text:
\subsection*{Setting the stage}

The pair-production of heavy quarks in the $q {\bar q}$-annihilation channel
corresponds to the scattering process,
\begin{equation}
\label{eq:qqQQ}
q(p_1) + {\bar q}(p_2) \:\:\rightarrow\:\: Q(p_3,m) + {\bar Q}(p_4,m) \, ,
\end{equation}
where $p_i$ denote the quark momenta and $m$ the mass of the heavy quark.
Energy-momentum conservation implies
\begin{equation}
\label{eq:engmom}
p_1^\mu+p_2^\mu = p_3^\mu+p_4^\mu \, .
\end{equation}
Following the notation of Ref.~\cite{Anastasiou:2000kg} we consider
the scattering amplitude ${\cal M}$ for the process~(\ref{eq:qqQQ})
at fixed values of the external parton momenta $p_i$, thus $p_1^2 =
p_2^2 = 0$ and $p_3^2 = p_4^2 = m^2$. It may be written as a
series expansion in the strong coupling $\alpha_{\rm s}$,
\begin{eqnarray}
  \label{eq:Mexp}
  | {\cal M} \rangle
  & = &
  4 \pi \alpha_{\rm s} \biggl[
  | {\cal M}^{(0)} \rangle
  + \biggl( {\alpha_{\rm s} \over 2 \pi} \biggr) | {\cal M}^{(1)} \rangle
  + \biggl( {\alpha_{\rm s} \over 2 \pi} \biggr)^2 | {\cal M}^{(2)} \rangle
  + {\cal O}(\alpha_{\rm s}^3)
  \biggr]
\, ,
\end{eqnarray}
where we define the expansion coefficients in powers of $\alpha_{\rm s}(\mu^2)
/ (2\pi)$ with $\mu$ being the renormalization scale. We work in
conventional dimensional regularization, $d=4-2 \epsilon$, in the
$\overline{\mathrm{MS}}$-scheme for the coupling constant renormalization. The heavy
mass $m$ on the other hand is always taken to be the pole mass.

We explicitly relate the bare (unrenormalized) coupling $\alpha_{\rm s}^{\rm{b}}$
to the renormalized coupling $\alpha_{\rm s}$ by
\begin{eqnarray}
\label{eq:alpha-s-renorm}
\alpha_{\rm s}^{\rm{b}} S_\epsilon \: = \: \alpha_{\rm s}
\biggl[
   1
   - {\beta_0 \over \epsilon} \biggl( {\alpha_{\rm s} \over 2 \pi} \biggr)
   + \left(
     {\beta_0^2 \over \epsilon^2}
     - {1 \over 2} {\beta_1 \over \epsilon}
     \right) \biggl( {\alpha_{\rm s} \over 2 \pi} \biggr)^2
  + {\cal O}(\alpha_{\rm s}^3)
  \biggr]
\, ,
\end{eqnarray}
where we put the factor $S_\epsilon=(4 \pi)^\epsilon \exp(-\epsilon \gamma_{\rm E}) = 1$
for simplicity and $\beta$ is the QCD $\beta$-function
\cite{vanRitbergen:1997va,Czakon:2004bu}
\begin{eqnarray}
\label{eq:betafct}
\beta_0 = {11 \over 6}\*{C^{}_A} - {2 \over 3}\*{T^{}_F}\*{n^{}_{\! f}} \, ,
\qquad
\beta_1 = {17 \over 6}\*{C^{}_A}^2 - {5 \over 3}\*{C^{}_A}\*{T^{}_F}\*{n^{}_{\! f}} - {C^{}_F}\*{T^{}_F}\*{n^{}_{\! f}} \, .
\end{eqnarray}
As we work in a general non-Abelian ${\rm{SU}}(N)$-gauge theory we set
${C^{}_A} = N$, ${C^{}_F} = (N^2-1)/2\*N$ and ${T^{}_F} = 1/2$.
Throughout this letter, $N$ denotes the number of colors and
${n^{}_{\! f}}$ the total number of flavors, which is the sum of
${n^{}_{\! l}}$ light and ${n^{}_{\! h}}$ heavy quarks.

