\,x+13)/[(x-1)(x+1)^3]\,x\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-6\,(x^2+1)/[(x-1)(x+1)^3]\,x\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+3\,(x^2+1)/[(x-1)(x+1)^3]\,x\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-(3\,\zeta(2)\,x^2+4\,x^2-4+3\,\zeta(2))/[(x-1)(x+1)^3]\,x\bigg] \nn \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
&& \nr + 6\,(x^2+1)\,\zeta(2)/[(x-1)(x+1)^3]\,x\,H(-1;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-(-41\,x^4+3\,\zeta(2)\,x^4+20\,x^3-6\,\zeta(2)\,x^3+6\,\zeta(2)\,x^2-6\,x^2\nn\\
%%%%%%%%%%%%%%%
&& \nr-6\,\zeta(2)\,x+20\,x-41+3\,\zeta(2))/[(x-1)^3(x+1)^3]\,x\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(13\,x^2+6\,x+13)/[(x-1)(x+1)^3]\,x\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+12\,(x^2+1)/[(x-1)(x+1)^3]\,x\,H(-1,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-6\,(x^2+1)/[(x-1)(x+1)^3]\,x\,H(0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+(13\,x^2+6\,x+13)/[(x-1)(x+1)^3]\,x\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-6\,(x^2+1)/[(x-1)(x+1)^3]\,x\,H(-1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+3\,(x^2+1)/[(x-1)(x+1)^3]\,x\,H(0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-(13\,\zeta(2)\,x^3+6\,\zeta(3)\,x^3+14\,x^3-7\,\zeta(2)\,x^2-6\,\zeta(3)\,x^2\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}+10\,x^2+6\,\zeta(3)\,x+7\,\zeta(2)\,x+10\,x-13\,\zeta(2)+14-6\,\zeta(3))\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}\hspace{.4cm}/[(x+1)^3(x-1)^2]\,x\Bigg\}. 
\eea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

$\bullet$ Figs.~\ref{ct-graph}~(c) and \ref{ct-graph}~(d):
\bea
\parbox{25mm}{
\begin{fmfgraph*}(24,15)
\fmfleft{i}
\fmfright{o1,o2}
\fmf{dashes}{vz,i}
\fmf{dbl_plain}{o1,v1,v2,v3,v4,vz,v5,v6,v7,v8,o2}
\fmffreeze
\fmfv{decor.shape=cross,decor.size=4mm}{v8}
\fmf{gluon}{v1,v8}
\end{fmfgraph*}
}
& = &
\parbox{25mm}{
\begin{fmfgraph*}(24,15)
\fmfleft{i}
\fmfright{o1,o2}
\fmf{dashes}{vz,i}
\fmf{dbl_plain}{o1,v1,v2,v3,v4,vz,v5,v6,v7,v8,o2}
\fmffreeze
\fmfv{decor.shape=cross,decor.size=4mm}{v1}
\fmf{gluon}{v1,v8}
\end{fmfgraph*}
}
\stackrel{\mathrm{def.}}{=}
\delta_{\mathrm{1F}}\times
\parbox{25mm}{
\begin{fmfgraph*}(24,15)
\fmfleft{i}
\fmfright{o1,o2}
\fmf{dashes}{vz,i}
\fmf{dbl_plain}{o1,v1,v2,v3,v4,vz,v5,v6,v7,v8,o2}
\fmffreeze
\fmf{gluon}{v1,v8}
\end{fmfgraph*}
} \nn \\
&=&
 \delta_{1\mathbf{F}}\times \frac{\alpha_S}{2\pi}\,C_F \,
 C(\epsilon)\left(\frac{\mu^2}{m^2}\right)^{\epsilon}\,\nn \\
&& \times\int \mathfrak{D}^Dk
\frac{\mathcal{N}_2}{\left[\left(p_1+k\right)^2 -
    m^2\right]\left[\left(p_2-k\right)^2 -m^2\right]k^2}, 
\eea
%
%
with
%
%
\be\label{ren2loopn2}
\mathcal{N}_2 =
-\gamma^\rho\left(\not\!k_{\phantom{1}}+\not\!p_1+m\right)
\left(v^Q\gamma^\mu +
  a^Q\gamma^\mu\gamma_5\right)\left(\not\!k_{\phantom{1}}-\not\!p_2 +
  m\right)\gamma_\rho. 
\ee
These diagrams yield 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bea
G_1^{(c,d)}\Bigl(s,\epsilon,\frac{\mu^2}{m^2}\Bigr) = && \nr \left(\frac{\alpha_S}{2\,\pi}\right)^2\;C_F \, C_A
\,C^2(\epsilon)\,\left(\frac{\mu^2}{m^2}\right)^{\epsilon}\,\Bigg\{\nn \\
%%%%%%%%%%%%%%%
&& \nr \phantom{+} \frac{1}{\epsilon^2}\bigg[ -1/2\,(x^2+1)/[(x-1)(x+1)]\,H(0;x)-1/4\bigg] \nn \\
%%%%%%%%%%%%%%%
&& \nr +  \frac{1}{\epsilon} \bigg[ -1/4\,(3\,x^2-2\,x+3)/[(x-1)(x+1)]\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+(x^2+1)/[(x-1)(x+1)]\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/2\,(x^2+1)/[(x-1)(x+1)]\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/2\,\zeta(2)\,(x^2+1)/[(x-1)(x+1)]\bigg] \nn \\
%%%%%%%%%%%%%%%
&& \nr-\zeta(2)\,(x^2+1)/[(x-1)(x+1)]\,H(-1;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/2\,(x^2+1)\,(\zeta(2)-4)/[(x-1)(x+1)]\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/2\,(3\,x^2-2\,x+3)/[(x-1)(x+1)]\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(x^2+1)/[(x-1)(x+1)]\,H(-1,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+(x^2+1)/[(x-1)(x+1)]\,H(0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/4\,(3\,x^2-2\,x+3)/[(x-1)(x+1)]\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+(x^2+1)/[(x-1)(x+1)]\,H(-1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/2\,(x^2+1)/[(x-1)(x+1)]\,H(0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/4\,(3\,\zeta(2)\,x^2+4\,\zeta(3)\,x^2-2\,\zeta(2)\,x+4\,\zeta(3)+3\,\zeta(2))\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.4cm}/[(x-1)(x+1)]\Bigg\}\nn \\
%%%%%%%%%%%%%%%
+ && \nr \left(\frac{\alpha_S}{2\,\pi}\right)^2\;C_F^2
\,C^2(\epsilon)\,\left(\frac{\mu^2}{m^2}\right)^{2\,\epsilon}\,\Bigg\{
\nn \\
%%%%%%%%%%%%%%%
&& \nr \phantom{+}\frac{1}{\epsilon^2}\bigg[ -3/2\,(x^2+1)/[(x-1)(x+1)]\,H(0;x)-3/4\bigg] \nn \\
%%%%%%%%%%%%%%%
&& \nr +  \frac{1}{\epsilon} \bigg[ -1/4\,(17\,x^2-6\,x+17)/[(x-1)(x+1)]\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+3\,(x^2+1)/[(x-1)(x+1)]\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-3/2\,(x^2+1)/[(x-1)(x+1)]\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/2\,(-2\,x^2+3\,\zeta(2)\,x^2+2+3\,\zeta(2))/[(x-1)(x+1)]\bigg] \nn \\
%%%%%%%%%%%%%%%
&& \nr -3\,\zeta(2)\,(x^2+1)/[(x-1)(x+1)]\,H(-1;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/2\,(3\,\zeta(2)\,x^2-26\,x^2+4\,x+3\,\zeta(2)-26)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}/[(x-1)(x+1)]\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/2\,(17\,x^2-6\,x+17)/[(x-1)(x+1)]\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-6\,(x^2+1)/[(x-1)(x+1)]\,H(-1,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+3\,(x^2+1)/[(x-1)(x+1)]\,H(0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/4\,(17\,x^2-6\,x+17)/[(x-1)(x+1)]\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+3\,(x^2+1)/[(x-1)(x+1)]\,H(-1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-3/2\,(x^2+1)/[(x-1)(x+1)]\,H(0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/4\,(17\,\zeta(2)\,x^2-8\,x^2+12\,\zeta(3)\,x^2-6\,\zeta(2)\,x+8\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}+17\,\zeta(2)+12\,\zeta(3))/[(x-1)(x+1)]\Bigg\},\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
G_2^{(c,d)}\Bigl(s,\epsilon,\frac{\mu^2}{m^2}\Bigr) = && \nr\left(\frac{\alpha_S}{2\,\pi}\right)^2\;
C_F\,C_A\,C^2(\epsilon)\,\left(\frac{\mu^2}{m^2}\right)^{\epsilon}\,
\Bigg\{ \nn \\
%%%%%%%%%%%%%%%
&& \nr \phantom{+}\frac{1}{\epsilon}\bigg[ -(3\,x^2-2\,x+3)/[(x+1)(x-1)^3]\,x\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2/(x-1)^2\,x\bigg] \nn \\
%%%%%%%%%%%%%%%
&& \nr -2\,(3\,x^2-2\,x+3)/[(x+1)(x-1)^3]\,x\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(3\,x^2-2\,x+3)/[(x+1)(x-1)^3]\,x\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-(3\,x^2-2\,x+3)/[(x+1)(x-1)^3]\,x\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+(3\,\zeta(2)\,x^2+4\,x^2-2\,\zeta(2)\,x+3\,\zeta(2)-4)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.4cm}/[(x+1)(x-1)^3]\,x\Bigg\}\nn \\
%%%%%%%%%%%%%%%
+ && \nr \left(\frac{\alpha_S}{2\,\pi}\right)^2\;C_F^2
\,C^2(\epsilon)\,\left(\frac{\mu^2}{m^2}\right)^{2\,\epsilon}\,\Bigg\{
\nn \\
%%%%%%%%%%%%%%%
&& \nr \phantom{+}\frac{1}{\epsilon} \bigg[ -3\,(3\,x^2-2\,x+3)/[(x+1)(x-1)^3]\,x\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+6/(x-1)^2\,x\bigg] \nn \\
%%%%%%%%%%%%%%%
&& \nr -10\,(3\,x^2-2\,x+3)/[(x+1)(x-1)^3]\,x\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+6\,(3\,x^2-2\,x+3)/[(x+1)(x-1)^3]\,x\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-3\,(3\,x^2-2\,x+3)/[(x+1)(x-1)^3]\,x\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+(20\,x^2+9\,\zeta(2)\,x^2-6\,\zeta(2)\,x-20+9\,\zeta(2))\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.4cm}/[(x+1)(x-1)^3]\,x\Bigg\}.
\eea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

$\bullet$ Fig.~\ref{ct-graph}~(e):
\bea
\parbox{25mm}{
\begin{fmfgraph*}(24,15)
\fmfleft{i}
\fmfright{o1,o2}
\fmf{dashes}{vz,i}
\fmf{dbl_plain}{o1,v1,v2,v3,v4,vz,v5,v6,v7,v8,o2}
\fmfforce{.76w,.5h}{vn}
\fmffreeze
\fmfv{decor.shape=cross,decor.size=4mm,label=$\delta_3$,label.dist=.1w}{vn}
\fmf{gluon}{v1,vn}
\fmf{gluon}{vn,v8}
\end{fmfgraph*}
}
&\stackrel{\mathrm{def.}}{=}&
\delta_3 \times \frac{\alpha_S}{2\pi}\,C_F \,
C(\epsilon)\,\left(\frac{\mu^2}{m^2}\right)^{\epsilon}\nn \\
&&\times\int \mathfrak{D}^Dk \frac{\mathcal{N}_3}{\left[\left(p_1+k\right)^2 -
    m^2\right]\left[\left(p_2-k\right)^2 -m^2\right]\left(k^2\right)^2},
\eea
%
%
with
%
%
\bea
\mathcal{N}_3 &=& \gamma^\rho\left(\not\!k_{\phantom{1}}+\not\!p_1+m\right)
\left(v^Q\gamma^\mu +
  a^Q\gamma^\mu\gamma_5\right)\left(\not\!k_{\phantom{1}}-\not\!p_2 +
  m\right)\gamma^\sigma \nn \\
&&\times\left(k^2 g_{\rho \sigma}-k_\rho k_\sigma \right).
\eea
This leads to the contributions
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bea
G_1^{(e)}\Bigl(s,\epsilon,\frac{\mu^2}{m^2}\Bigr) = && \nr \left(\frac{\alpha_S}{2\,\pi}\right)^2\;C_F\, T_R \, N_f
\,C^2(\epsilon)\,\left(\frac{\mu^2}{m^2}\right)^{\epsilon}\,\Bigg\{\nn
\\
%%%%%%%%%%%%%%%
&& \nr \phantom{+}\frac{1}{\epsilon^2}\bigg[ 2/3\,(x^2+1)/[(x-1)(x+1)]\,H(0;x)+1/3\bigg] \nn \\
%%%%%%%%%%%%%%%
&& \nr + \frac{1}{\epsilon}\bigg[ 1/3\,(3\,x^2-2\,x+3)/[(x-1)(x+1)]\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4/3\,(x^2+1)/[(x-1)(x+1)]\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2/3\,(x^2+1)/[(x-1)(x+1)]\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2/3\,\zeta(2)\,(x^2+1)/[(x-1)(x+1)]\bigg] \nn \\
%%%%%%%%%%%%%%%
&& \nr + 4/3\,\zeta(2)\,(x^2+1)/[(x-1)(x+1)]\,H(-1;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2/3\,(x^2+1)\,(-4+\zeta(2))/[(x-1)(x+1)]\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2/3\,(3\,x^2-2\,x+3)/[(x-1)(x+1)]\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+8/3\,(x^2+1)/[(x-1)(x+1)]\,H(-1,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4/3\,(x^2+1)/[(x-1)(x+1)]\,H(0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/3\,(3\,x^2-2\,x+3)/[(x-1)(x+1)]\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4/3\,(x^2+1)/[(x-1)(x+1)]\,H(-1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2/3\,(x^2+1)/[(x-1)(x+1)]\,H(0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/3\,(3\,\zeta(2)\,x^2+4\,\zeta(3)\,x^2-2\,\zeta(2)\,x+4\,\zeta(3)+3\,\zeta(2))\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}/[(x-1)(x+1)]\Bigg\}\nn \\
%%%%%%%%%%%%%%%
+ && \nr \left(\frac{\alpha_S}{2\,\pi}\right)^2\;C_F \, T_R
\,C^2(\epsilon)\,\left(\frac{\mu^2}{m^2}\right)^{\epsilon}\,\Bigg\{
\nn \\
%%%%%%%%%%%%%%%
&& \nr \phantom{+}\frac{1}{\epsilon^2}\bigg[ 2/3\,(x^2+1)/[(x-1)(x+1)]\,H(0;x)+1/3\bigg]\nn \\
%%%%%%%%%%%%%%%
&& \nr + \frac{1}{\epsilon}\bigg[ 1/3\,(3\,x^2-2\,x+3)/[(x-1)(x+1)]\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4/3\,(x^2+1)/[(x-1)(x+1)]\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2/3\,(x^2+1)/[(x-1)(x+1)]\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2/3\,\zeta(2)\,(x^2+1)/[(x-1)(x+1)]\bigg] \nn \\
%%%%%%%%%%%%%%%
&& \nr + 4/3\,\zeta(2)\,(x^2+1)/[(x-1)(x+1)]\,H(-1;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2/3\,(x^2+1)\,(-4+\zeta(2))/[(x-1)(x+1)]\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2/3\,(3\,x^2-2\,x+3)/[(x-1)(x+1)]\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+8/3\,(x^2+1)/[(x-1)(x+1)]\,H(-1,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4/3\,(x^2+1)/[(x-1)(x+1)]\,H(0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/3\,(3\,x^2-2\,x+3)/[(x-1)(x+1)]\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4/3\,(x^2+1)/[(x-1)(x+1)]\,H(-1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2/3\,(x^2+1)/[(x-1)(x+1)]\,H(0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/3\,(3\,\zeta(2)\,x^2+4\,\zeta(3)\,x^2-2\,\zeta(2)\,x+4\,\zeta(3)+3\,\zeta(2))\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}/[(x-1)(x+1)]\Bigg\} \nn \\
%%%%%%%%%%%%%%%
+ && \nr \left(\frac{\alpha_S}{2\,\pi}\right)^2\;C_F\, C_A
\,C^2(\epsilon)\,\left(\frac{\mu^2}{m^2}\right)^{\epsilon}\,\Bigg\{
\nn \\
%%%%%%%%%%%%%%%
&& \nr \phantom{+}\frac{1}{\epsilon^2}\bigg[ -5/6\,(x^2+1)/[(x-1)(x+1)]\,H(0;x)-5/12\bigg]\nn \\
%%%%%%%%%%%%%%%
&& \nr + \frac{1}{\epsilon} \bigg[ -5/12\,(3\,x^2-2\,x+3)/[(x-1)(x+1)]\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+5/3\,(x^2+1)/[(x-1)(x+1)]\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-5/6\,(x^2+1)/[(x-1)(x+1)]\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+5/6\,\zeta(2)\,(x^2+1)/[(x-1)(x+1)]\bigg]\nn \\
%%%%%%%%%%%%%%%
&& \nr + -5/3\,\zeta(2)\,(x^2+1)/[(x-1)(x+1)]\,H(-1;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+5/6\,(x^2+1)\,(-4+\zeta(2))/[(x-1)(x+1)]\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+5/6\,(3\,x^2-2\,x+3)/[(x-1)(x+1)]\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-10/3\,(x^2+1)/[(x-1)(x+1)]\,H(-1,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+5/3\,(x^2+1)/[(x-1)(x+1)]\,H(0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-5/12\,(3\,x^2-2\,x+3)/[(x-1)(x+1)]\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+5/3\,(x^2+1)/[(x-1)(x+1)]\,H(-1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-5/6\,(x^2+1)/[(x-1)(x+1)]\,H(0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+5/12\,(3\,\zeta(2)\,x^2+4\,\zeta(3)\,x^2-2\,\zeta(2)\,x+4\,\zeta(3)+3\,\zeta(2))\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.2cm}/[(x-1)(x+1)]\Bigg\} ,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
G_2^{(e)}\Bigl(s,\epsilon,\frac{\mu^2}{m^2}\Bigr) = && \nr \left(\frac{\alpha_S}{2\,\pi}\right)^2\;C_F\, T_R \, N_f
\,C^2(\epsilon)\,\left(\frac{\mu^2}{m^2}\right)^{\epsilon}\,\Bigg\{
\nn \\
%%%%%%%%%%%%%%%
&& \nr \phantom{+}\frac{1}{\epsilon}\bigg[ 4/3\,(3\,x^2-2\,x+3)/[(x+1)(x-1)^3]\,x\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-8/3/(x-1)^2\,x\bigg] \nn \\
%%%%%%%%%%%%%%%
&& \nr + 8/3\,(3\,x^2-2\,x+3)/[(x+1)(x-1)^3]\,x\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-8/3\,(3\,x^2-2\,x+3)/[(x+1)(x-1)^3]\,x\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4/3\,(3\,x^2-2\,x+3)/[(x+1)(x-1)^3]\,x\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4/3\,(4\,x^2+3\,\zeta(2)\,x^2-2\,\zeta(2)\,x-4+3\,\zeta(2))\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}/[(x+1)(x-1)^3]\,x\Bigg\}\nn \\
%%%%%%%%%%%%%%%
+ && \nr \left(\frac{\alpha_S}{2\,\pi}\right)^2\;C_F \, T_R
\,C^2(\epsilon)\,\left(\frac{\mu^2}{m^2}\right)^{\epsilon}\,\Bigg\{
\nn \\
%%%%%%%%%%%%%%%
&& \nr \phantom{+}\frac{1}{\epsilon}\bigg[ 4/3\,(3\,x^2-2\,x+3)/[(x+1)(x-1)^3]\,x\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-8/3/(x-1)^2\,x\bigg] \nn \\
%%%%%%%%%%%%%%%
&& \nr + 8/3\,(3\,x^2-2\,x+3)/[(x+1)(x-1)^3]\,x\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-8/3\,(3\,x^2-2\,x+3)/[(x+1)(x-1)^3]\,x\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4/3\,(3\,x^2-2\,x+3)/[(x+1)(x-1)^3]\,x\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4/3\,(4\,x^2+3\,\zeta(2)\,x^2-2\,\zeta(2)\,x-4+3\,\zeta(2))\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}/[(x+1)(x-1)^3]\,x\Bigg\} \nn \\
%%%%%%%%%%%%%%%
+ && \nr \left(\frac{\alpha_S}{2\,\pi}\right)^2\;C_F\, C_A
\,C^2(\epsilon)\,\left(\frac{\mu^2}{m^2}\right)^{\epsilon}\,\Bigg\{
\nn \\
%%%%%%%%%%%%%%%
&& \nr \phantom{+}\frac{1}{\epsilon} \bigg[ -5/3\,(3\,x^2-2\,x+3)/[(x+1)(x-1)^3]\,x\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+10/3/(x-1)^2\,x\bigg]\nn \\
%%%%%%%%%%%%%%%
&& \nr -10/3\,(3\,x^2-2\,x+3)/[(x+1)(x-1)^3]\,x\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+10/3\,(3\,x^2-2\,x+3)/[(x+1)(x-1)^3]\,x\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-5/3\,(3\,x^2-2\,x+3)/[(x+1)(x-1)^3]\,x\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+5/3\,(4\,x^2+3\,\zeta(2)\,x^2-2\,\zeta(2)\,x-4+3\,\zeta(2))\nn\\
%%%%%%%%%%%%%%%
&&\nr\hspace{1cm}/[(x+1)(x-1)^3]\,x\Bigg\}.
\eea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