The squared amplitude for the process~(\ref{eq:qqQQ})
summed over spins and colors is a function of the
Mandelstam variables $s$, $t$ and $u$ given by
\begin{equation}
\label{eq:Mandelstam}
s = (p_1+p_2)^2\, , \qquad
t  = (p_1-p_3)^2 - m^2\, , \qquad
u  = (p_1-p_4)^2 - m^2\, .
\end{equation}
Then it is convenient to define the function ${\cal A}(\epsilon, m, s, t, \mu)$
for the spin and color averaged amplitudes as
\begin{eqnarray}
\label{eq:Msqrd}
\overline{\sum |{\cal M}({q + \bar{q} \to  Q + \bar{Q}} )|^2}
&=&
{1 \over 4 N^2}\, {\cal A}(\epsilon, m, s, t, \mu)
\, ,
\end{eqnarray}
which has a perturbative expansion similar to Eq.~(\ref{eq:Mexp}),
\begin{equation}
\label{eq:Aexp}
{\cal A}(\epsilon, m, s, t, \mu) = 16 \pi^2 \alpha_{\rm s}^2
\left[
  {\cal A}^4
  + \biggl( {\alpha_{\rm s} \over 2 \pi} \biggr) {\cal A}^6
  + \biggl( {\alpha_{\rm s} \over 2 \pi} \biggr)^2 {\cal A}^8
  + {\cal O}(\alpha_{\rm s}^{3})
\right]
\, .
\end{equation}
In terms of the amplitudes the expansion coefficients in Eq.~(\ref{eq:Aexp})
may be expressed as
\begin{eqnarray}
\label{eq:A4def}
{\cal A}^4 &=&
\langle {\cal M}^{(0)} | {\cal M}^{(0)} \rangle \equiv 2(N^2-1) \left(\frac{t^2+u^2}{s^2} - \epsilon \right)
+ {\cal O}(m)\, , \\
\label{eq:A6def}
{\cal A}^6 &=& \left(
\langle {\cal M}^{(0)} | {\cal M}^{(1)} \rangle + \langle {\cal M}^{(1)} | {\cal M}^{(0)} \rangle
\right)\, , \\
\label{eq:A8def}
{\cal A}^8 &=& \left(
\langle {\cal M}^{(1)} | {\cal M}^{(1)} \rangle
+ \langle {\cal M}^{(0)} | {\cal M}^{(2)} \rangle + \langle {\cal M}^{(2)} | {\cal M}^{(0)} \rangle
\right)\, ,
\end{eqnarray}
where we have neglected powers in the heavy-quark mass $m$ in ${\cal
A}^4$. Expressions for ${\cal A}^6$ with the complete heavy-quark
mass dependence using dimensional regularization can be obtained
e.g. from Ref.~\cite{Korner:2002hy,Bernreuther:2004jv}. The
loop-by-loop contribution $\langle {\cal M}^{(1)} | {\cal M}^{(1)}
\rangle$ in dimensional regularization in ${\cal A}^8$ and also with
the full heavy-quark mass dependence can be computed with the help
of Ref.~\cite{Korner:2005rg}. In this letter, we provide for the
first time the real part of $\langle {\cal M}^{(0)} | {\cal M}^{(2)}
\rangle$ up to powers ${\cal O}(m)$ in the heavy-quark mass $m$.

\subsection*{Massive amplitudes from QCD factorization}

Let us briefly recall the key features of Ref.~\cite{Mitov:2006xs}
to calculate loop amplitudes with massive partons from massless
ones. The QCD factorization approach rests on the fact that a
massive amplitude ${\cal M}^{{\rm[p]},(m)}$ for any given physical
process shares essential properties in the small-mass limit with
the corresponding massless amplitude ${\cal M}^{{\rm[p]},(m=0)}$. The
latter one, ${\cal M}^{{\rm[p]},(m=0)}$, generally displays two
types of singularities, soft and collinear, related to the emission
of gluons with vanishing energy and to collinear parton radiation
off massless hard partons, respectively. These appear explicitly as
factorizing poles in $\epsilon$ in dimensional regularization after the
usual ultraviolet renormalization is performed. In the former case,
the soft singularities remain in ${\cal M}^{{\rm[p]},(m)}$ as single
poles in $\epsilon$ while some of the collinear singularities are now
screened by the mass $m$ of the heavy fields, which gives rise to
a logarithmic dependence on $m$, see e.g. Ref.~\cite{Catani:2000ef}.