$\bullet$ Fig.~\ref{ct-graph}~(f):
\bea
\parbox{25mm}{
\begin{fmfgraph*}(24,15)
\fmfleft{i}
\fmfright{o1,o2}
\fmf{dashes}{vz,i}
\fmf{dbl_plain}{o1,v1,v2,v3,v4,vz,v5,v6,v7,v8,o2}
\fmffreeze
\fmfv{decor.shape=cross,decor.size=4mm,label=$\delta_2^{(1l)}$,label.angle=90.,label.dist=0.1w}{vz}
\fmf{gluon}{v1,v8}
\end{fmfgraph*}
}
&=&
\delta_2^{(1l)}\times
\parbox{25mm}{
\begin{fmfgraph*}(24,15)
\fmfleft{i}
\fmfright{o1,o2}
\fmf{dashes}{vz,i}
\fmf{dbl_plain}{o1,v1,v2,v3,v4,vz,v5,v6,v7,v8,o2}
\fmffreeze
\fmf{gluon}{v1,v8}
\end{fmfgraph*}
} \nn \\
&\stackrel{\mathrm{def.}}{=}&\delta_2^{(1l)}\times
\frac{\alpha_S}{2\pi}\,C_F\, C(\epsilon)\,
\left(\frac{\mu^2}{m^2}\right)^{\epsilon}\nn \\
&&\times\int \mathfrak{D}^Dk
\frac{\mathcal{N}_2}{\left[\left(p_1+k\right)^2 -
    m^2\right]\left[\left(p_2-k\right)^2 -m^2\right]k^2}, 
\eea
%
%
with $\mathcal{N}_2$ as defined in Eq.(\ref{ren2loopn2}).
They contribute
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bea
G_1^{(f)}\Bigl(s,\epsilon,\frac{\mu^2}{m^2}\Bigr) = && \nr\left(\frac{\alpha_S}{2\,\pi}\right)^2\;
C_F^2\,C^2(\epsilon)\,\left(\frac{\mu^2}{m^2}\right)^{2\,\epsilon}\,
\Bigg\{ \nn \\
%%%%%%%%%%%%%%%
&& \nr \phantom{+}\frac{1}{\epsilon^2}\bigg[ -3/2\,(x^2+1)/[(x-1)(x+1)]\,H(0;x)-3/4\bigg]\nn \\
%%%%%%%%%%%%%%%
&& \nr + \frac{1}{\epsilon}\bigg[ -1/4\,(17\,x^2-6\,x+17)/[(x-1)(x+1)]\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+3\,(x^2+1)/[(x-1)(x+1)]\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-3/2\,(x^2+1)/[(x-1)(x+1)]\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/2\,(-2\,x^2+3\,\zeta(2)\,x^2+2+3\,\zeta(2))/[(x-1)(x+1)]\bigg] \nn \\
%%%%%%%%%%%%%%%
&& \nr -3\,\zeta(2)\,(x^2+1)/[(x-1)(x+1)]\,H(-1;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/2\,(-26\,x^2+3\,\zeta(2)\,x^2+4\,x+3\,\zeta(2)-26)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}/[(x-1)(x+1)]\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/2\,(17\,x^2-6\,x+17)/[(x-1)(x+1)]\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-6\,(x^2+1)/[(x-1)(x+1)]\,H(-1,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+3\,(x^2+1)/[(x-1)(x+1)]\,H(0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/4\,(17\,x^2-6\,x+17)/[(x-1)(x+1)]\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+3\,(x^2+1)/[(x-1)(x+1)]\,H(-1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-3/2\,(x^2+1)/[(x-1)(x+1)]\,H(0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/4\,(17\,\zeta(2)\,x^2-8\,x^2+12\,\zeta(3)\,x^2-6\,\zeta(2)\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}+12\,\zeta(3)+8+17\,\zeta(2))/[(x-1)(x+1)]\Bigg\},\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
G_2^{(f)}\Bigl(s,\epsilon,\frac{\mu^2}{m^2}\Bigr) = && \nr\left(\frac{\alpha_S}{2\,\pi}\right)^2\;
C_F^2\,C^2(\epsilon)\,\left(\frac{\mu^2}{m^2}\right)^{2\,\epsilon}\,
\Bigg\{\nn \\
%%%%%%%%%%%%%%%
&& \nr \phantom{+} \frac{1}{\epsilon}\bigg[ -3\,(3\,x^2-2\,x+3)/[(x+1)(x-1)^3]\,x\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+6/(x-1)^2\,x\bigg]\nn \\
%%%%%%%%%%%%%%%
&& \nr -10\,(3\,x^2-2\,x+3)/[(x+1)(x-1)^3]\,x\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+6\,(3\,x^2-2\,x+3)/[(x+1)(x-1)^3]\,x\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-3\,(3\,x^2-2\,x+3)/[(x+1)(x-1)^3]\,x\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+(9\,\zeta(2)\,x^2+20\,x^2-6\,\zeta(2)\,x-20+9\,\zeta(2))\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.4cm}/[(x+1)(x-1)^3]\,x\Bigg\}.
\eea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