Thus, in the small-mass limit the differences between a massless and
a massive amplitude can be thought of as a mere change in the
regularization scheme. As an upshot, QCD factorization provides a
direct relation between ${\cal M}^{{\rm[p]},(m)}$ and ${\cal
M}^{{\rm[p]},(m=0)}$ which can be cast in the remarkably simple and
suggestive relation
\begin{eqnarray}
\label{eq:Mm-M0}
{\cal M}^{{\rm[p]},(m)} &=&
\prod_{i\in\ \{{\rm all}\ {\rm legs}\}}\,
  \left(
    Z^{(m\vert0)}_{[i]}
  \right)^{1 \over 2}\,
  \times\
{\cal M}^{{\rm[p]},(m=0)}\, .
\end{eqnarray}

The function $Z^{(m\vert 0)}$ is process independent and depends
only on the external parton, i.e. quarks in the case at hand. For
external massive quarks $Q$ it is defined as the ratio of the
on-shell heavy-quark form factor and the massless on-shell one, both
being known~\cite{Bernreuther:2004ih,Moch:2005id,Gehrmann:2005pd} to
sufficient orders in $\alpha_{\rm s}$ and powers of $\epsilon$. An explicit
expression for
\begin{equation}
Z^{(m\vert0)}_{[Q]} \, = \, 1 + \sum\limits_{j=1}^{\infty} \left(
{\alpha_{\rm s} \over 2 \pi} \right)^j\, Z^{(j)} \, ,
\end{equation}
up to two loops is given
\footnote{Note the different normalization ${\alpha_{\rm s} /  (4 \pi)}$ of the
coupling used in that reference.}
in Ref.~\cite{Mitov:2006xs}. Exploiting the full predictive power of
the relation Eq.~(\ref{eq:Mm-M0}) and applying it to the process
Eq.~(\ref{eq:qqQQ}) we get
\begin{equation}
\label{eq:A8mtoA80} 2 {\rm Re}\, \langle {\cal M}^{(0)} | {\cal
M}^{(2)} \rangle^{(m)} = 2 {\rm Re}\, \langle {\cal M}^{(0)} |
{\cal M}^{(2)} \rangle^{(m=0)}
+ Z^{(1)} {\cal A}^{6,(m=0)}\ +\ 2 Z^{(2)} {\cal
A}^{4,(m=0)} \, ,
\end{equation}
which assumes the hierarchy of scales $m^2 \ll s,t,u$ , i.e. we
neglect terms ${\cal O}(m)$. Eq.~(\ref{eq:A8mtoA80}) predicts the
complete real part of the squared amplitude $\langle {\cal M}^{(0)}
| {\cal M}^{(2)} \rangle$ except for those terms, which are linear
in ${n^{}_{\! h}}$, i.e. the number of heavy quarks. These two-loop
contributions have been excluded explicitly from the definition
\cite{Mitov:2006xs} of $Z^{(m\vert 0)}$, as one needs additional
process dependent terms for their description. Their incorporation
for the case of Bhabha scattering was presented in
Ref.~\cite{Becher:2007cu} in agreement with the direct
calculation~\cite{Actis:2007gi}.

\subsection*{Direct calculation of the massive amplitude}

An alternative method is the direct calculation of all necessary
massive Feynman diagrams together with an expansion in the small
mass. The advantage of this approach is an independent check of
Eq.~(\ref{eq:A8mtoA80}) as well as of the corresponding massless
results. Moreover, it also allows for a relatively easy access to
all heavy-quark loop corrections. Presently, these have not been
obtained from the QCD factorization method, since they are related
to the process dependent contributions.

We performed our computation using the DiaGen/IdSolver system of one
of the authors (M.C.). After the diagram generation phase, all the
integrals have been reduced to a set of 145 masters with the help of
the Laporta algorithm \cite{Laporta:2001dd} extended by topological
symmetry properties, where, however, integrals which are related by
a $t \leftrightarrow u$ exchange of the Mandelstam variables are
considered as being independent. The evaluation of the masters
proceeded similarly to the methodology developed in
Ref.~\cite{Czakon:2004wm,Czakon:2006pa}. The main idea was to construct
Mellin-Barnes \cite{Smirnov:1999gc,Tausk:1999vh} representations for all the
integrals, followed by a subsequent analytic continuation with the MB package
\cite{Czakon:2005rk} and an expansion in the mass by closing
contours. The resulting integrals have then been transformed into
series representations, some two-fold, and resummed with the help of
XSummer~\cite{Moch:2005uc} in the cases of non-trivial dependence on
the kinematic variables. Some constants, though, were not given by
harmonic series and have been computed with the help of the PSLQ
algorithm~\cite{pslq:1992}. Needless to say that all of the above
steps were performed fully automatically.