$\bullet$ Fig.~\ref{ct-graph}~(g) is defined as the two-loop renormalization
constant $\delta_2^{(2l)}$ times the Born level amplitude and reads:
\bea
\parbox{25mm}{
\begin{fmfgraph*}(24,15)
\fmfleft{i}
\fmfright{o1,o2}
\fmf{dashes}{vz,i}
\fmf{dbl_plain}{o1,v1,v2,v3,v4,vz,v5,v6,v7,v8,o2}
\fmffreeze
\fmfv{decor.shape=cross,label=$\delta_2^{(2l)}$,label.angle=90.,label.dist=0.1w}{vz}
\end{fmfgraph*}
}
&\stackrel{\mathrm{def.}}{=}&
\delta_2^{(2l)}\times
\parbox{25mm}{
\begin{fmfgraph*}(24,15)
\fmfleft{i}
\fmfright{o1,o2}
\fmf{dashes}{vz,i}
\fmf{dbl_plain}{o1,v1,v2,v3,v4,vz,v5,v6,v7,v8,o2}
\fmffreeze
\end{fmfgraph*}
} \nn \\
&=& \delta_2^{(2l)} \times (-i)\left(v^Q \gamma^{\mu} + a^Q \gamma^\mu
  \gamma_5\right).
\eea
Its contribution to $G_1$ is:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bea
G_1^{(g)}\Bigl(s,\epsilon,\frac{\mu^2}{m^2}\Bigr) =  \,\delta_2^{(2l)}.
\eea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{fmfgroup}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\subsection{UV-Renormalized Two-Loop Form Factors\label{sec_2lren}}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Adding the terms given in Sections \ref{subsec_2l_cont} and \ref{subsec_sub_cont} we
obtain the UV-renormalized
axial vector form factors to second order in QCD. They still contain
terms proportional to $\epsilon^{-2}$ and $\epsilon^{-1}$ due to
infrared and collinear singularities in the loops. In this Subsection  we
put the renormalization scale $\mu$ equal to the on-shell mass
$m$ (The logarithms that are present if $\mu \not=m$ are given in
Section \ref{munotm}.). Then we get:
\bea
G_{1,R}^{(2l)}(s,\epsilon) & = &  C^2(\epsilon) \, \Bigg\{\nn \\
&&
\frac{1}{\epsilon^2}\; \left(C_F\; T_R\; N_f
\;  \tilde{c}_1\; + \;C_F \;T_R\; \tilde{c}_2 \;+ \;C_F\; C_A\; \tilde{c}_3\; + \;C_F^2\;
\tilde{c}_4\right)\nn \\
& + & \frac{1}{\epsilon}\; \left(C_F\; T_R\; N_f
\;  \tilde{c}_5\; + \;C_F \;T_R\; \tilde{c}_6 \;+ \;C_F\; C_A\; \tilde{c}_7\; + \;C_F^2\;
\tilde{c}_8\right)\nn \\
& + &  \left(C_F\; T_R\; N_f
\;  \tilde{c}_9\; + \;C_F \;T_R\; \tilde{c}_{10} \;+ \;C_F\; C_A\; \tilde{c}_{11}\; + \;C_F^2\;
\tilde{c}_{12}\right) \nn \\ 
& + & \mathcal{O}(\epsilon) \Bigg\} \label{g1_2l_ren_decomp} ,\\
%%%%
G_{2,R}^{(2l)}(s,\epsilon) & = & C^2(\epsilon)\,\Bigg\{\nn \\
&& \frac{1}{\epsilon^2}\;\left(C_F\; T_R\; N_f
\;  \tilde{d}_1\; + \;C_F \;T_R\; \tilde{d}_2 \;+ \;C_F\; C_A\; \tilde{d}_3\; + \;C_F^2\;
\tilde{d}_4\right)\nn \\
& + & \frac{1}{\epsilon}\;\left(C_F\; T_R\; N_f
\;  \tilde{d}_5\; + \;C_F \;T_R\; \tilde{d}_6 \;+ \;C_F\; C_A\; \tilde{d}_7\; + \;C_F^2\;
\tilde{d}_8\right)\nn \\
& + & \left(C_F\; T_R\; N_f
\;  \tilde{d}_9\; + \;C_F \;T_R\; \tilde{d}_{10} \;+ \;C_F\; C_A\; \tilde{d}_{11}\; + \;C_F^2\;
\tilde{d}_{12}\right) \nn \\
& + &  \mathcal{O}(\epsilon)\Bigg\} \label{g2_2l_ren_decomp} ,  
\eea
where:
\bea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\tilde{c}_1 = && \nr -\frac{4}{11}\,\tilde{c}_3 =
-\frac{3}{5}\,\tilde{c}_5 = 1/3\,(x^2+1)/[(x-1)(x+1)]\,H(0;x)-1/3,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\tilde{c}_2 = && \nr \tilde{c}_6 = 0,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\tilde{c}_4 = && \nr -(x^2+1)/[(x-1)(x+1)]\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+(x^2+1)^2/[(x+1)^2(x-1)^2]\,H(0,0;x)+1/2,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\tilde{c}_7 = && \nr
-1/36\,(x^2+1)\,(-67\,x^2+54\,\zeta(2)\,x^2+67-18\,\zeta(2))\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.2cm}/[(x+1)^2(x-1)^2]\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-(x^2+1)/[(x-1)(x+1)]\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+(x^2+1)^2/[(x+1)^2(x-1)^2]\,H(0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2/[(x-1)(x+1)]\,x^2\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(x^2+1)/[(x+1)^2(x-1)^2]\,x^2\,H(0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+(x^2+1)/[(x-1)(x+1)]\,H(1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-(x^2+1)^2/[(x+1)^2(x-1)^2]\,H(0,1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/36\,(54\,\zeta(2)\,x^4-49\,x^4-18\,\zeta(3)\,x^4-36\,\zeta(3)\,x^2+98\,x^2\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}-72\,\zeta(2)\,x^2-49+18\,\zeta(2)-18\,\zeta(3))/[(x+1)^2(x-1)^2],\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\tilde{c}_8 = && \nr
-1/2\,(2\,\zeta(2)\,x^4+7\,x^4-2\,x^3+4\,\zeta(2)\,x^2+2\,x+2\,\zeta(2)-7)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}/[(x+1)^2(x-1)^2]\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(x^2+1)/[(x-1)(x+1)]\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(x^2+1)^2/[(x+1)^2(x-1)^2]\,H(0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(x^2+1)\,(x^2-x+2)/[(x+1)^2(x-1)^2]\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4\,(x^2+1)^2/[(x+1)^2(x-1)^2]\,H(-1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+3\,(x^2+1)^2/[(x+1)^2(x-1)^2]\,H(0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+(2\,x^2+\zeta(2)\,x^2-2+\zeta(2))/[(x-1)(x+1)],\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\tilde{c}_9 = && \nr -4/3\,\zeta(2)\,(x^2+1)/[(x-1)(x+1)]\,H(-1;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/54\,(36\,\zeta(2)\,x^2+209\,x^2-78\,x+36\,\zeta(2)+209)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.2cm}/[(x-1)(x+1)]\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2/9\,(19\,x^2-6\,x+19)/[(x-1)(x+1)]\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-8/3\,(x^2+1)/[(x-1)(x+1)]\,H(-1,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4/3\,(x^2+1)/[(x-1)(x+1)]\,H(0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/9\,(19\,x^2-6\,x+19)/[(x-1)(x+1)]\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4/3\,(x^2+1)/[(x-1)(x+1)]\,H(-1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2/3\,(x^2+1)/[(x-1)(x+1)]\,H(0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/27\,(93\,\zeta(2)\,x^2+106\,x^2+36\,\zeta(3)\,x^2-18\,\zeta(2)\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}+36\,\zeta(3)-106+21\,\zeta(2))/[(x-1)(x+1)],\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\tilde{c}_{10} = && \nr
-1/54\,(36\,\zeta(2)\,x^4+265\,x^4+72\,\zeta(2)\,x^3-208\,x^3-144\,\zeta(2)\,x^2\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.2cm}+14\,x^2+72\,\zeta(2)\,x-208\,x+36\,\zeta(2)+265)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.2cm}/[(x+1)^3(x-1)]\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/9\,(19\,x^4-14\,x^3+14\,x^2-14\,x+19)/(x+1)^4\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2/3\,(x^4+2\,x^3-4\,x^2+2\,x+1)/[(x+1)^3(x-1)]\,H(0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/27\,(-383\,x^4+27\,\zeta(2)\,x^4+1026\,\zeta(2)\,x^3-1412\,x^3-2058\,x^2\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}+1278\,\zeta(2)\,x^2+1026\,\zeta(2)\,x-1412\,x-383+27\,\zeta(2))/(x+1)^4,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\tilde{c}_{11} = && \nr -2/3\,(34\,x^4+3\,x^3-33\,x^2+3\,x-1)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}/[(x+1)^2(x-1)^2]\,H(0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4\,(x^2+1)^2/[(x+1)^2(x-1)^2]\,H(-1,0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(11\,x^6+5\,x^5-15\,x^4+26\,x^3-27\,x^2+5\,x-1)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}/[(x+1)^3(x-1)^3]\,H(0,0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4\,(x^2+1)^2/[(x+1)^2(x-1)^2]\,H(1,0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-6\,(x^2+1)/[(x-1)(x+1)]\,H(1,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+6\,(x^2+1)^2/[(x+1)^2(x-1)^2]\,H(0,1,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/36\,(36\,\zeta(2)\,x^6-217\,x^6-144\,\zeta(2)\,x^5-282\,x^5+612\,\zeta(2)\,x^4\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.2cm}+451\,x^4-504\,\zeta(2)\,x^3+348\,x^3-467\,x^2+540\,\zeta(2)\,x^2-66\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.2cm}-144\,\zeta(2)\,x-36\,\zeta(2)+233)/[(x+1)^3(x-1)^3]\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/3\,(53\,x^4-141\,x^3+228\,x^2-141\,x-17)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}/[(x+1)^2(x-1)^2]\,H(-1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(7\,x^6-7\,x^5+33\,x^4-56\,x^3+25\,x^2-7\,x-1)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}/[(x+1)^3(x-1)^3]\,H(0,-1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/6\,(89\,x^6-402\,x^5+499\,x^4-354\,x^3+37\,x^2-24\,x+11)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}/[(x+1)^3(x-1)^3]\,H(0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(x^2+3)\,(x^2+1)/[(x+1)^2(x-1)^2]\,H(-1,0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(x^5-5\,x^4+18\,x^3-26\,x^2+x-5)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}/[(x+1)^3(x-1)^2]\,H(1,0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(2\,x^4+5\,x^3-14\,x^2+5\,x+6)/[(x+1)^2(x-1)^2]\,H(1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-(12\,x^5-x^4+10\,x^3-28\,x^2-2\,x-1)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.4cm}/[(x+1)^3(x-1)^3]\,x\,H(0,0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(x^6-5\,x^5+21\,x^4-34\,x^3+17\,x^2-5\,x-3)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}/[(x+1)^3(x-1)^3]\,H(0,1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+(-3\,x^5+4\,\zeta(2)\,x^5-x^4-8\,\zeta(2)\,x^4+40\,\zeta(2)\,x^3+2\,x^3-48\,\zeta(2)\,x^2\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.5cm}-2\,x^2+x+4\,\zeta(2)\,x+3-8\,\zeta(2))/[(x+1)^3(x-1)^2]\,H(1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-6\,(x^2+1)/[(x-1)(x+1)]\,H(-1,1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+6\,(x^2+1)^2/[(x+1)^2(x-1)^2]\,H(0,-1,1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4\,(x^2+1)^2/[(x+1)^2(x-1)^2]\,H(-1,0,1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(7\,x^5+7\,x^4+12\,x^3+4\,x^2+x+1)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}/[(x+1)^3(x-1)^2]\,H(0,0,1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4\,(x^2+1)^2/[(x+1)^2(x-1)^2]\,H(1,0,1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(x^2+1)/[(x-1)(x+1)]\,H(1,1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(x^2+1)^2/[(x+1)^2(x-1)^2]\,H(0,1,1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+8\,(x^2+1)/[(x+1)^2(x-1)]\,x\,H(0,1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-(x^2+1)\,(\zeta(2)\,x^2+2\,\zeta(3)\,x^2-\zeta(2)+2\,\zeta(3))\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.3cm}/[(x+1)^2(x-1)^2]\,H(1;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/540\,(-1917\,\zeta(2)^2\,x^6+3240\,\zeta(2)\,\ln(2)\,x^6-7975\,x^6\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.6cm}-5205\,\zeta(2)\,x^6+450\,\zeta(3)\,x^6-3564\,\zeta(2)^2\,x^5\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.6cm}-22680\,\zeta(3)\,x^5+33930\,\zeta(2)\,x^5-19440\,\zeta(2)\,\ln(2)\,x^5\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.6cm}+23925\,x^4+29160\,\zeta(2)\,\ln(2)\,x^4 \nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.6cm}+36270\,\zeta(3)\,x^4+12447\,\zeta(2)^2\,x^4-42405\,\zeta(2)\,x^4\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.6cm}-13716\,\zeta(2)^2\,x^3-8460\,\zeta(2)\,x^3+14445\,\zeta(2)^2\,x^2\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.6cm}-29160\,\zeta(2)\,\ln(2)\,x^2+47505\,\zeta(2)\,x^2-30690\,\zeta(3)\,x^2\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.6cm}-23925\,x^2-25470\,\zeta(2)\,x+22680\,\zeta(3)\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.6cm}+19440\,\zeta(2)\,\ln(2)\,x-3564\,\zeta(2)^2\,x-6030\,\zeta(3)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.6cm}-3240\,\zeta(2)\,\ln(2)+7975+81\,\zeta(2)^2+105\,\zeta(2))\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{2.6cm} /[(x+1)^3(x-1)^3]\nn\\
%%%%%%%%%%%%%%%
&& \nr+\zeta(2)\,(x^2+1)^2/[(x+1)^2(x-1)^2]\,H(0,1;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+52/3\,(x^2+1)/[(x-1)(x+1)]\,H(-1,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/3\,(10\,\zeta(2)\,x^4-6\,\zeta(3)\,x^4-81\,\zeta(2)\,x^3-12\,\zeta(3)\,x^2+108\,\zeta(2)\,x^2\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.2cm}-81\,\zeta(2)\,x+26\,\zeta(2)-6\,\zeta(3))/[(x+1)^2(x-1)^2]\,H(-1;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/216\,(1044\,\zeta(2)\,x^6-1620\,\zeta(3)\,x^6+2545\,x^6+3024\,\zeta(3)\,x^5\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.6cm}-570\,x^5-12528\,\zeta(2)\,x^5-13716\,\zeta(3)\,x^4-2545\,x^4\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.6cm}+18396\,\zeta(2)\,x^4-12744\,\zeta(2)\,x^3+1140\,x^3+25920\,\zeta(3)\,x^3\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.6cm}+3060\,\zeta(2)\,x^2-2545\,x^2-13500\,\zeta(3)\,x^2-570\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.6cm}-2808\,\zeta(2)\,x+3024\,\zeta(3)\,x-1404\,\zeta(3)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.6cm}+396\,\zeta(2)+2545)/[(x+1)^3(x-1)^3]\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+\zeta(2)\,(7\,x^6+6\,x^5-29\,x^4+24\,x^3-31\,x^2+6\,x+5)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.1cm}/[(x+1)^3(x-1)^3]\,H(0,-1;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/9\,(31\,x^4-105\,x^3-36\,\zeta(2)\,x^2+105\,x-36\,\zeta(2)-31)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}/[(x+1)^2(x-1)^2]\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-10\,(x^2+1)^2/[(x+1)^2(x-1)^2]\,H(0,-1,-1,0;x),\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\tilde{c}_{12} = && \nr 8\,(x^2+1)^2/[(x+1)^2(x-1)^2]\,H(-1,0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-8\,(x^2+1)^2/[(x+1)^2(x-1)^2]\,H(1,0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+8\,(x^2+1)^2/[(x+1)^2(x-1)^2]\,H(1,0,1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,\zeta(2)\,(5\,x^4-27\,x^3+36\,x^2-27\,x+7)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(x+1)^2(x-1)^2]\,H(-1;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,\zeta(2)\,(5\,x^6+6\,x^5-31\,x^4+24\,x^3-29\,x^2+6\,x+7)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.4cm}/[(x+1)^3(x-1)^3]\,H(0,-1;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/8\,(85\,x^6-64\,\zeta(3)\,x^6+136\,\zeta(2)\,x^6-160\,\zeta(3)\,x^5-6\,x^5\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.2cm}-296\,\zeta(2)\,x^5+1056\,\zeta(3)\,x^4-85\,x^4-568\,\zeta(2)\,x^4\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.2cm}+944\,\zeta(2)\,x^3+12\,x^3-1536\,\zeta(3)\,x^3-85\,x^2\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.2cm}-616\,\zeta(2)\,x^2+1056\,\zeta(3)\,x^2+24\,\zeta(2)\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.2cm}-160\,\zeta(3)\,x-6\,x-64\,\zeta(3)+85-8\,\zeta(2))\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.2cm}/[(x+1)^3(x-1)^3]\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/2\,(-55\,x^4+8\,\zeta(2)\,x^4+82\,x^3+16\,\zeta(2)\,x^2-82\,x+8\,\zeta(2)+55)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}/[(x+1)^2(x-1)^2]\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(8\,x^4-27\,x^3+60\,x^2-27\,x+6)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}/[(x+1)^2(x-1)^2]\,H(0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(11\,x^6-2\,x^5+33\,x^4-56\,x^3+23\,x^2-2\,x+1)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}/[(x+1)^3(x-1)^3]\,H(0,0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4\,(x^2+1)/[(x-1)(x+1)]\,H(-1,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/4\,(229\,x^6+52\,\zeta(2)\,x^6-266\,x^5-16\,\zeta(2)\,x^5+124\,\zeta(2)\,x^4\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.1cm}+15\,x^4-320\,\zeta(2)\,x^3+196\,x^3+84\,\zeta(2)\,x^2-189\,x^2+70\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.1cm}-16\,\zeta(2)\,x-55+12\,\zeta(2))/[(x+1)^3(x-1)^3]\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(3\,x^4-15\,x^3+76\,x^2-15\,x+5)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}/[(x+1)^2(x-1)^2]\,H(-1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+16\,(x^2+1)^2/[(x+1)^2(x-1)^2]\,H(-1,-1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(7\,x^6+10\,x^5-65\,x^4+104\,x^3-75\,x^2+10\,x-3)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}/[(x+1)^3(x-1)^3]\,H(0,-1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(5\,x^6-10\,x^5-60\,x^4+59\,x^3-16\,x^2-7\,x+5)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}/[(x+1)^3(x-1)^3]\,H(0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-12\,(x^2+1)^2/[(x+1)^2(x-1)^2]\,H(-1,0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+(27\,x^6-2\,x^5+27\,x^4-56\,x^3-7\,x^2-2\,x-7)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.3cm}/[(x+1)^3(x-1)^3]\,H(0,0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+8\,(x^5-2\,x^4+18\,x^3-14\,x^2+4\,x+1)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}/[(x+1)^3(x-1)^2]\,H(1,0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(8\,x^4-19\,x^3+46\,x^2-19\,x+8)/[(x+1)^2(x-1)^2]\,H(1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4\,(x^6-x^5+14\,x^4-20\,x^3+14\,x^2-x+1)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}/[(x+1)^3(x-1)^3]\,H(0,1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4\,(\zeta(2)\,x^5+4\,x^5+2\,x^4-5\,\zeta(2)\,x^4+34\,\zeta(2)\,x^3-2\,x^3+2\,x^2\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.8cm}-30\,\zeta(2)\,x^2-2\,x+7\,\zeta(2)\,x-4+\zeta(2))\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.8cm}/[(x+1)^3(x-1)^2]\,H(1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(5\,x^4-10\,x^3+18\,x^2-10\,x+5)/[(x+1)^2(x-1)^2]\,H(0,1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4\,(3\,x^5+3\,x^4+6\,x^3-2\,x^2-x-1)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.5cm}/[(x+1)^3(x-1)^2]\,H(0,0,1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4\,\zeta(3)\,(x^2+1)^2/[(x+1)^2(x-1)^2]\,H(1;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/20\,(-362\,\zeta(2)^2\,x^6+240\,\zeta(2)\,\ln(2)\,x^6-230\,x^6-275\,\zeta(2)\,x^6\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.4cm}+140\,\zeta(3)\,x^6+144\,\zeta(2)^2\,x^5+600\,\zeta(3)\,x^5+230\,\zeta(2)\,x^5\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.4cm}-1440\,\zeta(2)\,\ln(2)\,x^5+690\,x^4+2160\,\zeta(2)\,\ln(2)\,x^4\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.4cm}-2980\,\zeta(3)\,x^4-1198\,\zeta(2)^2\,x^4+255\,\zeta(2)\,x^4\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.4cm}+2656\,\zeta(2)^2\,x^3+500\,\zeta(2)\,x^3-954\,\zeta(2)^2\,x^2\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.4cm}-2160\,\zeta(2)\,\ln(2)\,x^2-165\,\zeta(2)\,x^2+3060\,\zeta(3)\,x^2\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.4cm}-690\,x^2-730\,\zeta(2)\,x-600\,\zeta(3)\,x+1440\,\zeta(2)\,\ln(2)\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.4cm}+144\,\zeta(2)^2\,x-220\,\zeta(3)-240\,\zeta(2)\,\ln(2)+230\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.4cm}-118\,\zeta(2)^2+185\,\zeta(2))/[(x+1)^3(x-1)^3]\nn\\
%%%%%%%%%%%%%%%
&& \nr+4\,(x^2+1)^2/[(x+1)^2(x-1)^2]\,H(0,-1,-1,0;x), 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\eea
\bea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\tilde{d}_j = && \nr 0 \quad \mbox{for} \quad j=1\dots7,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\tilde{d}_8 = && \nr -2\,(5\,x^2-2\,x+5)/[(x-1)^3(x+1)]\,x\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4\,(3\,x^2-2\,x+3)\,(x^2+1)/[(x+1)^2(x-1)^4]\,x\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4/(x-1)^2\,x,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\tilde{d}_9 = && \nr -2/9\,(51\,x^2-26\,x+51)/[(x+1)(x-1)^3]\,x\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+8/3\,(3\,x^2-2\,x+3)/[(x+1)(x-1)^3]\,x\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4/3\,(3\,x^2-2\,x+3)/[(x+1)(x-1)^3]\,x\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4/9\,(19\,x^2+9\,\zeta(2)\,x^2-6\,\zeta(2)\,x-19+9\,\zeta(2)) \nn \\
%%%%%%%%%%%%%%%
&& \nr \hspace{1cm}/[(x+1)(x-1)^3]\,x,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\tilde{d}_{10} = && \nr
-2/9\,(51\,x^4-176\,x^3-70\,x^2-72\,\zeta(2)\,x^2-176\,x+51)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}/[(x+1)^3(x-1)^3]\,x\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4/3\,(3\,x^4-14\,x^3-2\,x^2-14\,x+3)/[(x+1)^4(x-1)^2]\,x\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+16/[(x+1)^3(x-1)^3]\,x^3\,H(0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4/9\,(-8\,x^4+45\,\zeta(2)\,x^4-90\,\zeta(2)\,x^3-80\,x^3-144\,x^2+18\,\zeta(2)\,x^2\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.1cm}\:-90\,\zeta(2)\,x-80\,x-8+45\,\zeta(2))/[(x+1)^4(x-1)^2]\,x,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\tilde{d}_{11} = && \nr -18\,\zeta(2)\,(x^2-4\,x+1)\,(x^2+1)/[(x+1)^2(x-1)^4]\,x\,H(-1;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+24\,\zeta(2)\,(x^2+1)\,(x^2-5\,x+1)/[(x+1)^3(x-1)^5]\,x^2\,H(0,-1;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/18\,(324\,\zeta(2)\,x^6+687\,x^6+1008\,\zeta(3)\,x^5-3384\,\zeta(2)\,x^5+26\,x^5\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.4cm}-4752\,\zeta(3)\,x^4+3960\,\zeta(2)\,x^4-687\,x^4-7056\,\zeta(2)\,x^3\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.4cm}+9216\,\zeta(3)\,x^3-52\,x^3-687\,x^2+1620\,\zeta(2)\,x^2\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.4cm}-4752\,\zeta(3)\,x^2+1008\,\zeta(3)\,x+26\,x-648\,\zeta(2)\,x+687)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.4cm}/[(x+1)^3(x-1)^5]\,x\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2/3\,(3\,x^2-88\,x+3)/[(x+1)(x-1)^3]\,x\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-48\,(x^2-x+1)/[(x+1)^2(x-1)^4]\,x^2\,H(0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+8\,(5\,x^4-14\,x^3+30\,x^2-14\,x+5)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}/[(x+1)^3(x-1)^5]\,x^2\,H(0,0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/3\,(39\,x^6+48\,\zeta(2)\,x^5+184\,x^5-252\,\zeta(2)\,x^4-87\,x^4+48\,\zeta(2)\,x^3\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.1cm}-152\,x^3+81\,x^2-252\,\zeta(2)\,x^2-32\,x+48\,\zeta(2)\,x-33)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.1cm}/[(x+1)^3(x-1)^5]\,x\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(9\,x^4-92\,x^3+130\,x^2-92\,x+9)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}/[(x+1)^2(x-1)^4]\,x\,H(-1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-8\,(7\,x^4-29\,x^3+62\,x^2-29\,x+7)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}/[(x+1)^3(x-1)^5]\,x^2\,H(0,-1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(9\,x^4-112\,x^3+146\,x^2-196\,x+9)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}/[(x+1)^3(x-1)^5]\,x^3\,H(0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4\,(x^4-5\,x^3+38\,x^2-5\,x+1)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(x+1)^3(x-1)^5]\,x^2\,H(0,0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-16\,(3\,x^2-8\,x+3)/[(x+1)^3(x-1)^3]\,x^2\,H(1,0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-24\,(3\,x^2-2\,x+3)/[(x+1)^2(x-1)^4]\,x^2\,H(1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+8\,(5\,x^4-14\,x^3+42\,x^2-14\,x+5)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}/[(x+1)^3(x-1)^5]\,x^2\,H(0,1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4\,(x^4+12\,\zeta(2)\,x^3-32\,\zeta(2)\,x^2-2\,x^2+12\,\zeta(2)\,x+1)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}/[(x+1)^3(x-1)^3]\,x\,H(1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-16/[(x+1)^2(x-1)^2]\,x^2\,H(0,1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-32/[(x+1)^3(x-1)^3]\,x^3\,H(0,0,1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/45\,(1080\,\zeta(2)\,\ln(2)\,x^6+540\,\zeta(3)\,x^6-1395\,\zeta(2)\,x^6-1210\,x^6\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.4cm}-7380\,\zeta(3)\,x^5-4320\,\zeta(2)\,\ln(2)\,x^5-1188\,\zeta(2)^2\,x^5\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.4cm}+10500\,\zeta(2)\,x^5+3630\,x^4+10980\,\zeta(3)\,x^4+4878\,\zeta(2)^2\,x^4\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.4cm}-7425\,\zeta(2)\,x^4+5400\,\zeta(2)\,\ln(2)\,x^4-3360\,\zeta(2)\,x^3\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.4cm}-3168\,\zeta(2)^2\,x^3+4878\,\zeta(2)^2\,x^2-10980\,\zeta(3)\,x^2\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.4cm}+7875\,\zeta(2)\,x^2-3630\,x^2-5400\,\zeta(2)\,\ln(2)\,x^2\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.4cm}+4320\,\zeta(2)\,\ln(2)\,x-1188\,\zeta(2)^2\,x-7140\,\zeta(2)\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.4cm}+7380\,\zeta(3)\,x-540\,\zeta(3)-1080\,\zeta(2)\,\ln(2)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.4cm}+1210+945\,\zeta(2))/[(x+1)^3(x-1)^5]\,x,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\tilde{d}_{12} = && \nr 36\,\zeta(2)\,(x^2-4\,x+1)\,(x^2+1)/[(x+1)^2(x-1)^4]\,x\,H(-1;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-48\,\zeta(2)\,(x^2+1)\,(x^2-5\,x+1)/[(x+1)^3(x-1)^5]\,x^2\,H(0,-1;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/2\,(61\,x^6+96\,\zeta(2)\,x^6-22\,x^5-216\,\zeta(2)\,x^5-32\,\zeta(3)\,x^5\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.2cm}-1032\,\zeta(2)\,x^4-61\,x^4+1056\,\zeta(3)\,x^4+448\,\zeta(2)\,x^3+44\,x^3\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.2cm}-1664\,\zeta(3)\,x^3+1056\,\zeta(3)\,x^2-480\,\zeta(2)\,x^2-61\,x^2\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.2cm}-32\,\zeta(3)\,x-22\,x+56\,\zeta(2)\,x+61-24\,\zeta(2))\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.2cm}/[(x+1)^3(x-1)^5]\,x\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(25\,x^2-86\,x+25)/[(x-1)^3(x+1)]\,x\,H(-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+8\,(3\,x^4-13\,x^3+64\,x^2-13\,x+3)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}/[(x+1)^2(x-1)^4]\,x\,H(0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-16\,(x^4+19\,x^3-28\,x^2+19\,x+1)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.8cm}/[(x+1)^3(x-1)^5]\,x^2\,H(0,0,-1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+(125\,x^6-230\,x^5+187\,x^4+96\,\zeta(2)\,x^4-432\,\zeta(2)\,x^3+204\,x^3\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.5cm}+96\,\zeta(2)\,x^2-269\,x^2+26\,x-43)/[(x+1)^3(x-1)^5]\,x\,H(0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4\,(15\,x^4-16\,x^3+198\,x^2-16\,x+15)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}/[(x+1)^2(x-1)^4]\,x\,H(-1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-16\,(x^4-37\,x^3+54\,x^2-37\,x+1)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.8cm}/[(x+1)^3(x-1)^5]\,x^2\,H(0,-1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4\,(14\,x^5+175\,x^4-56\,x^3+14\,x^2+6\,x-9)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.6cm}/[(x+1)^3(x-1)^5]\,x\,H(0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-8\,(x^4-5\,x^3+38\,x^2-5\,x+1)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(x+1)^3(x-1)^5]\,x^2\,H(0,0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-32\,(x^2-17\,x+1)/[(x+1)^3(x-1)^3]\,x^2\,H(1,0,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+8\,(3\,x^4-7\,x^3+64\,x^2-7\,x+3)/[(x+1)^2(x-1)^4]\,x\,H(1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-16\,(x^4+19\,x^3-16\,x^2+19\,x+1)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.8cm}/[(x+1)^3(x-1)^5]\,x^2\,H(0,1,0,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+32\,(x^4-\zeta(2)\,x^3+x^3+17\,\zeta(2)\,x^2-\zeta(2)\,x+x+1)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.8cm}/[(x+1)^3(x-1)^3]\,x\,H(1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-8\,(3\,x^4-6\,x^3+14\,x^2-6\,x+3)/[(x+1)^2(x-1)^4]\,x\,H(0,1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr+64/[(x+1)^3(x-1)^3]\,x^3\,H(0,0,1,0;x)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/5\,(240\,\zeta(2)\,\ln(2)\,x^6-60\,\zeta(3)\,x^6-235\,\zeta(2)\,x^6-85\,x^6\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.2cm}+320\,\zeta(3)\,x^5-960\,\zeta(2)\,\ln(2)\,x^5+8\,\zeta(2)^2\,x^5\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.2cm}-30\,\zeta(2)\,x^5+255\,x^4-3340\,\zeta(3)\,x^4-1160\,\zeta(2)^2\,x^4\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.2cm}+1595\,\zeta(2)\,x^4+1200\,\zeta(2)\,\ln(2)\,x^4+540\,\zeta(2)\,x^3\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.2cm}+3240\,\zeta(2)^2\,x^3-1160\,\zeta(2)^2\,x^2+3340\,\zeta(3)\,x^2-1525\,\zeta(2)\,x^2\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.2cm}-255\,x^2-1200\,\zeta(2)\,\ln(2)\,x^2+960\,\zeta(2)\,\ln(2)\,x+8\,\zeta(2)^2\,x\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.2cm}-510\,\zeta(2)\,x-320\,\zeta(3)\,x+60\,\zeta(3)-240\,\zeta(2)\,\ln(2)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.2cm}+85+165\,\zeta(2))/[(x+1)^3(x-1)^5]\,x. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\eea

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\boldmath
%\subsection{Form factors $G_{i,R}^{\,(2l)}$ for $\mu \not{\! \! =} \,
%  m$ \label{munotm}}
\subsection{Form Factors for $\mu \not{\! \! =} \, m$ \label{munotm}}
\unboldmath
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this Section we give the expressions for the renormalized
axial vector two-loop form factors
for the case of $\mu \not{\! \! =} \, m$.

At the one-loop level 
we do not have an explicit dependence on the logarithm of the ratio of the 
renormalization scale and the mass of the heavy quark, 
because an overall factor $(\mu^2/m^2)^{\epsilon}$ can be taken
out, see Eqs.~(\ref{g1_1l_ren_decomp},\ref{g2_1l_ren_decomp}).

At the two-loop level, such a dependence results from the coupling constant 
renormalization, first appearing at this level.
Factoring an overall $(\mu^2/m^2)^{2 \epsilon}$, we have:
\bea
G_{i,R}^{(2l)}\Bigl(\epsilon,s,\frac{\mu^2}{m^2}\Bigr) &=& 
C^2(\epsilon) \, \left( \frac{\mu^2}{m^2} \right)^{2 \epsilon} 
\Biggl\{
G_{i,R}^{(2l)}(\epsilon,s) 
+ {\mathcal O}_{i}^{(2l)}(\epsilon,s) \ln{\left( \frac{\mu^2}{m^2} \right)}
\nn\\
& & \hspace*{30mm}
+ {\mathcal P}_{i}^{(2l)}(s) \ln^2{\left( \frac{\mu^2}{m^2} \right)}
\Biggr\} 
\, ,
\label{G1G2_2l_munotm}
\eea
where the functions $G_{i,R}^{(2l)}(\epsilon,s)$ are given in 
Eqs.~(\ref{g1_2l_ren_decomp},\ref{g2_2l_ren_decomp}) and the functions 
${\mathcal O}_{i}^{(2l)}(\epsilon,s)$ and ${\mathcal P}_{i}^{(2l)}(s)$ 
can be obtained either from the renormalization group \cite{bbghlmr}
or by explicit expansion. We find:  
\bea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\mathcal O}_{1}^{(2l)}(\epsilon,s) = && - C_F\,[\,T_R\,(N_f+1)\,-11/4 \,N_c\,]\nn\\
%%%%%%%%%%%%%%%
&& \times \Bigg\{ \phantom{+}\,\frac{2}{3\,\epsilon} \, \bigg\{\,(x^2+1)/[(x-1)(x+1)]\,H(0;x)-1\,\bigg\}\nn \\
%%%%%%%%%%%%%%%
&& \hspace{.7cm}+ 1/3\,(3\,x^2-2\,x+3)/[(x-1)(x+1)]\,H(0;x)\nn\\
%%%%%%%%%%%%%%%
&& \hspace{.7cm}- 4/3\,(x^2+1)/[(x-1)(x+1)]\,H(-1,0;x)\nn \\
%%%%%%%%%%%%%%%
&& \hspace{.7cm}+ 2/3\,(x^2+1)/[(x-1)(x+1)]\,H(0,0;x)\nn \\
%%%%%%%%%%%%%%%
&& \hspace{.7cm}- 2/3\,(x^2\,\zeta(2)+2\,x^2+\zeta(2)-2)/[(x-1)(x+1)]\Bigg\},\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\mathcal P}_{1}^{(2l)}(s) = && C_F\,[\,T_R\,(N_f+1)\,-11/4 \,N_c\,]\nn\\
%%%%%%%%%%%%%%%
&&\times 1/3\;\{(x^2+1)/[(x-1)\,(x+1)]\,H(0;x)-1\},\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\mathcal O}_{2}^{(2l)}(\epsilon,s) = && - C_F\,[\,T_R\,(N_f+1)\,-11/4 \,N_c\,]\nn\\
%%%%%%%%%%%%%%%
&& \times 4/3\,\{(3\,x^2-2\,x+3)/[(x-1)^3 (x+1)]\,x\,H(0;x)\nn \\
%%%%%%%%%%%%%%%
&& \hspace{1.3cm}-2\,(x-1)^2\,x\},\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\mathcal P}_{2}^{(2l)}(s) = && 0 . 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\eea