\begin{figure}[ht]
  \parbox[t]{9cm}{\center \parbox[c][4cm][c]{8cm}{
    \epsfig{file=np.eps,width=8cm}}
    \caption{\label{fig:np}
      The most complicated non-planar topologies, together with the multiplicities of
      the respective master integrals for $q{\bar q} \to Q {\bar Q}$
      scattering. The thick lines are massive.
    }}
  \hspace{1cm}
  \parbox[t]{6cm}{\center \parbox[c][4cm][c]{5cm}{
    \epsfig{file=z2.eps,width=5cm}}
    \caption{\label{fig:z2}
     A heavy-quark loop contribution to the wave-function renormalization of
     the light quarks.}}
\end{figure}
In our calculation, we have only been able to obtain the leading
color term and the full dependence on the number of light and heavy
fermion species. The reason is that the terms subleading in color
contain contributions from non-planar graphs. Fig.~\ref{fig:np}
shows the bottleneck cases with 6- and 8-fold Mellin-Barnes
representations, for the six and seven liners respectively. The
representations have been obtained similarly as in
Ref.~\cite{Tausk:1999vh}, i.e. from the Feynman-parametric
representation of the two-loop graphs and not by integrating
loop-by-loop, the strategy adopted for planars. In this way, the
mass expansion generates integrals that are at worst as complex as
the massless ones, plus some new terms that behave as $1/\sqrt{m^2}$
and must cancel in the complete result. The sheer amount of
remaining difficult 4-fold integrals turned out to be presently
intractable with the developed software.

Finally, as mentioned above, we renormalized the mass of the heavy quark and the
external wave functions in the on-shell scheme
(besides the $\overline{\mathrm{MS}}$ renormalization of the coupling constant).
The former results are now known up to three-loop level from Ref.~\cite{Melnikov:2000qh,Melnikov:2000zc}.
A peculiarity of the light quark states is that they obtain contributions to
the wave-function renormalization constants from heavy-quark loops, see
Fig.~\ref{fig:z2}.
In fact, we have
\begin{equation}
\label{eq:Z2wavefct}
Z_2 = 1+ \left(\frac{\alpha_s}{2\pi}\right)^2 {C^{}_F} {T^{}_F} {n^{}_{\! h}}
\left[
  \frac{1}{4\epsilon}
  - \frac{1}{2} \log \left( \frac{m^2}{\mu^2} \right)
  - \frac{5}{24}
\right]\, .
\end{equation}

\subsection*{Results}

We are now in a position to present our result
for $q{\bar q} \to Q {\bar Q}$ scattering for the
interference of the two-loop and Born amplitude,
\begin{eqnarray}
  \label{eq:ReM0x2}
{\lefteqn{
  2 {\rm Re}\, \langle {\cal M}^{(0)} | {\cal M}^{(2)} \rangle = }}
\\
& &
2 (N^2-1) \biggl(
N^2 A + B  + {1 \over N^2} C
+ N {n^{}_{\! l}} D_l + N {n^{}_{\! h}} D_h
+ {{n^{}_{\! l}} \over N} E_l + {{n^{}_{\! h}} \over N} E_h + ({n^{}_{\! l}}+ {n^{}_{\! h}})^2 F
\biggr)
\, ,
\nonumber
\end{eqnarray}
which we choose to express by grouping terms according to the power
of the number of colors $N$ and the numbers of ${n^{}_{\! l}}$ light and
${n^{}_{\! h}}=1$ heavy quarks with ${n^{}_{\! f}}={n^{}_{\! l}}+{n^{}_{\! h}}$ total flavors. As detailed
above, the coefficients $A$, $D_l$, $E_l$ and $F$ have been computed
with both methods, i.e. by employing our universal multiplicative
relation~(\ref{eq:Mm-M0}) between the massive and massless
amplitudes as well as a direct evaluation of the loop integrals in
the small-mass expansion. The terms linear in ${n^{}_{\! h}}$, $D_h$ and $E_h$
could also be easily obtained from a direct Feynman diagram
calculation, while for $B$ and $C$ the approach based on
factorization proved more powerful.