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\section{Analytical Continuation above Threshold\label{sec_analytic}}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The results for the renormalized form factors can be analytically
continued into the timelike region $s>0$ and in particular above the
physical threshold $s>4$ by using the substitution \cite{bbghlmr}
\be
x\rightarrow-y+i\epsilon,
\ee
with
\be
y=\frac{\sqrt{s}-\sqrt{s-4}}{\sqrt{s}+\sqrt{s-4}}.
\ee
For $s>4$ the form factors become complex due to
absorptive parts. We write:
\bea
G_1(s+i\epsilon,\epsilon) & = & \Re \, G_1(s,\epsilon)+ i\pi \, \Im \,
G_1(s,\epsilon),\\
G_1(s+i\epsilon,\epsilon) & = & \Re \, G_1(s,\epsilon)+ i\pi \, \Im \,
G_1(s,\epsilon).
\eea
These imaginary parts arise from the analytical continuation of the
harmonic polylogarithms with rightmost index 0.
In the following two Subsections we shall give the real and imaginary
parts of $G_{1,2}$ at one and two loops, putting $\mu = m$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\begin{boldmath}
\subsection{One-Loop Form Factors above Threshold \label{subsec_1l_anacon}}
\end{boldmath}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The analytical continuation of the coefficients
$\tilde{a}_i$ and $\tilde{b}_i$ in
Eqs.~(\ref{g1_1l_ren_decomp},\ref{g2_1l_ren_decomp}) is:
\bea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{a}_1  = && \nr (y^2+1)/[(y-1)\,(y+1)]\,H(0;y)-1,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{a}_2  = && \nr 1/2\,(3\,y^2+2\,y+3)/[(y-1)\,(y+1)]\,H(0;y) \nn \\
%%%%%%%%%%%%%%%
&& \nr+(y^2+1)/[(y-1)\,(y+1)]\,H(0,0;y) \nn \\
%%%%%%%%%%%%%%%
&& \nr+2\,(y^2+1)/[(y-1)\,(y+1)]\,H(1,0;y) \nn \\
%%%%%%%%%%%%%%%
&& \nr-2\,(y^2+2\,\zeta(2)\,y^2-1+2\,\zeta(2))/[(y-1)\,(y+1)],\\  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{a}_3  = && \nr -4\,(y^2+1)\,(-1+\zeta(2))/[(y-1)\,(y+1)]\,H(0;y) \nn \\
%%%%%%%%%%%%%%%
&& \nr+1/2\,(3\,y^2+2\,y+3)/[(y-1)\,(y+1)]\,H(0,0;y) \nn \\
%%%%%%%%%%%%%%%
&& \nr+(y^2+1)/[(y-1)\,(y+1)]\,H(0,0,0;y) \nn \\
%%%%%%%%%%%%%%%
&& \nr+2\,(y^2+1)/[(y-1)\,(y+1)]\,H(1,0,0;y) \nn \\
%%%%%%%%%%%%%%%
&& \nr+(3\,y^2+2\,y+3)/[(y-1)\,(y+1)]\,H(1,0;y) \nn \\
%%%%%%%%%%%%%%%
&& \nr+2\,(y^2+1)/[(y-1)\,(y+1)]\,H(0,1,0;y) \nn \\
%%%%%%%%%%%%%%%
&& \nr+4\,(y^2+1)/[(y-1)\,(y+1)]\,H(1,1,0;y) \nn \\
%%%%%%%%%%%%%%%
&& \nr-8\,\zeta(2)\,(y^2+1)/[(y-1)\,(y+1)]\,H(1;y) \nn \\
%%%%%%%%%%%%%%%
&& \nr-2\,(\zeta(3)\,y^2+3\,\zeta(2)\,y^2+2\,y^2+2\,\zeta(2)\,y-2+3\,\zeta(2)+\zeta(3))\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.7cm}/[(y-1)\,(y+1)],\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{a}_1 = && \nr (y^2+1)/[(y-1)\,(y+1)],\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{a}_2 = && \nr (y^2+1)/[(y-1)\,(y+1)]\,H(0;y) \nn \\
%%%%%%%%%%%%%%%
&& \nr+2\,(y^2+1)/[(y-1)\,(y+1)]\,H(1;y) \nn \\
%%%%%%%%%%%%%%%
&& \nr+1/2\,(3\,y^2+2\,y+3)/[(y-1)\,(y+1)],\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{a}_3 = && \nr 1/2\,(3\,y^2+2\,y+3)/[(y-1)\,(y+1)]\,H(0;y) \nn \\
%%%%%%%%%%%%%%%
&& \nr+(y^2+1)/[(y-1)\,(y+1)]\,H(0,0;y) \nn \\
%%%%%%%%%%%%%%%
&& \nr+2\,(y^2+1)/[(y-1)\,(y+1)]\,H(1,0;y) \nn \\
%%%%%%%%%%%%%%%
&& \nr+(3\,y^2+2\,y+3)/[(y-1)\,(y+1)]\,H(1;y) \nn \\
%%%%%%%%%%%%%%%
&& \nr+2\,(y^2+1)/[(y-1)\,(y+1)]\,H(0,1;y) \nn \\
%%%%%%%%%%%%%%%
&& \nr+4\,(y^2+1)/[(y-1)\,(y+1)]\,H(1,1;y) \nn \\
%%%%%%%%%%%%%%%
&& \nr-2\,(y^2+1)\,(\zeta(2)-2)/[(y-1)\,(y+1)],\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{b}_2  = && \nr
-2\,(3\,y^2+2\,y+3)/[(y-1)\,(y+1)^3]\,y\,H(0;y) +4/(y+1)^2\,y,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{b}_3  = && \nr -4\,(3\,y^2+2\,y+3)/[(y-1)\,(y+1)^3]\,y\,H(0;y) \nn \\
%%%%%%%%%%%%%%%
&& \nr-2\,(3\,y^2+2\,y+3)/[(y-1)\,(y+1)^3]\,y\,H(0,0;y) \nn \\
%%%%%%%%%%%%%%%
&& \nr-4\,(3\,y^2+2\,y+3)/[(y-1)\,(y+1)^3]\,y\,H(1,0;y) \nn \\
%%%%%%%%%%%%%%%
&& \nr+8\,(y^2+3\,\zeta(2)\,y^2+2\,\zeta(2)\,y+3\,\zeta(2)-1)\nn\\
%%%%%%%%%%%%%%%
&& \nr\hspace{.7cm}/[(y-1)\,(y+1)^3]\,y,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{b}_2  = && \nr -2\,(3\,y^2+2\,y+3)/[(y-1)\,(y+1)^3]\,y,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{b}_3  = && \nr -2\,(3\,y^2+2\,y+3)/[(y-1)\,(y+1)^3]\,y\,H(0;y) \nn \\
%%%%%%%%%%%%%%%
&& \nr-4\,(3\,y^2+2\,y+3)/[(y-1)\,(y+1)^3]\,y\,H(1;y) \nn \\
%%%%%%%%%%%%%%%
&& \nr-4\,(3\,y^2+2\,y+3)/[(y-1)\,(y+1)^3]\,y.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\eea
The above expressions for the singular term and the term of order 
$\epsilon^0$ of $G_1$ agree with those of \cite{Jersak:1981sp}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\begin{boldmath}
\subsection{Two-Loop Form Factors above Threshold \label{subsec_2l_anacon}}
\end{boldmath}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The real and imaginary parts of the coefficients $\tilde{c}_i$ and $\tilde{d}_i$
in Eq.~(\ref{g1_2l_ren_decomp}) are:
\bea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{c}_{1}  = && \nr -\frac{4}{11}\, \Re \, \tilde{c}_3 =-\frac{3}{5}\, \Re \, \tilde{c}_5=
\nn \\
%%%%%%%%%%%%%%%
&& \nr 1/3\,(y^2+1)/[(y-1)(y+1)]\,H(0;y)-1/3,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{c}_{2}  = && \nr \Re \, \tilde{c}_{6} = 0,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{c}_{4}  = && \nr -(y^2+1)/[(y-1)(y+1)]\,H(0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+(y^2+1)^2/[(y-1)^2(y+1)^2]\,H(0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/2\,(-y^4+6\,\zeta(2)\,y^4+2\,y^2+12\,\zeta(2)\,y^2-1+6\,\zeta(2))\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^2(y+1)^2],\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{c}_{7}  = && \nr
1/36\,(y^2+1)\,(67\,y^2+162\,\zeta(2)\,y^2-67+18\,\zeta(2))\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^2(y+1)^2]\,H(0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-(y^2+1)/[(y+1)(y-1)]\,H(-1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+(y^2+1)^2/[(y-1)^2(y+1)^2]\,H(0,-1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2/[(y+1)(y-1)]\,y^2\,H(0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(y^2+1)/[(y-1)^2(y+1)^2]\,y^2\,H(0,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+(y^2+1)/[(y+1)(y-1)]\,H(1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-(y^2+1)^2/[(y-1)^2(y+1)^2]\,H(0,1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/36\,(49\,y^4+18\,\zeta(3)\,y^4+162\,\zeta(2)\,y^4-144\,\zeta(2)\,y^2+36\,\zeta(3)\,y^2\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.4cm} -98\,y^2-18\,\zeta(2)+18\,\zeta(3)+49)/[(y-1)^2(y+1)^2],\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{c}_{8}  = && \nr
-1/2\,(20\,\zeta(2)\,y^4+7\,y^4+2\,y^3+40\,\zeta(2)\,y^2-2\,y-7+20\,\zeta(2))\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y+1)^2(y-1)^2]\,H(0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(y^2+1)\,(y^2+y+2)/[(y+1)^2(y-1)^2]\,H(0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+3\,(y^2+1)^2/[(y+1)^2(y-1)^2]\,H(0,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4\,(y^2+1)^2/[(y+1)^2(y-1)^2]\,H(1,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(y^2+1)/[(y+1)(y-1)]\,H(1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(y^2+1)^2/[(y-1)^2(y+1)^2]\,H(0,1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-12\,\zeta(2)\,(y^2+1)^2/[(y+1)^2(y-1)^2]\,H(1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-(5\,\zeta(2)\,y^4-2\,y^4+6\,\zeta(2)\,y^3+18\,\zeta(2)\,y^2+4\,y^2+6\,\zeta(2)\,y\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{0.6cm} -2+13\,\zeta(2))/[(y+1)^2(y-1)^2],\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{c}_{9}  = && \nr
1/54\,(72\,\zeta(2)\,y^2-209\,y^2-78\,y+72\,\zeta(2)-209)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)(y+1)]\,H(0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/9\,(19\,y^2+6\,y+19)/[(y-1)(y+1)]\,H(0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2/3\,(y^2+1)/[(y-1)(y+1)]\,H(0,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4/3\,(y^2+1)/[(y-1)(y+1)]\,H(1,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2/9\,(19\,y^2+6\,y+19)/[(y-1)(y+1)]\,H(1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4/3\,(y^2+1)/[(y-1)(y+1)]\,H(0,1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-8/3\,(y^2+1)/[(y-1)(y+1)]\,H(1,1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+16/3\,\zeta(2)\,(y^2+1)/[(y-1)(y+1)]\,H(1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2/27\,(18\,\zeta(3)\,y^2+53\,y^2+132\,\zeta(2)\,y^2+36\,\zeta(2)\,y-53\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.4cm} +96\,\zeta(2)+18\,\zeta(3))/[(y-1)(y+1)],\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{c}_{10}  = && \nr
1/54\,(-265\,y^4+72\,\zeta(2)\,y^4-208\,y^3-144\,\zeta(2)\,y^3-14\,y^2\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
-288\,\zeta(2)\,y^2-208\,y-144\,\zeta(2)\,y-265+72\,\zeta(2))\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{2cm} /[(y-1)^3(y+1)]\,H(0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/9\,(19\,y^4+14\,y^3+14\,y^2+14\,y+19)/(y-1)^4\,H(0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2/3\,(y^4-2\,y^3-4\,y^2-2\,y+1)/[(y-1)^3(y+1)]\,H(0,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/27\,(-383\,y^2+198\,\zeta(2)\,y^2-504\,\zeta(2)\,y+646\,y\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.4cm} -383+198\,\zeta(2))/(y-1)^2,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{c}_{11}  = && \nr
(-11\,\zeta(2)\,y^4+2\,\zeta(3)\,y^4+30\,\zeta(2)\,y^3+84\,\zeta(2)\,y^2+4\,\zeta(3)\,y^2\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} +30\,\zeta(2)\,y-37\,\zeta(2)+2\,\zeta(3))/[(y-1)^2(y+1)^2]\,H(-1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-\zeta(2)\,(7\,y^6+30\,y^5+127\,y^4+204\,y^3+101\,y^2+30\,y-19)\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^3]\,H(0,-1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/216\,(8568\,\zeta(2)\,y^6+1620\,\zeta(3)\,y^6-2545\,y^6-570\,y^5\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.4cm}
+3024\,\zeta(3)\,y^5+30888\,\zeta(2)\,y^5+35496\,\zeta(2)\,y^4+2545\,y^4\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.4cm}
+13716\,\zeta(3)\,y^4+1140\,y^3+25920\,\zeta(3)\,y^3+25488\,\zeta(2)\,y^3\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.4cm}
+2545\,y^2+936\,\zeta(2)\,y^2+13500\,\zeta(3)\,y^2-216\,\zeta(2)\,y\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1.4cm}
-570\,y+3024\,\zeta(3)\,y+792\,\zeta(2)-2545+1404\,\zeta(3))\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{2.4cm} /[(y-1)^3(y+1)^3]\,H(0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+(3\,y^5+2\,\zeta(2)\,y^5+22\,\zeta(2)\,y^4-y^4-2\,y^3+68\,\zeta(2)\,y^3-2\,y^2\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} +108\,\zeta(2)\,y^2+2\,\zeta(2)\,y-y+22\,\zeta(2)+3)\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{2cm} /[(y-1)^3(y+1)^2]\,H(-1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(y^2+1)/[(y+1)(y-1)]\,H(-1,-1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(y^2+1)^2/[(y-1)^2(y+1)^2]\,H(0,-1,-1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4\,(y^2+1)^2/[(y-1)^2(y+1)^2]\,H(-1,0,-1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(7\,y^5-7\,y^4+12\,y^3-4\,y^2+y-1)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^2]\,H(0,0,-1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4\,(y^2+1)^2/[(y-1)^2(y+1)^2]\,H(1,0,-1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-6\,(y^2+1)/[(y+1)(y-1)]\,H(1,-1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+6\,(y^2+1)^2/[(y-1)^2(y+1)^2]\,H(0,1,-1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/36\,(-217\,y^6+1332\,\zeta(2)\,y^6+282\,y^5+252\,\zeta(2)\,y^5\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
+1692\,\zeta(2)\,y^4+451\,y^4-348\,y^3+3528\,\zeta(2)\,y^3-467\,y^2\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
+324\,\zeta(2)\,y^2+66\,y+252\,\zeta(2)\,y-36\,\zeta(2)+233)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{2cm} /[(y-1)^3(y+1)^3]\,H(0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-8\,(y^2+1)/[(y-1)^2(y+1)]\,y\,H(0,-1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(2\,y^4-5\,y^3-14\,y^2-5\,y+6)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^2(y+1)^2]\,H(-1,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(y^6+5\,y^5+21\,y^4+34\,y^3+17\,y^2+5\,y-3)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^3]\,H(0,-1,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/6\,(89\,y^6+402\,y^5+499\,y^4+354\,y^3+37\,y^2+24\,y+11)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^3]\,H(0,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(y^5+5\,y^4+18\,y^3+26\,y^2+y+5)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^2]\,H(-1,0,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-(12\,y^5+y^4+10\,y^3+28\,y^2-2\,y+1)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^3]\,y\,H(0,0,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(y^2+3)\,(y^2+1)/[(y-1)^2(y+1)^2]\,H(1,0,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/3\,(53\,y^4+141\,y^3+228\,y^2+141\,y-17)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^2(y+1)^2]\,H(1,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(7\,y^6+7\,y^5+33\,y^4+56\,y^3+25\,y^2+7\,y-1)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^3]\,H(0,1,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/9\,(54\,\zeta(2)\,y^4+31\,y^4+105\,y^3+180\,\zeta(2)\,y^2-105\,y\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} -31+126\,\zeta(2))/[(y-1)^2(y+1)^2]\,H(1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-6\,(y^2+1)/[(y+1)(y-1)]\,H(-1,1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+6\,(y^2+1)^2/[(y-1)^2(y+1)^2]\,H(0,-1,1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2/3\,(34\,y^4-3\,y^3-33\,y^2-3\,y-1)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^2(y+1)^2]\,H(0,1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4\,(y^2+1)^2/[(y-1)^2(y+1)^2]\,H(-1,0,1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(11\,y^6-5\,y^5-15\,y^4-26\,y^3-27\,y^2-5\,y-1)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^3]\,H(0,0,1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4\,(y^2+1)^2/[(y-1)^2(y+1)^2]\,H(1,0,1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-10\,(y^2+1)^2/[(y-1)^2(y+1)^2]\,H(0,1,1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/3\,(6\,\zeta(3)\,y^4+149\,\zeta(2)\,y^4+342\,\zeta(2)\,y^3+12\,\zeta(3)\,y^2\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
+576\,\zeta(2)\,y^2+342\,\zeta(2)\,y-77\,\zeta(2)+6\,\zeta(3))\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{2cm} /[(y-1)^2(y+1)^2]\,H(1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+\zeta(2)\,(35\,y^6+48\,y^5+227\,y^4+360\,y^3+181\,y^2+48\,y-11)\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^3]\,H(0,1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/540\,(450\,\zeta(3)\,y^6+3240\,\zeta(2)\,\ln(2)\,y^6-7975\,y^6\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
-13257\,\zeta(2)^2\,y^6+4560\,\zeta(2)\,y^6-3726\,\zeta(2)^2\,y^5\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
+19440\,\zeta(2)\,\ln(2)\,y^5-46620\,\zeta(2)\,y^5+22680\,\zeta(3)\,y^5\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
-62700\,\zeta(2)\,y^4+36270\,\zeta(3)\,y^4+29160\,\zeta(2)\,\ln(2)\,y^4\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
-23193\,\zeta(2)^2\,y^4+23925\,y^4+24120\,\zeta(2)\,y^3\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
-31644\,\zeta(2)^2\,y^3-29160\,\zeta(2)\,\ln(2)\,y^2+68520\,\zeta(2)\,y^2\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
-30690\,\zeta(3)\,y^2-23925\,y^2-8235\,\zeta(2)^2\,y^2\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
-19440\,\zeta(2)\,\ln(2)\,y-3726\,\zeta(2)^2\,y-22680\,\zeta(3)\,y\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
+22500\,\zeta(2)\,y+1701\,\zeta(2)^2-6030\,\zeta(3)+7975\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} -3240\,\zeta(2)\,\ln(2)-10380\,\zeta(2))\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{2cm} /[(y-1)^3(y+1)^3]\nn \\
%%%%%%%%%%%%%%%
&& \nr+52/3\,(y^2+1)/[(y-1)(y+1)]\,H(1,1,0;y),\nn\\
%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{c}_{12}  = && \nr 2\,(5\,y^4+10\,y^3+18\,y^2+10\,y+5)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y+1)^2(y-1)^2]\,H(0,-1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4\,(3\,y^5-3\,y^4+6\,y^3+2\,y^2-y+1)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^2]\,H(0,0,-1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/4\,(272\,\zeta(2)\,y^6-229\,y^6+8\,\zeta(2)\,y^5-266\,y^5-15\,y^4\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
+200\,\zeta(2)\,y^4+196\,y^3+352\,\zeta(2)\,y^3-168\,\zeta(2)\,y^2+189\,y^2\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} +8\,\zeta(2)\,y+70\,y-96\,\zeta(2)+55)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{2cm} /[(y-1)^3(y+1)^3]\,H(0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(8\,y^4+19\,y^3+46\,y^2+19\,y+8)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y+1)^2(y-1)^2]\,H(-1,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4\,(y^6+y^5+14\,y^4+20\,y^3+14\,y^2+y+1)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^3]\,H(0,-1,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(5\,y^6+10\,y^5-60\,y^4-59\,y^3-16\,y^2+7\,y+5)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^3]\,H(0,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-8\,(y^5+2\,y^4+18\,y^3+14\,y^2+4\,y-1)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^2]\,H(-1,0,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+(27\,y^6+2\,y^5+27\,y^4+56\,y^3-7\,y^2+2\,y-7)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^3]\,H(0,0,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+12\,(y^2+1)^2/[(y+1)^2(y-1)^2]\,H(1,0,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(3\,y^4+15\,y^3+76\,y^2+15\,y+5)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y+1)^2(y-1)^2]\,H(1,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(7\,y^6-10\,y^5-65\,y^4-104\,y^3-75\,y^2-10\,y-3)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^3]\,H(0,1,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+16\,(y^2+1)^2/[(y+1)^2(y-1)^2]\,H(1,1,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/2\,(-55\,y^4+80\,\zeta(2)\,y^4-82\,y^3+160\,\zeta(2)\,y^2+82\,y\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} +55+80\,\zeta(2))/[(y+1)^2(y-1)^2]\,H(1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(8\,y^4+27\,y^3+60\,y^2+27\,y+6)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y+1)^2(y-1)^2]\,H(0,1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(11\,y^6+2\,y^5+33\,y^4+56\,y^3+23\,y^2+2\,y+1)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^3]\,H(0,0,1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4\,\zeta(2)\,(7\,y^4+36\,y^3+132\,y^2+36\,y+11)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y+1)^2(y-1)^2]\,H(1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-8\,\zeta(2)\,(4\,y^6-6\,y^5-41\,y^4-72\,y^3-49\,y^2-6\,y-4)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^3]\,H(0,1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-48\,\zeta(2)\,(y^2+1)^2/[(y+1)^2(y-1)^2]\,H(1,1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/10\,(70\,\zeta(3)\,y^6+120\,\zeta(2)\,\ln(2)\,y^6-115\,y^6-196\,\zeta(2)^2\,y^6\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
+1580\,\zeta(2)\,y^6+18\,\zeta(2)^2\,y^5+720\,\zeta(2)\,\ln(2)\,y^5\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
+1880\,\zeta(2)\,y^5-300\,\zeta(3)\,y^5+240\,\zeta(2)\,y^4-1490\,\zeta(3)\,y^4\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
+1080\,\zeta(2)\,\ln(2)\,y^4-74\,\zeta(2)^2\,y^4+345\,y^4-1720\,\zeta(2)\,y^3\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
+232\,\zeta(2)^2\,y^3-1080\,\zeta(2)\,\ln(2)\,y^2-1500\,\zeta(2)\,y^2\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
+1530\,\zeta(3)\,y^2-345\,y^2+258\,\zeta(2)^2\,y^2-720\,\zeta(2)\,\ln(2)\,y\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
+18\,\zeta(2)^2\,y+300\,\zeta(3)\,y-160\,\zeta(2)\,y+136\,\zeta(2)^2\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
-110\,\zeta(3)+115-120\,\zeta(2)\,\ln(2)-320\,\zeta(2))\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{2cm} /[(y-1)^3(y+1)^3]\nn \\
%%%%%%%%%%%%%%%
&& \nr+8\,(y^2+1)^2/[(y-1)^2(y+1)^2]\,H(-1,0,-1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(-24\,\zeta(2)\,y^4+2\,\zeta(3)\,y^4-57\,\zeta(2)\,y^3+4\,\zeta(3)\,y^2\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
-138\,\zeta(2)\,y^2-57\,\zeta(2)\,y-24\,\zeta(2)+2\,\zeta(3))\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{2cm} /[(y+1)^2(y-1)^2]\,H(-1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-8\,(y^2+1)^2/[(y-1)^2(y+1)^2]\,H(-1,0,1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+8\,(y^2+1)^2/[(y-1)^2(y+1)^2]\,H(1,0,1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4\,(y^2+1)^2/[(y-1)^2(y+1)^2]\,H(0,1,1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/8\,(-85\,y^6+104\,\zeta(2)\,y^6+64\,\zeta(3)\,y^6-6\,y^5-160\,\zeta(3)\,y^5\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
+184\,\zeta(2)\,y^5-1056\,\zeta(3)\,y^4-2312\,\zeta(2)\,y^4+85\,y^4+12\,y^3\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
-1888\,\zeta(2)\,y^3-1536\,\zeta(3)\,y^3+85\,y^2-1056\,\zeta(3)\,y^2\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
-152\,\zeta(2)\,y^2-6\,y+360\,\zeta(2)\,y-160\,\zeta(3)\,y+64\,\zeta(3)\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} +248\,\zeta(2)-85)/[(y-1)^3(y+1)^3]\,H(0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-12\,\zeta(2)\,(y^6+y^5+14\,y^4+20\,y^3+14\,y^2+y+1)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^3]\,H(0,-1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4\,(y^2+1)/[(y-1)(y+1)]\,H(1,1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4\,(-4\,y^5+5\,\zeta(2)\,y^5+2\,y^4+7\,\zeta(2)\,y^4+2\,y^3+74\,\zeta(2)\,y^3+2\,y^2\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
+54\,\zeta(2)\,y^2+17\,\zeta(2)\,y+2\,y-4-5\,\zeta(2))\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^2]\,H(-1,0;y),\nn\\
%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\eea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
and
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{c}_{1}  = && \nr -\frac{4}{11}\, \Im \, \tilde{c}_3
=-\frac{3}{5}\, \Im \, \tilde{c}_5 = 1/3\,(y^2+1)/[(y-1)(y+1)],\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{c}_{2}  = && \nr \Im \, \tilde{c}_6 = 0,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{c}_{4}  = && \nr (y^2+1)^2/[(y-1)^2(y+1)^2]\,H(0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-(y^2+1)/[(y-1)(y+1)],\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{c}_{7}  = && \nr -(y^2+1)/[(y-1)(y+1)]\,H(-1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+(y^2+1)^2/[(y-1)^2(y+1)^2]\,H(0,-1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2/[(y-1)(y+1)]\,y^2\,H(0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(y^2+1)/[(y-1)^2(y+1)^2]\,y^2\,H(0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+(y^2+1)/[(y-1)(y+1)]\,H(1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-(y^2+1)^2/[(y-1)^2(y+1)^2]\,H(0,1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/36\,(y^2+1)\,(67\,y^2+18\,\zeta(2)\,y^2+18\,\zeta(2)-67)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^2(y+1)^2],\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{c}_{8}  = && \nr 2\,(y^2+y+2)\,(y^2+1)/[(y+1)^2(y-1)^2]\,H(0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+3\,(y^2+1)^2/[(y-1)^2(y+1)^2]\,H(0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4\,(y^2+1)^2/[(y+1)^2(y-1)^2]\,H(1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(y^2+1)/[(y-1)(y+1)]\,H(1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(y^2+1)^2/[(y-1)^2(y+1)^2]\,H(0,1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/2\,(7\,y^4+8\,\zeta(2)\,y^4+2\,y^3+16\,\zeta(2)\,y^2-2\,y-7+8\,\zeta(2))\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y+1)^2(y-1)^2],\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{c}_{9}  = && \nr -1/9\,(19\,y^2+6\,y+19)/[(y-1)(y+1)]\,H(0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2/3\,(y^2+1)/[(y-1)(y+1)]\,H(0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4/3\,(y^2+1)/[(y+1)(y-1)]\,H(1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2/9\,(19\,y^2+6\,y+19)/[(y-1)(y+1)]\,H(1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4/3\,(y^2+1)/[(y-1)(y+1)]\,H(0,1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-8/3\,(y^2+1)/[(y-1)(y+1)]\,H(1,1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/54\,(209\,y^2+78\,y+209)/[(y-1)(y+1)],\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{c}_{10}  = && \nr 1/9\,(19\,y^4+14\,y^3+14\,y^2+14\,y+19)/(y-1)^4\,H(0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2/3\,(y^4-2\,y^3-4\,y^2-2\,y+1)/[(y-1)^3(y+1)]\,H(0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/54\,(265\,y^4+208\,y^3+14\,y^2+208\,y+265)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)],\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{c}_{11}  = && \nr 52/3\,(y^2+1)/[(y-1)(y+1)]\,H(1,1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(y^2+1)/[(y-1)(y+1)]\,H(-1,-1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(y^2+1)^2/[(y-1)^2(y+1)^2]\,H(0,-1,-1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-(2\,\zeta(2)\,y^4-3\,y^4-2\,y^3+4\,\zeta(2)\,y^2+2\,y+2\,\zeta(2)+3)\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^2(y+1)^2]\,H(-1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-8\,(y^2+1)/[(y-1)^2(y+1)]\,y\,H(0,-1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4\,(y^2+1)^2/[(y-1)^2(y+1)^2]\,H(-1,0,-1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(7\,y^5-7\,y^4+12\,y^3-4\,y^2+y-1)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^2]\,H(0,0,-1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4\,(y^2+1)^2/[(y-1)^2(y+1)^2]\,H(1,0,-1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-6\,(y^2+1)/[(y-1)(y+1)]\,H(1,-1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+6\,(y^2+1)^2/[(y-1)^2(y+1)^2]\,H(0,1,-1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(y^6+5\,y^5+21\,y^4+34\,y^3+17\,y^2+5\,y-3)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^3]\,H(0,-1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/6\,(89\,y^6+402\,y^5+499\,y^4+354\,y^3+37\,y^2+24\,y+11)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^3]\,H(0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(y^5+5\,y^4+18\,y^3+26\,y^2+y+5)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^2]\,H(-1,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/36\,(468\,\zeta(2)\,y^5-217\,y^5-288\,\zeta(2)\,y^4+499\,y^4-48\,y^3\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
+1260\,\zeta(2)\,y^3-300\,y^2+252\,\zeta(2)\,y^2+216\,\zeta(2)\,y-167\,y\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} -36\,\zeta(2)+233)/[(y-1)^3(y+1)^2]\,H(0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(2\,y^4-5\,y^3-14\,y^2-5\,y+6)/[(y-1)^2(y+1)^2]\,H(-1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-(12\,y^5+y^4+10\,y^3+28\,y^2-2\,y+1)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^3]\,y\,H(0,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/3\,(53\,y^4+141\,y^3+228\,y^2+141\,y-17)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^2(y+1)^2]\,H(1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(7\,y^6+7\,y^5+33\,y^4+56\,y^3+25\,y^2+7\,y-1)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^3]\,H(0,1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/9\,(18\,\zeta(2)\,y^4+31\,y^4+105\,y^3+36\,\zeta(2)\,y^2-105\,y\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} +18\,\zeta(2)-31)/[(y-1)^2(y+1)^2]\,H(1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-6\,(y^2+1)/[(y-1)(y+1)]\,H(-1,1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+6\,(y^2+1)^2/[(y-1)^2(y+1)^2]\,H(0,-1,1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2/3\,(34\,y^4-3\,y^3-33\,y^2-3\,y-1)/[(y-1)^2(y+1)^2]\,H(0,1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4\,(y^2+1)^2/[(y-1)^2(y+1)^2]\,H(-1,0,1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(y^2+3)\,(y^2+1)/[(y-1)^2(y+1)^2]\,H(1,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(11\,y^6-5\,y^5-15\,y^4-26\,y^3-27\,y^2-5\,y-1)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^3]\,H(0,0,1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4\,(y^2+1)^2/[(y-1)^2(y+1)^2]\,H(1,0,1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-10\,(y^2+1)^2/[(y-1)^2(y+1)^2]\,H(0,1,1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/216\,(-2545\,y^6+1620\,\zeta(3)\,y^6+2160\,\zeta(2)\,y^6+3024\,\zeta(3)\,y^5\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
-570\,y^5+1944\,\zeta(2)\,y^5-432\,\zeta(2)\,y^4+13716\,\zeta(3)\,y^4\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
+2545\,y^4+1140\,y^3+25920\,\zeta(3)\,y^3+13500\,\zeta(3)\,y^2\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
+2545\,y^2-1728\,\zeta(2)\,y^2-1944\,\zeta(2)\,y+3024\,\zeta(3)\,y-570\,y\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} +1404\,\zeta(3)-2545)/[(y-1)^3(y+1)^3],\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{c}_{12}  = && \nr
4\,(-4\,y^4+\zeta(2)\,y^4-2\,y^3+2\,\zeta(2)\,y^2+2\,y+\zeta(2)+4)\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y+1)^2(y-1)^2]\,H(-1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(5\,y^4+10\,y^3+18\,y^2+10\,y+5)/[(y+1)^2(y-1)^2]\,H(0,-1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+8\,(y^2+1)^2/[(y-1)^2(y+1)^2]\,H(-1,0,-1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4\,(3\,y^5-3\,y^4+6\,y^3+2\,y^2-y+1)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^2]\,H(0,0,-1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/4\,(-229\,y^5+56\,\zeta(2)\,y^5-64\,\zeta(2)\,y^4-37\,y^4+48\,\zeta(2)\,y^3\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
+22\,y^3+174\,y^2-144\,\zeta(2)\,y^2+15\,y+32\,\zeta(2)\,y\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} -40\,\zeta(2)+55)/[(y-1)^3(y+1)^2]\,H(0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(8\,y^4+19\,y^3+46\,y^2+19\,y+8)/[(y+1)^2(y-1)^2]\,H(-1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4\,(y^6+y^5+14\,y^4+20\,y^3+14\,y^2+y+1)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^3]\,H(0,-1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(5\,y^6+10\,y^5-60\,y^4-59\,y^3-16\,y^2+7\,y+5)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^3]\,H(0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-8\,(y^5+2\,y^4+18\,y^3+14\,y^2+4\,y-1)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^2]\,H(-1,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+(27\,y^6+2\,y^5+27\,y^4+56\,y^3-7\,y^2+2\,y-7)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^3]\,H(0,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+12\,(y^2+1)^2/[(y+1)^2(y-1)^2]\,H(1,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(3\,y^4+15\,y^3+76\,y^2+15\,y+5)/[(y+1)^2(y-1)^2]\,H(1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(7\,y^6-10\,y^5-65\,y^4-104\,y^3-75\,y^2-10\,y-3)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^3]\,H(0,1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+16\,(y^2+1)^2/[(y+1)^2(y-1)^2]\,H(1,1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/2\,(32\,\zeta(2)\,y^4-55\,y^4-82\,y^3+64\,\zeta(2)\,y^2+82\,y\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} +32\,\zeta(2)+55)/[(y+1)^2(y-1)^2]\,H(1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(8\,y^4+27\,y^3+60\,y^2+27\,y+6)/[(y+1)^2(y-1)^2]\,H(0,1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-8\,(y^2+1)^2/[(y-1)^2(y+1)^2]\,H(-1,0,1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(11\,y^6+2\,y^5+33\,y^4+56\,y^3+23\,y^2+2\,y+1)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^3]\,H(0,0,1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+8\,(y^2+1)^2/[(y-1)^2(y+1)^2]\,H(1,0,1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4\,(y^2+1)/[(y-1)(y+1)]\,H(1,1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4\,(y^2+1)^2/[(y-1)^2(y+1)^2]\,H(0,1,1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/8\,(-56\,\zeta(2)\,y^6+64\,\zeta(3)\,y^6-85\,y^6-6\,y^5-160\,\zeta(3)\,y^5\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
-136\,\zeta(2)\,y^5+85\,y^4-392\,\zeta(2)\,y^4-1056\,\zeta(3)\,y^4\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
-1536\,\zeta(3)\,y^3+12\,y^3-1056\,\zeta(3)\,y^2+360\,\zeta(2)\,y^2+85\,y^2\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
+136\,\zeta(2)\,y-6\,y-160\,\zeta(3)\,y+64\,\zeta(3)-85+88\,\zeta(2))\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^3], 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\eea
respectively
\bea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{d}_j  = && \nr 0 \quad \mbox{for}\quad j=1\ldots 7 , \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{d}_{8}  =&& \nr 2\,(5\,y^2+2\,y+5)/[(y-1)(y+1)^3]\,y\,H(0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4\,(3\,y^2+2\,y+3)\,(y^2+1)/[(y-1)^2(y+1)^4]\,y\,H(0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4\,(-y^4+9\,\zeta(2)\,y^4+6\,\zeta(2)\,y^3+18\,\zeta(2)\,y^2+2\,y^2+6\,\zeta(2)\,y\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} -1+9\,\zeta(2))/[(y-1)^2(y+1)^4]\,y,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{d}_{9}  = && \nr 2/9\,(51\,y^2+26\,y+51)/[(y-1)(y+1)^3]\,y\,H(0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4/3\,(3\,y^2+2\,y+3)/[(y-1)(y+1)^3]\,y\,H(0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+8/3\,(3\,y^2+2\,y+3)/[(y-1)(y+1)^3]\,y\,H(1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4/9\,(36\,\zeta(2)\,y^2+19\,y^2+24\,\zeta(2)\,y-19+36\,\zeta(2))\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)(y+1)^3]\,y,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{d}_{10}  = && \nr
2/9\,(51\,y^4+176\,y^3+144\,\zeta(2)\,y^2-70\,y^2+176\,y+51)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^3]\,y\,H(0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4/3\,(3\,y^4+14\,y^3-2\,y^2+14\,y+3)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^4(y+1)^2]\,y\,H(0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-16/[(y-1)^3(y+1)^3]\,y^3\,H(0,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-8/9\,(-4\,y^2+9\,\zeta(2)\,y^2+32\,y-4+9\,\zeta(2))\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^2(y+1)^2]\,y,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{d}_{11}  = && \nr -72\,\zeta(2)\,(3\,y^2+2\,y+3)/[(y-1)^2(y+1)^4]\,y^2\,H(-1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+24\,\zeta(2)\,(5\,y^4+14\,y^3+42\,y^2+14\,y+5)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^5]\,y^2\,H(0,-1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/18\,(648\,\zeta(2)\,y^6-687\,y^6+26\,y^5+1008\,\zeta(3)\,y^5+8712\,\zeta(2)\,y^5\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
+11808\,\zeta(2)\,y^4+4752\,\zeta(3)\,y^4+687\,y^4+9216\,\zeta(3)\,y^3\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
-52\,y^3+14112\,\zeta(2)\,y^3+687\,y^2+4752\,\zeta(3)\,y^2\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
-648\,\zeta(2)\,y^2+1008\,\zeta(3)\,y+26\,y-648\,\zeta(2)\,y-687)\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{2cm} /[(y-1)^3(y+1)^5]\,y\,H(0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4\,(y^4+24\,\zeta(2)\,y^3+64\,\zeta(2)\,y^2-2\,y^2+24\,\zeta(2)\,y+1)\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^3]\,y\,H(-1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+16/[(y-1)^2(y+1)^2]\,y^2\,H(0,-1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-32/[(y-1)^3(y+1)^3]\,y^3\,H(0,0,-1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/3\,(-39\,y^6+184\,y^5+84\,\zeta(2)\,y^5+87\,y^4+432\,\zeta(2)\,y^4\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
+1416\,\zeta(2)\,y^3-152\,y^3+432\,\zeta(2)\,y^2-81\,y^2-32\,y\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} +84\,\zeta(2)\,y+33)/[(y-1)^3(y+1)^5]\,y\,H(0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+24\,(3\,y^2+2\,y+3)/[(y-1)^2(y+1)^4]\,y^2\,H(-1,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-8\,(5\,y^4+14\,y^3+42\,y^2+14\,y+5)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^5]\,y^2\,H(0,-1,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(9\,y^4+112\,y^3+146\,y^2+196\,y+9)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^5]\,y^3\,H(0,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+16\,(3\,y^2+8\,y+3)/[(y-1)^3(y+1)^3]\,y^2\,H(-1,0,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4\,(y^4+5\,y^3+38\,y^2+5\,y+1)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^5]\,y^2\,H(0,0,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(9\,y^4+92\,y^3+130\,y^2+92\,y+9)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^2(y+1)^4]\,y\,H(1,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+8\,(7\,y^4+29\,y^3+62\,y^2+29\,y+7)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^5]\,y^2\,H(0,1,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2/3\,(3\,y^2+88\,y+3)/[(y-1)(y+1)^3]\,y\,H(1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+48\,(y^2+y+1)/[(y-1)^2(y+1)^4]\,y^2\,H(0,1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-8\,(5\,y^4+14\,y^3+30\,y^2+14\,y+5)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^5]\,y^2\,H(0,0,1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+12\,\zeta(2)\,(3\,y^4+40\,y^3+62\,y^2+40\,y+3)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^2(y+1)^4]\,y\,H(1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-48\,\zeta(2)\,(4\,y^4+17\,y^3+32\,y^2+17\,y+4)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^5]\,y^2\,H(0,1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2/45\,(605\,y^6-540\,\zeta(2)\,\ln(2)\,y^6-270\,\zeta(3)\,y^6-180\,\zeta(2)\,y^6\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
+621\,\zeta(2)^2\,y^5-3690\,\zeta(3)\,y^5-2160\,\zeta(2)\,\ln(2)\,y^5\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
+9390\,\zeta(2)\,y^5-1815\,y^4+5670\,\zeta(2)\,y^4+3906\,\zeta(2)^2\,y^4\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
-2700\,\zeta(2)\,\ln(2)\,y^4-5490\,\zeta(3)\,y^4-5100\,\zeta(2)\,y^3\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
+4626\,\zeta(2)^2\,y^3-5760\,\zeta(2)\,y^2+2700\,\zeta(2)\,\ln(2)\,y^2\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
+5490\,\zeta(3)\,y^2+3906\,\zeta(2)^2\,y^2+1815\,y^2+621\,\zeta(2)^2\,y\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
+2160\,\zeta(2)\,\ln(2)\,y-4290\,\zeta(2)\,y+3690\,\zeta(3)\,y\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
+540\,\zeta(2)\,\ln(2)+270\,\zeta(2)+270\,\zeta(3)-605)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{2cm} /[(y-1)^3(y+1)^5]\,y,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{d}_{12}  = && \nr
-24\,\zeta(2)\,(3\,y^4+7\,y^3+64\,y^2+7\,y+3)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^2(y+1)^4]\,y\,H(-1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-48\,\zeta(2)\,(y^4-19\,y^3-16\,y^2-19\,y+1)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^5]\,y^2\,H(0,-1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/2\,(61\,y^6+96\,\zeta(2)\,y^6-120\,\zeta(2)\,y^5+22\,y^5+32\,\zeta(3)\,y^5\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
-61\,y^4+1056\,\zeta(3)\,y^4+3168\,\zeta(2)\,y^4-44\,y^3+1664\,\zeta(3)\,y^3\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
+896\,\zeta(2)\,y^3-144\,\zeta(2)\,y^2+1056\,\zeta(3)\,y^2-61\,y^2\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
+32\,\zeta(3)\,y+22\,y-200\,\zeta(2)\,y+61-240\,\zeta(2))\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^5]\,y\,H(0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-32\,(-y^4+2\,\zeta(2)\,y^3+y^3+34\,\zeta(2)\,y^2+2\,\zeta(2)\,y+y-1)\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^3]\,y\,H(-1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-8\,(3\,y^4+6\,y^3+14\,y^2+6\,y+3)/[(y-1)^2(y+1)^4]\,y\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} \,H(0,-1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+64/[(y-1)^3(y+1)^3]\,y^3\,H(0,0,-1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+(-125\,y^6+24\,\zeta(2)\,y^5-230\,y^5+24\,\zeta(2)\,y^4-187\,y^4\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
+480\,\zeta(2)\,y^3+204\,y^3+24\,\zeta(2)\,y^2+269\,y^2+26\,y\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} +24\,\zeta(2)\,y+43)/[(y-1)^3(y+1)^5]\,y\,H(0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+8\,(3\,y^4+7\,y^3+64\,y^2+7\,y+3)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^2(y+1)^4]\,y\,H(-1,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+16\,(y^4-19\,y^3-16\,y^2-19\,y+1)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^5]\,y^2\,H(0,-1,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4\,(14\,y^5-175\,y^4-56\,y^3-14\,y^2+6\,y+9)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^5]\,y\,H(0,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+32\,(y^2+17\,y+1)/[(y-1)^3(y+1)^3]\,y^2\,H(-1,0,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-8\,(y^4+5\,y^3+38\,y^2+5\,y+1)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^5]\,y^2\,H(0,0,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4\,(15\,y^4+16\,y^3+198\,y^2+16\,y+15)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^2(y+1)^4]\,y\,H(1,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+16\,(y^4+37\,y^3+54\,y^2+37\,y+1)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^5]\,y^2\,H(0,1,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(25\,y^2+86\,y+25)/[(y+1)^3(y-1)]\,y\,H(1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+8\,(3\,y^4+13\,y^3+64\,y^2+13\,y+3)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^2(y+1)^4]\,y\,H(0,1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+16\,(y^4-19\,y^3-28\,y^2-19\,y+1)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^5]\,y^2\,H(0,0,1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+24\,\zeta(2)\,(9\,y^4+14\,y^3+102\,y^2+14\,y+9)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^2(y+1)^4]\,y\,H(1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-192\,\zeta(2)\,(8\,y^2+13\,y+8)/[(y-1)^3(y+1)^5]\,y^3\,H(0,1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/5\,(85\,y^6-240\,\zeta(2)\,\ln(2)\,y^6+60\,\zeta(3)\,y^6-1640\,\zeta(2)\,y^6\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
+68\,\zeta(2)^2\,y^5+320\,\zeta(3)\,y^5-960\,\zeta(2)\,\ln(2)\,y^5\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
-3480\,\zeta(2)\,y^5-255\,y^4-4400\,\zeta(2)\,y^4+20\,\zeta(2)^2\,y^4\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
-1200\,\zeta(2)\,\ln(2)\,y^4+3340\,\zeta(3)\,y^4+3600\,\zeta(2)\,y^3\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
-960\,\zeta(2)^2\,y^3+5560\,\zeta(2)\,y^2+1200\,\zeta(2)\,\ln(2)\,y^2\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
-3340\,\zeta(3)\,y^2+20\,\zeta(2)^2\,y^2+255\,y^2+68\,\zeta(2)^2\,y\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
+960\,\zeta(2)\,\ln(2)\,y-120\,\zeta(2)\,y-320\,\zeta(3)\,y+240\,\zeta(2)\,\ln(2)\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} +480\,\zeta(2)-60\,\zeta(3)-85)/[(y-1)^3(y+1)^5]\,y,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\eea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
and
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{d}_j  = && \nr 0 \quad \mbox{for} \quad j = 1 \ldots 7,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{d}_{8}  = && \nr -4\,(3\,y^2+2\,y+3)\,(y^2+1)/[(y-1)^2(y+1)^4]\,y\,H(0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2\,(5\,y^2+2\,y+5)/[(y-1)(y+1)^3]\,y,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{d}_{9}  = && \nr 4/3\,(3\,y^2+2\,y+3)/[(y-1)(y+1)^3]\,y\,H(0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+8/3\,(3\,y^2+2\,y+3)/[(y-1)(y+1)^3]\,y\,H(1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2/9\,(51\,y^2+26\,y+51)/[(y-1)(y+1)^3]\,y,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{d}_{10}  = && \nr -4/3\,(3\,y^4+14\,y^3-2\,y^2+14\,y+3)/[(y-1)^4(y+1)^2]\,y\,H(0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-16/[(y-1)^3(y+1)^3]\,y^3\,H(0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+2/9\,(51\,y^4+176\,y^3-70\,y^2+176\,y+51)/[(y-1)^3(y+1)^3]\,y,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{d}_{11}  = && \nr -4/[(y-1)(y+1)]\,y\,H(-1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+16/[(y-1)^2(y+1)^2]\,y^2\,H(0,-1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-32/[(y-1)^3(y+1)^3]\,y^3\,H(0,0,-1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/3\,(39\,y^5-223\,y^4-60\,\zeta(2)\,y^4-252\,\zeta(2)\,y^3+136\,y^3\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
-252\,\zeta(2)\,y^2+16\,y^2-60\,\zeta(2)\,y+65\,y-33)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{2cm} /[(y-1)^3(y+1)^4]\,y\,H(0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+24\,(3\,y^2+2\,y+3)/[(y-1)^2(y+1)^4]\,y^2\,H(-1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-8\,(5\,y^4+14\,y^3+42\,y^2+14\,y+5)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^5]\,y^2\,H(0,-1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(9\,y^4+112\,y^3+146\,y^2+196\,y+9)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^5]\,y^3\,H(0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+16\,(3\,y^2+8\,y+3)/[(y-1)^3(y+1)^3]\,y^2\,H(-1,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4\,(y^4+5\,y^3+38\,y^2+5\,y+1)/[(y-1)^3(y+1)^5]\,y^2\,H(0,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(9\,y^4+92\,y^3+130\,y^2+92\,y+9)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^2(y+1)^4]\,y\,H(1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+8\,(7\,y^4+29\,y^3+62\,y^2+29\,y+7)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^5]\,y^2\,H(0,1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2/3\,(3\,y^2+88\,y+3)/[(y-1)(y+1)^3]\,y\,H(1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+48\,(y^2+y+1)/[(y-1)^2(y+1)^4]\,y^2\,H(0,1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-8\,(5\,y^4+14\,y^3+30\,y^2+14\,y+5)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^5]\,y^2\,H(0,0,1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-1/18\,(687\,y^6-26\,y^5-648\,\zeta(2)\,y^5-1008\,\zeta(3)\,y^5-4752\,\zeta(3)\,y^4\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
-1296\,\zeta(2)\,y^4-687\,y^4-9216\,\zeta(3)\,y^3+52\,y^3\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
+1296\,\zeta(2)\,y^2-4752\,\zeta(3)\,y^2-687\,y^2+648\,\zeta(2)\,y\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} -1008\,\zeta(3)\,y-26\,y+687)/[(y-1)^3(y+1)^5]\,y,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{d}_{12}  = && \nr 32\,(y^2+y+1)/[(y+1)^3(y-1)]\,y\,H(-1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-8\,(3\,y^4+6\,y^3+14\,y^2+6\,y+3)/[(y-1)^2(y+1)^4]\,y\,H(0,-1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+64/[(y-1)^3(y+1)^3]\,y^3\,H(0,0,-1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-(125\,y^5-8\,\zeta(2)\,y^4+105\,y^4+64\,\zeta(2)\,y^3+82\,y^3-286\,y^2\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} +64\,\zeta(2)\,y^2+17\,y-8\,\zeta(2)\,y-43)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{2cm} /[(y-1)^3(y+1)^4]\,y\,H(0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+8\,(3\,y^4+7\,y^3+64\,y^2+7\,y+3)/[(y-1)^2(y+1)^4]\,y\,H(-1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+16\,(y^4-19\,y^3-16\,y^2-19\,y+1)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^5]\,y^2\,H(0,-1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+4\,(14\,y^5-175\,y^4-56\,y^3-14\,y^2+6\,y+9)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^5]\,y\,H(0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+32\,(y^2+17\,y+1)/[(y-1)^3(y+1)^3]\,y^2\,H(-1,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-8\,(y^4+5\,y^3+38\,y^2+5\,y+1)/[(y-1)^3(y+1)^5]\,y^2\,H(0,0,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-4\,(15\,y^4+16\,y^3+198\,y^2+16\,y+15)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^2(y+1)^4]\,y\,H(1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+16\,(y^4+37\,y^3+54\,y^2+37\,y+1)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^5]\,y^2\,H(0,1,0;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr-2\,(25\,y^2+86\,y+25)/[(y+1)^3(y-1)]\,y\,H(1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+8\,(3\,y^4+13\,y^3+64\,y^2+13\,y+3)/[(y-1)^2(y+1)^4]\,y\,H(0,1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+16\,(y^4-19\,y^3-28\,y^2-19\,y+1)\nn \\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm} /[(y-1)^3(y+1)^5]\,y^2\,H(0,0,1;y)\nn\\
%%%%%%%%%%%%%%%
&& \nr+1/2\,(61\,y^6+96\,\zeta(2)\,y^6+104\,\zeta(2)\,y^5+32\,\zeta(3)\,y^5+22\,y^5\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
+368\,\zeta(2)\,y^4-61\,y^4+1056\,\zeta(3)\,y^4+1664\,\zeta(3)\,y^3-44\,y^3\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
-368\,\zeta(2)\,y^2+1056\,\zeta(3)\,y^2-61\,y^2-104\,\zeta(2)\,y+22\,y\nn
\\
%%%%%%%%%%%%%%%
&& \nr\hspace{1cm}
+32\,\zeta(3)\,y-96\,\zeta(2)+61)/[(y-1)^3(y+1)^5]\,y. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\eea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\begin{boldmath}
\subsection{Threshold Expansions \label{subsec_threshold_expansion}}
\end{boldmath}
In this Section we  expand the 
one and 
two loop form factors near threshold $S\sim4m^2$, i.e. $y \rightarrow 1$ in powers of
\be
\beta = \sqrt{1-\frac{4m^2}{S}} \, .
\ee
Up to and including terms
of order $\beta^0$ 
the non-vanishing real and imaginary parts of the coefficents $\tilde{a}_i$, $\tilde{b}_i$, $\tilde{c}_j$ and $\tilde{d}_j$
($i=1\dots 3$, $j=1 \dots 12$) are:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{a}_2    &=& \frac{3\,\zeta (2)}{\beta}\, \,-\,2
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{a}_3    &=&  -\frac{6\,\zeta (2)}{\beta}\, \left( -1+\ln(\beta) +\ln(2)  \right)
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{a}_1    &=& -{\frac {1}{2\,\beta}},\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{a}_2    &=&   \frac{1}{\beta}\,\left( -1+\ln(\beta) +\ln(2)  \right)
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{a}_3    &=&   \frac{1}{\beta}\,\Big( - \ln^2 (\beta)-2+2\,
\ln(\beta) -\ln^2 (2) \nn \\
&& -2\,\ln(2) \ln(\beta) +2\,\ln(2) +\zeta (2)  \Big)
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{b}_2    &=& -1,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{b}_3    &=& -\frac{6\,\zeta (2)}{\beta}\, +2
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{b}_2    &=& \frac{1}{\beta,}\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{b}_3    &=& \frac{1}{\beta}\,\left( 2-2\,\ln(\beta) -2\,\ln(2)  \right)
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{c}_4    &=& -\frac{3\,\zeta (2)}{4\, \beta^2}\, -\frac{3}{2}\,\zeta (2) 
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{c}_8    &=& \frac{3\,\zeta (2)}{\beta^2}  \left(
  -1+\ln(\beta) +\ln(2)  \right) \nn \\
&& +3\,\zeta (2)  \left( 2\,\ln(\beta) -1+2\,\ln(2)  \right),\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{c}_9    &=& \frac{2 \, \zeta (2)}{3\,\beta} \left( -11+6\,\ln(\beta) +6\,\ln(2)  \right) +{\frac {14}{9}}
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{c}_{10} &=& {\frac {80}{9}}-\frac{16}{3}\,\zeta (2) 
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{c}_{11} &=& - \frac{\zeta (2)}{6\,\beta}  \left(
  66\,\ln(\beta) -97+66\,\ln(2)  \right)-\frac{9}{2}\,\zeta
\left( 3 \right) -{\frac {101}{18}}\nn\\
&&-18\,\zeta (2) \ln(2)+{\frac {89}{6}}\,\zeta (2) -4\,\zeta (2) \ln(\beta)  
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{c}_{12} &=& \frac{3\,\zeta (2)}{2\,\beta^2} \Big( -4\,
  \ln^2 (2)+3\,\zeta (2) +8\,\ln(2) -8\,\ln(2) \ln(\beta)\nn \\
&&  -4+8\,\ln(\beta) -4\, \ln^2 (\beta) \Big) - \frac{6\,\zeta
  (2)}{\beta}-12\,\zeta (2)  \ln^2 (2)\nn \\
&& -24\,\zeta (2)\ln(2) \ln(\beta) -12\,\zeta ( 2)  \ln^2 (\beta) +{\frac {23}{6}}\nn \\
&& +9\, \zeta^2 (2) -8\,\zeta (2) -{\frac {27}{4}}\,\zeta (3)
+2\,\zeta (2) \ln(\beta) + 21\,\zeta (2) \ln(2) 
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{c}_1    &=& -\frac{1}{6\,\beta},\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{c}_3    &=& \frac{11}{24\,\beta},\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{c}_5    &=& \frac{5}{18\,\beta},\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{c}_7    &=& -\frac{31}{72\,\beta},\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{c}_8    &=& -\frac{3\,\zeta (2)}{2\,\beta^2}+\frac{1}{\beta}-3\,\zeta (2) 
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{c}_9    &=& \frac{1}{\beta}\Big( {\frac
    {62}{27}}-{\frac {22}{9}}\,\ln(2) +\frac{2}{3}\, \ln^2
  (2)-{\frac {22}{9}}\,\ln(\beta) \nn \\
&& +\frac{2}{3}\, \ln^2 (\beta)+\frac{4}{3}\,\ln(2) \ln(\beta)  \Big) 
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{c}_{11} &=& \frac{1}{\beta}\,\Big( {\frac {97}{18}}\,\ln(\beta) -{\frac
    {239}{54}}-{\frac {11}{6}}\, \ln^2 (\beta)-{\frac {11}{6}}\, \ln^2
  2\nn \\
&& +{\frac {97}{18}}\,\ln(2) -\frac{11}{3}\,\ln(2) \ln(\beta)  \Big)+2\,\zeta (2) 
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{c}_{12} &=& \frac{6\,\zeta (2)}{\beta^2}
\left(-1+\ln(\beta) +\ln(2)  \right) + \frac{1}{\beta}\, \left( 2-2\,\ln(\beta)
  -2\,\ln(2)  \right) \nn \\
&& +\zeta (2)  \left( 12\,\ln(2) +12\,\ln(\beta) -1 \right) 
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{d}_8    &=&  \frac{3\,\zeta (2)}{\beta^2}+\frac{3}{2}\,\zeta (2) 
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{d}_9    &=&  \frac{4\,\zeta (2)}{\beta}-{\frac {11}{9}}
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{d}_{10} &=& -\frac{8}{3}\,\zeta (2) +{\frac {49}{9}}
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{d}_{11} &=& -\frac{11\,\zeta (2)}{\beta} + {\frac {26}{3}}\,\zeta (2) -18\,\zeta
\left( 2 \right) \ln(2) \nn \\
&&-8\,\zeta (2) \ln(\beta) -\frac{15}{2}\,\zeta (3) +{\frac {151}{36}}
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{d}_{12} &=& -\frac{6\,\zeta (2)}{\beta^2}  \left( 2\,\ln(\beta) -1+2\,\ln(2)  \right)- \frac{3\,\zeta (2)}{\beta}-15\,\zeta (2) \ln(2) \nn \\
&& -\frac{9}{2}\,\zeta (2) -{\frac {45}{4}}\,\zeta (3) -20\,\zeta (2) \ln(\beta) +{\frac {41}{12}}
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{d}_8    &=& \frac{1}{2\,\beta},\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{d}_9    &=&  \frac{1}{\beta}\,\left( \frac{4}{3}\,\ln(2) +\frac{4}{3}\,\ln(\beta) -{\frac {16}{9}} \right)
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{d}_{11} &=&   \frac{1}{\beta}\,\left( -\frac{11}{3}\,\ln(2) +{\frac {32}{9}}-\frac{11}{3}\,\ln(\beta)  \right)+4\,\zeta (2) 
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{d}_{12} &=&  \frac{6\,\zeta (2)}{\beta^2}+ \frac{1}{\beta}\,\left( -3-\ln(2) -\ln(\beta)  \right) +10\,\zeta (2) 
.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\eea
%
These expansions are relevant, for instance,  for studies of $Q\bar{Q}$ production near
threshold. For a study using the one-loop axial form factors see \cite{KuhnTeubner}.
%
\begin{boldmath}
\subsection{Asymptotic Expansions \label{subsec_asymptotic_expansion}}
\end{boldmath}
In this section, we write down the expansions of the form factors in
the limit $S \gg m^2$, i.e. $y \rightarrow 0$. Putting
$r=\frac{S}{m^2}$ and $L=\ln r$, 
and keeping terms up to $r^{-2}$ we find for the coefficients that
are non-vanishing to that order:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{a}_1    &=&  \frac{1}{r^2}\,\left( -3+2\,L \right) -\frac{2}{r}-1+L,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{a}_2    &=& \frac{1}{r^2}\,\left( -8+13\,L+8\,\zeta (2)
  -{L}^{2} \right) + \frac{1}{r}\,\left( -1+5\,L \right)
\nn \\
&& -2+\frac{3}{2}\,L-\frac{1}{2}\,{L}^{2}+4\,\zeta (2) 
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{a}_3    &=&  \frac{1}{r^2}\,\left( 4\,\zeta (3)
  +52\,\zeta (2)
  +\frac{1}{3}\,{L}^{3}-\frac{13}{2}\,{L}^{2}-5+32\,L-8\,L\zeta (2)
\right) \nn \\
&& + \frac{1}{r}\,\left( 20\,\zeta (2) -\frac{5}{2}\,{L}^{2}-3+6\,L \right)
\nn \\
&& +6\,\zeta (2) -4+2\,\zeta (3) -\frac{3}{4}\,{L}^{2}+4\,L+\frac{1}{6}\,{L}^{3}-4\,L\zeta (2) 
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{a}_1    &=& -\frac{2}{r^2}-1
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{a}_2    &=&  \frac{1}{r^2}\,\left( -13+2\,L \right) -\frac{5}{r}-{\frac {3}{2}}+L
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{a}_3    &=&  \frac{1}{r^2}\, \left( 13\,L-32-{L}^{2}+4\,\zeta \left( 2
  \right)  \right)+ \frac{1}{r}\,\left( -6+5\,L \right)
\nn \\
&& +2\,\zeta (2) -\frac{1}{2}\,{L}^{2}-4+\frac{3}{2}\,L
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{b}_2    &=&  \frac{1}{r^2}\,\left( 12-4\,L \right) +  \frac{1}{r}\,\left( 4-6\,L \right)
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{b}_3    &=&  \frac{1}{r^2}\,\left( 12-32\,L+2\,{L}^{2}-16\,\zeta (2)  \right) + \frac{1}{r}\left( 8-12\,L+3\,{L}^{2}-24\,\zeta (2)  \right) 
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{b}_2    &=& \frac{4}{r^2}+\frac{6}{r}
\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{b}_3    &=& \frac{1}{r^2}\,\left( 32-4\,L \right) +  \frac{1}{r}\,\left( -6\,L+12 \right)
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{c}_1    &=&  \frac{1}{r^2}\,\left( -1+\frac{2}{3}\,L \right)-\frac{2}{3\,r}-{\frac {1}{3}}+\frac{1}{3}\,L
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{c}_3    &=&  \frac{1}{r^2}\,\left( {\frac {11}{4}}-{\frac {11}{6}}\,L \right) +\frac{11}{6\,r}+{\frac {11}{12}}-{\frac {11}{12}}\,L
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{c}_4    &=&  \frac{1}{r^2}\,\left( 5+2\,{L}^{2}-12\,\zeta \left( 2
  \right) -5\,L \right) + 
\frac{1}{r}\,\left( -2\,L+2 \right)\nn \\
&& +\frac{1}{2}\,{L}^{2}-3\,\zeta (2) +{\frac {1}{2}}-L
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{c}_5    &=&  \frac{1}{r^2}\,\left( {\frac {5}{3}}-{\frac {10}{9}}\,L \right) +\frac{10}{9\,r}+{\frac {5}{9}}-\frac{5}{9}\,L
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{c}_7    &=&  \frac{1}{r^2}\,\left( \frac{13}{2}\,\zeta (2)
  -2\,\zeta (3) -{L}^{2}+\frac{1}{3}\,{L}^{3}+{\frac{47}{9}}\,L-6\,L\zeta (2) -{\frac {55}{12}} \right)\nn \\
&& + \frac{1}{r}\,\left( -{\frac {67}{18}}+\zeta (2)  \right) +\frac{1}{2}\,\zeta (2) -{\frac {49}{36}}-\frac{1}{2}\,L\zeta (2) +{\frac {67}{36}}\,L-\frac{1}{2}\,\zeta (3) 
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{c}_8    &=&  \frac{1}{r^2}\,\left( -2\,{L}^{3}+16-116\,\zeta \left( 2
  \right) +{\frac {37}{2}}\,{L}^{2}+40\,L\zeta (2)
  -{\frac {79}{2}}\,L \right) \nn \\
&& + \frac{1}{r}\,\left( -38\,\zeta (2) +6\,{L}^{2}-9\,L+5 \right)
\nn \\
&& +2-\frac{1}{2}\,{L}^{3}-\frac{7}{2}\,L+2\,{L}^{2}-13\,\zeta (2) +10\,L\zeta (2) 
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{c}_9    &=&  \frac{1}{r^2}\,\left( -{\frac {356}{9}}\,\zeta \left( 2
  \right) -\frac{8}{3}\,\zeta (3) -\frac{2}{9}\,{L}^{3}+\frac{8}{3}\,L\zeta
  \left( 2 \right) +{\frac {49}{9}}\,{L}^{2}+{\frac {23}{2}}-{\frac
    {959}{27}}\,L \right) \nn \\
&& + \frac{1}{r}\,\left( -{\frac {89}{9}}\,L+\frac{5}{3}\,{L}^{2}+{\frac {59}{27}}-{\frac
    {32}{3}}\,\zeta (2)  \right) \nn \\
&& -\frac{1}{9}\,{L}^{3}+{\frac {19}{18}}\,{L}^{2}-{\frac {209}{54}}\,L-{\frac {64}{9}}\,\zeta (2) -\frac{4}{3}\,\zeta (3) +{\frac {106}{27}}+\frac{4}{3}\,L\zeta (2) 
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{c}_{10} &=&   \frac{1}{r^2}\,\left( -\frac{16}{3}\,L\zeta (2)
  +{\frac {1153}{18}}+{\frac {229}{9}}\,{L}^{2}+12\,\zeta (2) +\frac{4}{9}\,{L}^{3}-{\frac {2230}{27}}\,L \right)\nn
\\
&& + \frac{1}{r}\,\left( {\frac {385}{27}}-{\frac {161}{9}}\,L+{\frac
    {17}{3}}\,{L}^{2}+\frac{4}{3}\,\zeta (2)  \right)
\nn \\
&& +{\frac {19}{18}}\,{L}^{2}-{\frac {22}{3}}\,\zeta (2) -{\frac {265}{54}}\,L+{\frac {383}{27}}-\frac{1}{9}\,{L}^{3}+\frac{4}{3}\,L\zeta (2) 
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{c}_{11} &=&   \frac{1}{r^2}\,\left( {\frac {241}{4}}+{\frac {98}{5}}\,
  \zeta^2(2)+144\,\zeta \left( 2
  \right) \ln(2)+{\frac {59}{3}}\,L\zeta (2) +{\frac
    {173}{18}}\,{L}^{2}+{\frac {17}{18}}\,{L}^{3}\right.\nn \\
&& \left.-{\frac
    {4111}{18}}\,\zeta (2) -110\,L\zeta (3)
  +{\frac {1319}{6}}\,\zeta (3) -10\,{L}^{2}\zeta \left(
    2 \right) +{\frac {15619}{108}}\,L \right)\nn \\
&& +  \frac{1}{r}\,\left( \frac{1}{24}\,{L}^{4}+{\frac {69}{10}}\, \zeta^2(2)-14\,L\zeta (3) +{\frac
    {1189}{36}}\,L+{\frac {5}{12}}\,{L}^{2}-{\frac {137}{3}}\,\zeta
  \left( 2 \right)  \right. \nn \\
&& \left.+55\,\zeta (3)+36\,\zeta \left( 2
  \right) \ln(2)+7\,L\zeta (2) -\frac{7}{2}\,{L}^{2}\zeta (2) -{\frac {157}{108}} \right)\nn \\
&& -{\frac {1595}{108}}-\frac{11}{3}\,L\zeta (2) -{\frac
  {233}{72}}\,{L}^{2}+{\frac {11}{36}}\,{L}^{3}+{\frac {67}{6}}\,\zeta(3) -{\frac {63}{20}}\,  \zeta^2 (2)\nn \\
&& +\frac{1}{2}\,{L}^{2}\zeta (2) +{\frac {173}{9}}\,\zeta (2) -\frac{13}{2}\,L\zeta (3) +6\,\zeta (2) \ln(2)+{\frac {2545}{216}}\,L
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{c}_{12} &=&  \frac{1}{r^2}\,\left( {\frac {841}{4}}+{\frac {351}{5}}\,
  \zeta^2(2)-288\,\zeta \left( 2
  \right) \ln(2)+{L}^{4}+350\,L\zeta (2) -625\,\zeta
  \left( 2 \right) \right. \nn \\
&& \left.-148\,L\zeta (3) +95\,{L}^{2}-{\frac
    {85}{6}}\,{L}^{3}+190\,\zeta (3) -55\,{L}^{2}\zeta
  \left( 2 \right) +{\frac {37}{2}}\,L \right) \nn \\
&& +  \frac{1}{r}\,\left( -\frac{16}{3}\,{L}^{3}-74\,\zeta (2)
  -\frac{1}{12}\,{L}^{4}+\frac{9}{5}\, \zeta^2 (2)-20\,L\zeta (3) -{\frac {91}{4}}\,L\right. \nn \\
&& \left. +{\frac
    {59}{4}}\,{L}^{2}+{L}^{2}\zeta (2) +10\,\zeta \left(
    3 \right) -72\,\zeta (2) \ln(2)+113\,L\zeta \left( 2
  \right) +{\frac {107}{4}} \right)\nn \\
&& +{\frac {23}{2}}-\frac{5}{3}\,{L}^{3}+31\,L\zeta (2)
+{\frac {55}{8}}\,{L}^{2}-32\,\zeta (2) -11\,\zeta (3) +{\frac {68}{5}}\,  \zeta^2 (2)\nn \\
&& -12\,{L}^{2}\zeta (2) +{\frac {7}{24}}\,{L}^{4}+8\,L\zeta (3) -12\,\zeta (2) \ln(2)-{\frac {85}{8}}\,L
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{c}_1    &=& -\frac{2}{3\,r^2}-{\frac {1}{3}}
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{c}_3    &=& \frac{11}{6\,r^2}+{\frac {11}{12}}
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{c}_4    &=&  \frac{1}{r^2}\,\left( 5-4\,L \right)+\frac{2}{r}+1-L
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{c}_5    &=& \frac{10}{9\,r^2}+{\frac {5}{9}}
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{c}_7    &=&  \frac{1}{r^2}\,\left( 2\,\zeta (2) -{\frac {47}{9}}+2\,L-{L}^{2} \right) +\frac{1}{2}\,\zeta (2) -{\frac {67}{36}}
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{c}_8    &=&   \frac{1}{r^2}\,\left( -16\,\zeta (2) +6\,{L}^{2}+{\frac {79}{2}}-37\,L \right)+  \frac{1}{r}\,\left(
  9-12\,L \right)\nn \\
&& +{\frac {7}{2}}-4\,L-4\,\zeta (2) +\frac{3}{2}\,{L}^{2}
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{c}_9    &=&  \frac{1}{r^2}\,\left( {\frac
    {959}{27}}+\frac{2}{3}\,{L}^{2}-{\frac {98}{9}}\,L \right)
+ \frac{1}{r}\left( -\frac{10}{3}\,L+{\frac {89}{9}} \right)\nn \\
&& -{\frac {19}{9}}\,L+{\frac {209}{54}}+\frac{1}{3}\,{L}^{2},\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{c}_{10} &=&   \frac{1}{r^2}\,\left( -\frac{4}{3}\,{L}^{2}-{\frac
    {458}{9}}\,L+{\frac {2230}{27}} \right)+  \frac{1}{r}\,\left(
  {\frac {161}{9}}-{\frac {34}{3}}\,L \right)\nn \\
&& +\frac{1}{3}\,{L}^{2}-{\frac {19}{9}}\,L+{\frac {265}{54}}
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{c}_{11} &=&  \frac{1}{r^2}\,\left( 110\,\zeta (3) -{\frac
    {15619}{108}}-{\frac {173}{9}}\,L+20\,L\zeta (2)
  -31\,\zeta (2) -{\frac {17}{6}}\,{L}^{2} \right)
\nn \\
&& +  \frac{1}{r}\,\left( -\frac{1}{6}\,{L}^{3}-7\,\zeta (2) +5\,L\zeta \left(
    2 \right) -{\frac {1189}{36}}+14\,\zeta (3) -\frac{5}{6}\,L
\right)\nn \\
&& -{\frac {11}{12}}\,{L}^{2}+{\frac {233}{36}}\,L+\frac{13}{2}\,\zeta (3) -L\zeta (2) -{\frac {2545}{216}}
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{c}_{12} &=&  
\frac{1}{r^2}\,\left( 62\,L\zeta (2) +{\frac{85}{2}}\,{L}^{2}-{\frac {37}{2}}+148\,\zeta (3)
  -4\,{L}^{3}-180\,\zeta (2) -190\,L \right)\nn \\
&& +  \frac{1}{r}\,\left( 20\,\zeta (3) -49\,\zeta (2)
  +16\,{L}^{2}+\frac{1}{3}\,{L}^{3}+2\,L\zeta \left(2 \right) +{\frac
    {91}{4}}-{\frac {59}{2}}\,L \right)\nn \\
&& +{\frac{85}{8}}-8\,\zeta (3) +5\,{L}^{2}-{\frac {55}{4}}\,L-\frac{7}{6}\,{L}^{3}+10\,L\zeta (2) -11\,\zeta (2) 
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{d}_8    &=&  \frac{1}{r^2}\, \left( -20-4\,{L}^{2}+24\,\zeta \left( 2
  \right) +28\,L \right)\nn \\
&& +  \frac{1}{r}\,\left( -6\,{L}^{2}+10\,L-4+36\,\zeta (2)  \right)
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{d}_9    &=&   \frac{1}{r^2}\,\left( {\frac {196}{9}}\,L+{\frac{32}{3}}
  \,\zeta (2) - {\frac{44}{3}}-\frac{4}{3}\,{L}^{2} \right)\nn \\
&& +  \frac{1}{r}\,\left( -{\frac {76}{9}}+16\,\zeta (2) -2\,{L}^{2}+{\frac {34}{3}}\,L \right)
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{d}_{10} &=&  \frac{1}{r^2}\, \left( -44-{\frac
    {52}{3}}\,{L}^{2}-16\,\zeta (2) +{\frac {628}{9}}\,L
\right)\nn \\
&& +  \frac{1}{r}\left( -8\,\zeta (2) +{\frac {32}{9}}-2\,{L}^{2}+{\frac {34}{3}}\,L \right)
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{d}_{11} &=&   \frac{1}{r^2}\,\left( -96\,\zeta (2) \ln
  2+{\frac {157}{3}}-{\frac {43}{3}}\,{L}^{2}-{\frac
    {401}{9}}\,L-36\,L\zeta (2) -{\frac {138}{5}}\,
  \zeta^2(2)\right. \nn \\
&& \left.-\frac{1}{6}\,{L}^{4}+56\,L\zeta
  \left( 3 \right) +14\,{L}^{2}\zeta (2) +{\frac
    {680}{3}}\,\zeta (2) -164\,\zeta (3)
\right)\nn \\
&& +  \frac{1}{r}\,\Big( -12\,\zeta (3) -{\frac
    {229}{6}}\,L-12\,\zeta (2) \nn \\
&& -24\,\zeta (2) \ln(2)+\frac{11}{2}\,{L}^{2}+{\frac {242}{9}} \Big)
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Re \, \tilde{d}_{12} &=&   \frac{1}{r^2}\,\bigg( -115+43\,L+360\,\zeta \left( 2
  \right) -100\,L\zeta (2) +16\,L\zeta (3)
  -12\,{L}^{2}\zeta (2)  \nn \\
&& +192\,\zeta (2) \ln
  2+{\frac {68}{5}}\, \zeta^2 (2)+\frac{1}{3}\,{L}^{4}-64\,\zeta (3) +4\,{L}^{3}-67\,{L}^{2}
\bigg)\nn \\
&& +  \frac{1}{r}\,\left( 96\,\zeta (2) +48\,\zeta (2) \ln
  2+{\frac{61}{2}}\,L+6\,{L}^{3}-17\right. \nn \\
&& \left. -12\,\zeta (3) -{\frac {43}{2}}\,{L}^{2}-120\,L\zeta (2)  \right)
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{d}_8    &=&   \frac{1}{r^2}\,\left( -28+8\,L \right)+  \frac{1}{r}\,\left( -10+12\,L \right)
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{d}_9    &=&  \frac{1}{r^2}\,\left( -{\frac {196}{9}}+\frac{8}{3}\,L \right) +  \frac{1}{r}\,\left( -{\frac {34}{3}}+4\,L \right)
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{d}_{10} &=&   \frac{1}{r^2}\,\left( -{\frac {628}{9}}+{\frac{104}{3}}\,L \right) +  \frac{1}{r}\,\left( -{\frac {34}{3}}+4\,L \right)
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{d}_{11} &=&   \frac{1}{r^2}\,\left( {\frac{401}{9}}+\frac{2}{3}\,{L}^{3}+{\frac {86}{3}}\,L-20\,L\zeta
  \left( 2 \right) +36\,\zeta (2) -56\,\zeta \left( 3
  \right)  \right)\nn \\
&& + \frac{1}{r}\,\left( {\frac {229}{6}}-11\,L \right) 
,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Im \, \tilde{d}_{12} &=&   \frac{1}{r^2}\,\left( -43-16\,\zeta (3)
  +52\,\zeta (2) -\frac{4}{3}\,{L}^{3}+8\,L\zeta (2)
  -12\,{L}^{2}+134\,L \right)\nn \\
&& +  \frac{1}{r}\,\left( 43\,L-{\frac {61}{2}}+48\,\zeta (2) -18\,{L}^{2} \right)
.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\eea
%
The limit $r \to \infty$ corresponds to the massless limit $m \to 0$.
Therefore the chirality-flipping form factors $F_2$ and $G_2$ must
be of order $1/r$, and those terms in the chirality-conserving
form factor $F_1$ that survive the limit  $r \to \infty$ must be equal
to the corresponding terms in $G_1$. The asymptotic expansions given
above and in \cite{bbghlmr} satisfy these constraints.

All the results of this Section can be obtained in an electronic
form by downloading the source of this manuscript from
http://www.arxiv.org.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Summary}

In this paper we calculated the axial vector form factors $G_1$ and
$G_2$ to second order in the QCD coupling, excluding the contributions
from Fig.~\ref{triangle-graph}~(a) and (b). The results for the two
form factors were obtained keeping the full dependence  on the mass of
the heavy quark, as well as on the arbitrary momentum transfer. \par
The renormalized
form factors are expressed in terms of the ${\rm \overline{MS}}$ coupling
$\alpha_s$ for $(N_f +1)$ quark flavors and of the on-shell mass of the
heavy quark.
The expressions for the unsubtracted as well as the UV-renormalized form
factors are given in  closed analytic form as a Laurent expansion in
$\epsilon=(4-D)/2$. The coefficients of this expansion have a suitable
representation in terms of 1-dimensional harmonic polylogarithms.
Poles in $\epsilon$, which correspond to soft and collinear divergences,
are still present  in $G_{1,2,R}$,  like in $F_{1,2,R}$.
Depending on the
observable considered, these divergences must cancel either among
each other or against  the divergences arising from the real radiation,
which in this paper was not  taken into account.
\par
As already discussed in the introduction and in
our preceeding paper \cite{bbghlmr} our results for the vector and axial 
vector form
factors are part of the order $\alpha_S^2 $ QCD corrections
to the differential electroproduction cross sections of heavy quarks 
and, moreover,
have a number of immediate  applications. These include studies of the 
extrapolation of the continuum amplitudes
to the ${\bar Q} Q$ production threshold and studies concerning the 
generic singularity
structure of QCD amplitudes with massive quarks.
Of general interest is the computation of the
NNLO QCD corrections to the forward-backward asymmetry of heavy quarks.
We shall report on this in a future publication.


\section*{Acknowledgement}

We are grateful to J.~Vermaseren for his kind assistance in the use
of the algebra manipulating program {\tt FORM}~\cite{FORM}, by which
many of our calculations were carried out.
This work was partially supported by the European Union under
contract HPRN-CT-2000-00149, by Deutsche Forschungsgemeinschaft (DFG), 
SFB/TR9, by DFG-Graduiertenkolleg RWTH Aachen, by the Swiss 
National Funds (SNF) under contract 200021-101874,  and by the USA DoE under 
the grant DE-FG03-91ER40662, Task J. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{thebibliography}{99} 
\def    \np     #1#2#3{{\it Nucl. Phys.} {\bf #1} (19#2) #3}
\def    \nptwoth     #1#2#3{{\it Nucl. Phys.} {\bf #1} (20#2) #3}
\def    \nptwothnz     #1#2#3{{\it Nucl. Phys.} {\bf #1} (19#2) #3}
\def    \prep   #1#2#3{{\it Phys. Rep.} {\bf #1}  (19#2) #3}
\def    \pl     #1#2#3{{\it Phys. Lett.} {\bf #1} (19#2) #3}
\def    \pltwoth     #1#2#3{{\it Phys. Lett.} {\bf #1} (20#2) #3}
\def    \plold  #1#2#3{{\it Phys. Lett.} {\bf #1B} (19#2) #3}
\def    \prl    #1#2#3{{\it Phys. Rev. Lett.} {\bf #1}  (19#2) #3} 
\def    \pr     #1#2#3{{\it Phys. Rev.} {\bf #1}  (19#2) #3}
\def    \prd    #1#2#3{{\it Phys. Rev.} {\bf D#1}  (19#2) #3} 
\def    \zeit   #1#2#3{{\it Z. Phys.} {\bf C#1}  (19#2) #3}
\def    \cmp    #1#2#3{{\it Comm. Math. Phys.} {\bf #1}  (19#2) #3}
\def    \ibid   #1#2#3{{\it ibid.} {\bf #1} (19#2) #3}
\def    \nc     #1#2#3{{\it Nuovo Cim.} {\bf #1} (19#2) #3}
\def    \acta   #1#2#3{{\it Acta Phys. Polon.} {\bf #1} (19#2) #3}
\def    \tmp    #1#2#3{{\it Theor. Math. Phys.} {\bf #1} (19#2) #3}
\def    \eur    #1#2#3{{\it Eur. Phys.  J.} {\bf C#1} (19#2) #3}
\def    \comp    #1#2#3{{\it Comput. Phys. Commun.} {\bf #1} (20#2) #3}
\def    \hepph  #1 {{\tt hep-ph/#1}}
\def    \hepex  #1 {{\tt hep-ex/#1}}
\def    \mathph  #1 {{\tt math-ph/#1}}
\parskip 0pt
\itemsep=0pt


%%%%%%%%%%%%%% FB-Asym %%%%%%%%%%%%%%%%%%%
%
\bibitem{bbghlmr}
W.~Bernreuther, R.~Bonciani, T.~Gehrmann, R.~Heinesch, T.~Leineweber, P.~Mastrolia and E.~Remiddi,
%``Two-loop QCD Corrections to the Heavy Quark Form Factors: the Vector
%Contributions,''
\nptwoth{B706}{05}{245} 
(\hepph{0406046}).
%Nucl.\ Phys.\ B {\bf 706} (2005) 245
%{\tt hep-ph/0406046}, to appear in Nucl.\ Phys.\ B.
%%CITATION = HEP-PH 0406046;%%


\bibitem{EWWorkingGroup}
 The LEP collaborations ALEPH, DELPHI, L3, OPAL, the LEP Electroweak
 Working Group, and the SLD Heavy Flavour Group, {\it A Combination of
 Preliminary Electroweak Measurements and Constraints on the Standard
 Model}, report LEPEWWG/2003-01, April 2003.

%\cite{Freitas:2004mn}
\bibitem{Freitas:2004mn}
A.~Freitas and K.~Moenig,
%``Corrections to quark asymmetries at LEP,''
(\hepph{0411304}).
%%CITATION = HEP-PH 0411304;%%

\bibitem{tesla}
 J.~A.~Aguilar-Saavedra {\it et al.}  [ECFA/DESY LC Physics Working Group
 Collaboration], ``TESLA Technical Design Report Part III: Physics at an
 $e^+e^-$ Linear Collider'', DESY-report 2001-011 (\hepph{0106315}).
%%CITATION = HEP-PH 0106315;%%

\bibitem{triangle}
W.~Bernreuther, R.~Bonciani, T.~Gehrmann,
R.~Heinesch, T.~Leineweber and
E.~Remiddi, in preparation.

\bibitem{HV}
G.~'t Hooft and M.~J.~G.~Veltman,
%``Regularization And Renormalization Of Gauge Fields,''
\nptwothnz{B44}{72}{189}
%Nucl.\ Phys.\ B {\bf 44} (1972) 189.
%%CITATION = NUPHA,B44,189;%%

\bibitem{Larin}
S.~A.~Larin,
%``The Renormalization of the axial anomaly 
% in dimensional regularization,''
\pl{B303}{93}{113}
%Phys.\ Lett.\ B {\bf 303} (1993) 113
(\hepph{9302240}).
%%CITATION = HEP-PH 9302240;%%

%%%%%%%%%%%%%%% Harm-Polylog %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bibitem{Polylog}
  E. Remiddi and J. A. M. Vermaseren, {\it Int. J. Mod. Phys.} 
  {\bf A15} (2000) 725 ({\tt hep-ph/9905237}). 
%%CITATION = HEP-PH 9905237;%%

\bibitem{Polylog3}
  T. Gehrmann and E. Remiddi, \comp{141}{01}{296} 
  ({\tt hep-ph/0107173}).
%%CITATION = HEP-PH 0107173;%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bibitem{Broadhurst}
D.~J.~Broadhurst, N.~Gray and K.~Schilcher,
\zeit{52}{91}{111}
%Z.\ Phys.\ C {\bf 52} (1991) 111.
%%CITATION = ZEPYA,C52,111;%%

\bibitem{Melnikov}
K.~Melnikov and T.~van Ritbergen, \nptwoth{B591}{00}{515}
%Nucl.\ Phys.\ B {\bf 591} (2000) 515
(\hepph{0005131}).
%%CITATION = HEP-PH 0005131;%%

%%%%%%%%%%%%%% 2-loop %%%%%%%%%%%%%%%%%%%

\bibitem{RoPieRem1}
  R. Bonciani, P. Mastrolia and E. Remiddi, \nptwoth{B661}{03}{289} 
[Erratum-ibid. \nptwoth{B702}{04}{259}] (\hepph{0301170}).
%%CITATION = HEP-PH 0301170;%%

\bibitem{RoPieRem2}
  R. Bonciani, P. Mastrolia and E. Remiddi, \nptwoth{B676}{04}{399}
  (\hepph{0307295}).
%%CITATION = HEP-PH 0307295;%%

\bibitem{RoPieRem3}
  R. Bonciani, P. Mastrolia and E. Remiddi, \nptwoth{B690}{04}{138}
  (\hepph{0311145}).
%%CITATION = HEP-PH 0311145;%%

%%%%%%%%%%%%%%%%

\bibitem{Lap}
  S. Laporta and E. Remiddi, \pl{B379}{96}{283} (\hepph{9602417}).\\
%%CITATION = APPOA,B28,959;%%
  S.~Laporta, {\it Int.\ J.\ Mod.\ Phys.}  {\bf A 15} (2000) 5087
 (\hepph{0102033}).
%%CITATION = HEP-PH 0102033;%%


%%%%%%%%%%%%%%%%%% Dim Reg %%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\bibitem{DimReg}
%  G. 't Hooft and M. Veltman, \np{B44}{72}{189}.\\
%%%CITATION = NUPHA,B44,189;%%
%  C. G. Bollini and J. J. Giambiagi, \plold{40}{72}{566}; 
%  \nc{12B}{72}{20}. \\
%%%CITATION = NUCIA,B12,20;%%
%  J. Ashmore, {\it Lett. Nuovo Cimento} {\bf 4} (1972) 289.\\
%%%CITATION = NCLTA,4,289;%%
%  G. M. Cicuta and E. Montaldi, {\it Lett. Nuovo Cimento} {\bf 4} 
%  (1972) 289. \\
%%%CITATION = NCLTA,4,329;%%
%  R. Gastmans and R. Meuldermans, \np{B63}{73}{277}.
%%%CITATION = NUPHA,B63,277;%%


%%%%%%%%%%%%%%%%%% IBP %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

\bibitem{Chet}
  F.V. Tkachov, \pl{B100}{81}{65}.\\
%%CITATION = PHLTA,B100,65;%%
  K.G. Chetyrkin and F.V. Tkachov, \np{B192}{81}{159}.
%%CITATION = NUPHA,B192,159;%%


%%%%%%%%%%%%%%%%%% LI %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bibitem{Rem3}
  T. Gehrmann and E. Remiddi, \nptwoth{B580}{00}{485} 
  ({\tt hep-ph/9912329}). 
%%CITATION = HEP-PH 9912329;%%
  

%%%%%%%%%%%%%%%%%% Diff. Eqs. Method %%%%%%%%%%%%%%

\bibitem{Kot}
  A. V. Kotikov, \pl{B254}{91}{158}.
%%CITATION = PHLTA,B254,158;%%

\bibitem{Rem1} 
  E. Remiddi, \nc{110A}{97}{1435} ({\tt hep-th/9711188}).
%%CITATION = HEP-TH 9711188;%%

\bibitem{Rem2}
  M. Caffo, H. Czy\.{z}, S. Laporta and E. Remiddi, 
  \acta{B29}{98}{2627} ({\tt hep-ph/9807119}).\\
%%CITATION = HEP-TH 9807119;%%
  M. Caffo, H. Czy\.{z}, S. Laporta and E. Remiddi, 
  \nc{A111}{98}{365} ({\tt hep-ph/9805118}).
%%CITATION = HEP-TH 9805118;%%

%\cite{Jersak:1981sp}
\bibitem{Jersak:1981sp}
J.~Jersak, E.~Laermann and P.~M.~Zerwas,
%``Electroweak Production Of Heavy Quarks In E+ E- Annihilation,''
\prd{25}{82}{1218}
%Phys.\ Rev.\ D {\bf 25} (1982) 1218
[Erratum-ibid. \prd{36}{87}{310}].
%[Erratum-ibid.\ D {\bf 36} (1987) 310].
%%CITATION = PHRVA,D25,1218;%%

\bibitem{KuhnTeubner}
J.~H.~K\"uhn and T.~Teubner,
%``Axial contributions at the top threshold,''
\eur{9}{99}{221}
%Eur.\ Phys.\ J.\ C {\bf 9} (1999) 221
(\hepph{9903322}).
%%CITATION = HEP-PH 9903322;%%


%%%%%%%%%%%%%%%%%%% Programs %%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bibitem{FORM} J.A.M.\ Vermaseren, Symbolic Manipulation with
               {\tt FORM}, Version 2, CAN, Amsterdam, 1991; 
               New features of {\tt FORM}, (\mathph{0010025}).
%%CITATION = MATH-TH 0010025;%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{thebibliography}

\end{fmffile}

\end{document} 






hep-ph0412259.extracted_bib
\begin{thebibliography}{99} 
\def    \np     #1#2#3{{\it Nucl. Phys.} {\bf #1} (19#2) #3}
\def    \nptwoth     #1#2#3{{\it Nucl. Phys.} {\bf #1} (20#2) #3}
\def    \nptwothnz     #1#2#3{{\it Nucl. Phys.} {\bf #1} (19#2) #3}
\def    \prep   #1#2#3{{\it Phys. Rep.} {\bf #1}  (19#2) #3}
\def    \pl     #1#2#3{{\it Phys. Lett.} {\bf #1} (19#2) #3}
\def    \pltwoth     #1#2#3{{\it Phys. Lett.} {\bf #1} (20#2) #3}
\def    \plold  #1#2#3{{\it Phys. Lett.} {\bf #1B} (19#2) #3}
\def    \prl    #1#2#3{{\it Phys. Rev. Lett.} {\bf #1}  (19#2) #3} 
\def    \pr     #1#2#3{{\it Phys. Rev.} {\bf #1}  (19#2) #3}
\def    \prd    #1#2#3{{\it Phys. Rev.} {\bf D#1}  (19#2) #3} 
\def    \zeit   #1#2#3{{\it Z. Phys.} {\bf C#1}  (19#2) #3}
\def    \cmp    #1#2#3{{\it Comm. Math. Phys.} {\bf #1}  (19#2) #3}
\def    \ibid   #1#2#3{{\it ibid.} {\bf #1} (19#2) #3}
\def    \nc     #1#2#3{{\it Nuovo Cim.} {\bf #1} (19#2) #3}
\def    \acta   #1#2#3{{\it Acta Phys. Polon.} {\bf #1} (19#2) #3}
\def    \tmp    #1#2#3{{\it Theor. Math. Phys.} {\bf #1} (19#2) #3}
\def    \eur    #1#2#3{{\it Eur. Phys.  J.} {\bf C#1} (19#2) #3}
\def    \comp    #1#2#3{{\it Comput. Phys. Commun.} {\bf #1} (20#2) #3}
\def    \hepph  #1 {{\tt hep-ph/#1}}
\def    \hepex  #1 {{\tt hep-ex/#1}}
\def    \mathph  #1 {{\tt math-ph/#1}}
\parskip 0pt
\itemsep=0pt


%%%%%%%%%%%%%% FB-Asym %%%%%%%%%%%%%%%%%%%
%
\bibitem{bbghlmr}
W.~Bernreuther, R.~Bonciani, T.~Gehrmann, R.~Heinesch, T.~Leineweber, P.~Mastrolia and E.~Remiddi,
%``Two-loop QCD Corrections to the Heavy Quark Form Factors: the Vector
%Contributions,''
\nptwoth{B706}{05}{245} 
(\hepph{0406046}).
%Nucl.\ Phys.\ B {\bf 706} (2005) 245
%{\tt hep-ph/0406046}, to appear in Nucl.\ Phys.\ B.
%%CITATION = HEP-PH 0406046;%%


\bibitem{EWWorkingGroup}
 The LEP collaborations ALEPH, DELPHI, L3, OPAL, the LEP Electroweak
 Working Group, and the SLD Heavy Flavour Group, {\it A Combination of
 Preliminary Electroweak Measurements and Constraints on the Standard
 Model}, report LEPEWWG/2003-01, April 2003.

%\cite{Freitas:2004mn}
\bibitem{Freitas:2004mn}
A.~Freitas and K.~Moenig,
%``Corrections to quark asymmetries at LEP,''
(\hepph{0411304}).
%%CITATION = HEP-PH 0411304;%%

\bibitem{tesla}
 J.~A.~Aguilar-Saavedra {\it et al.}  [ECFA/DESY LC Physics Working Group
 Collaboration], ``TESLA Technical Design Report Part III: Physics at an
 $e^+e^-$ Linear Collider'', DESY-report 2001-011 (\hepph{0106315}).
%%CITATION = HEP-PH 0106315;%%

\bibitem{triangle}
W.~Bernreuther, R.~Bonciani, T.~Gehrmann,
R.~Heinesch, T.~Leineweber and
E.~Remiddi, in preparation.

\bibitem{HV}
G.~'t Hooft and M.~J.~G.~Veltman,
%``Regularization And Renormalization Of Gauge Fields,''
\nptwothnz{B44}{72}{189}
%Nucl.\ Phys.\ B {\bf 44} (1972) 189.
%%CITATION = NUPHA,B44,189;%%

\bibitem{Larin}
S.~A.~Larin,
%``The Renormalization of the axial anomaly 
% in dimensional regularization,''
\pl{B303}{93}{113}
%Phys.\ Lett.\ B {\bf 303} (1993) 113
(\hepph{9302240}).
%%CITATION = HEP-PH 9302240;%%

%%%%%%%%%%%%%%% Harm-Polylog %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bibitem{Polylog}
  E. Remiddi and J. A. M. Vermaseren, {\it Int. J. Mod. Phys.} 
  {\bf A15} (2000) 725 ({\tt hep-ph/9905237}). 
%%CITATION = HEP-PH 9905237;%%

\bibitem{Polylog3}
  T. Gehrmann and E. Remiddi, \comp{141}{01}{296} 
  ({\tt hep-ph/0107173}).
%%CITATION = HEP-PH 0107173;%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bibitem{Broadhurst}
D.~J.~Broadhurst, N.~Gray and K.~Schilcher,
\zeit{52}{91}{111}
%Z.\ Phys.\ C {\bf 52} (1991) 111.
%%CITATION = ZEPYA,C52,111;%%

\bibitem{Melnikov}
K.~Melnikov and T.~van Ritbergen, \nptwoth{B591}{00}{515}
%Nucl.\ Phys.\ B {\bf 591} (2000) 515
(\hepph{0005131}).
%%CITATION = HEP-PH 0005131;%%

%%%%%%%%%%%%%% 2-loop %%%%%%%%%%%%%%%%%%%

\bibitem{RoPieRem1}
  R. Bonciani, P. Mastrolia and E. Remiddi, \nptwoth{B661}{03}{289} 
[Erratum-ibid. \nptwoth{B702}{04}{259}] (\hepph{0301170}).
%%CITATION = HEP-PH 0301170;%%

\bibitem{RoPieRem2}
  R. Bonciani, P. Mastrolia and E. Remiddi, \nptwoth{B676}{04}{399}
  (\hepph{0307295}).
%%CITATION = HEP-PH 0307295;%%

\bibitem{RoPieRem3}
  R. Bonciani, P. Mastrolia and E. Remiddi, \nptwoth{B690}{04}{138}
  (\hepph{0311145}).
%%CITATION = HEP-PH 0311145;%%

%%%%%%%%%%%%%%%%

\bibitem{Lap}
  S. Laporta and E. Remiddi, \pl{B379}{96}{283} (\hepph{9602417}).\\
%%CITATION = APPOA,B28,959;%%
  S.~Laporta, {\it Int.\ J.\ Mod.\ Phys.}  {\bf A 15} (2000) 5087
 (\hepph{0102033}).
%%CITATION = HEP-PH 0102033;%%


%%%%%%%%%%%%%%%%%% Dim Reg %%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\bibitem{DimReg}
%  G. 't Hooft and M. Veltman, \np{B44}{72}{189}.\\
%%%CITATION = NUPHA,B44,189;%%
%  C. G. Bollini and J. J. Giambiagi, \plold{40}{72}{566}; 
%  \nc{12B}{72}{20}. \\
%%%CITATION = NUCIA,B12,20;%%
%  J. Ashmore, {\it Lett. Nuovo Cimento} {\bf 4} (1972) 289.\\
%%%CITATION = NCLTA,4,289;%%
%  G. M. Cicuta and E. Montaldi, {\it Lett. Nuovo Cimento} {\bf 4} 
%  (1972) 289. \\
%%%CITATION = NCLTA,4,329;%%
%  R. Gastmans and R. Meuldermans, \np{B63}{73}{277}.
%%%CITATION = NUPHA,B63,277;%%


%%%%%%%%%%%%%%%%%% IBP %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

\bibitem{Chet}
  F.V. Tkachov, \pl{B100}{81}{65}.\\
%%CITATION = PHLTA,B100,65;%%
  K.G. Chetyrkin and F.V. Tkachov, \np{B192}{81}{159}.
%%CITATION = NUPHA,B192,159;%%


%%%%%%%%%%%%%%%%%% LI %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bibitem{Rem3}
  T. Gehrmann and E. Remiddi, \nptwoth{B580}{00}{485} 
  ({\tt hep-ph/9912329}). 
%%CITATION = HEP-PH 9912329;%%
  

%%%%%%%%%%%%%%%%%% Diff. Eqs. Method %%%%%%%%%%%%%%

\bibitem{Kot}
  A. V. Kotikov, \pl{B254}{91}{158}.
%%CITATION = PHLTA,B254,158;%%

\bibitem{Rem1} 
  E. Remiddi, \nc{110A}{97}{1435} ({\tt hep-th/9711188}).
%%CITATION = HEP-TH 9711188;%%

\bibitem{Rem2}
  M. Caffo, H. Czy\.{z}, S. Laporta and E. Remiddi, 
  \acta{B29}{98}{2627} ({\tt hep-ph/9807119}).\\
%%CITATION = HEP-TH 9807119;%%
  M. Caffo, H. Czy\.{z}, S. Laporta and E. Remiddi, 
  \nc{A111}{98}{365} ({\tt hep-ph/9805118}).
%%CITATION = HEP-TH 9805118;%%

%\cite{Jersak:1981sp}
\bibitem{Jersak:1981sp}
J.~Jersak, E.~Laermann and P.~M.~Zerwas,
%``Electroweak Production Of Heavy Quarks In E+ E- Annihilation,''
\prd{25}{82}{1218}
%Phys.\ Rev.\ D {\bf 25} (1982) 1218
[Erratum-ibid. \prd{36}{87}{310}].
%[Erratum-ibid.\ D {\bf 36} (1987) 310].
%%CITATION = PHRVA,D25,1218;%%

\bibitem{KuhnTeubner}
J.~H.~K\"uhn and T.~Teubner,
%``Axial contributions at the top threshold,''
\eur{9}{99}{221}
%Eur.\ Phys.\ J.\ C {\bf 9} (1999) 221
(\hepph{9903322}).
%%CITATION = HEP-PH 9903322;%%


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\bibitem{FORM} J.A.M.\ Vermaseren, Symbolic Manipulation with
               {\tt FORM}, Version 2, CAN, Amsterdam, 1991; 
               New features of {\tt FORM}, (\mathph{0010025}).
%%CITATION = MATH-TH 0010025;%%

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\end{thebibliography}