As the only dimensionless kinematic variable in
our problem, we choose $x=-t/s$ and keep the dependence on the renormalization
scale, $\mu$, explicit. We also introduce the following compact notation
\begin{equation}
\mathrm{L}_m = \log\left( \frac{m^2}{s} \right)\, , \;\;\;\;
\mathrm{L}_s = \log\left( \frac{s}{\mu^2} \right)\, , \;\;\;\;
\mathrm{L}_x = \log\left( x \right)\, , \;\;\;\;
\mathrm{L}_y = \log\left( 1-x \right)\, .
\end{equation}

The different components now read
{\small{
\begin{eqnarray}
A &=&
\frac{1}{\epsilon^4}\left\{
\frac{x^2}{2}-\frac{x}{2}+\frac{1}{4}
\right\}
+
\frac{1}{\epsilon^3}\left\{
\mathrm{L}_m\left[
x^2-x+\frac{1}{2}
\right]
+
\mathrm{L}_s\left[
-x^2+x-\frac{1}{2}
\right]
+
\frac{21 x^2}{4}-\frac{21 x}{4}
+\mathrm{L}_x \left(-2 x^2
\right. \right. \nonumber \\ && \left. \left.
+2
   x-1\right)+\frac{19}{8}
\right\}
+
\frac{1}{\epsilon^2}\left\{
\mathrm{L}_m \mathrm{L}_s\left[
-2 x^2+2 x-1
\right]
+
\mathrm{L}_s^2\left[
x^2-x+\frac{1}{2}
\right]
+
\mathrm{L}_m\left[
\frac{29 x^2}{6}-\frac{29 x}{6}
\right. \right. \nonumber \\ && \left. \left.
+\mathrm{L}_x \left(-2 x^2+2
   x-1\right)+\frac{23}{12}
\right]
+
\mathrm{L}_s\left[
-\frac{19 x^2}{6}+\frac{19 x}{6}+\mathrm{L}_x \left(4 x^2-4
   x+2\right)
-\frac{13}{12}
\right]
+
\left(2 x^2
\right. \right. \nonumber \\ && \left. \left.
-\frac{5 x}{2}+\frac{5}{4}\right) \mathrm{L}_x^2+\left(-\frac{26
   x^2}{3}+\frac{55 x}{6}-\frac{23}{6}\right) \mathrm{L}_x+\frac{173
   x^2}{72}-\frac{173 x}{72}+\pi ^2
   \left(-\frac{x^2}{6}+\frac{x}{6}-\frac{1}{12}\right)
\right. \nonumber \\ && \left.
-\frac{205}{144}
\right\}
+
\frac{1}{\epsilon}\left\{
\mathrm{L}_m^3\left[
-\frac{x^2}{3}+\frac{x}{3}-\frac{1}{6}
\right]
+
\mathrm{L}_m \mathrm{L}_s^2\left[
2 x^2-2 x+1
\right]
+
\mathrm{L}_s^3\left[
-\frac{2 x^2}{3}+\frac{2 x}{3}-\frac{1}{3}
\right]
\right. \nonumber \\ && \left.
+
\mathrm{L}_m^2\left[
-x^2+x+\mathrm{L}_x \left(x^2-x+\frac{1}{2}\right)-\frac{1}{2}
\right]
+
\mathrm{L}_m \mathrm{L}_s\left[
-\frac{7 x^2}{3}+\frac{7 x}{3}+\mathrm{L}_x \left(4 x^2-4 x+2\right)-\frac{1}{6}
\right]
\right. \nonumber \\ && \left.
+
\mathrm{L}_s^2\left[
-\frac{x^2}{2}+\frac{x}{2}+\mathrm{L}_x \left(-4 x^2+4 x-2\right)-\frac{3}{4}
\right]
+
\mathrm{L}_m\left[
\left(\frac{1}{4}-\frac{x}{2}\right) \mathrm{L}_x^2+\left(\frac{3
   x}{2}-x^2\right) \mathrm{L}_x-\frac{47 x^2}{12}
\right. \right. \nonumber \\ && \left. \left.
+\frac{47 x}{12}-\frac{35}{8}
\right]
+
\mathrm{L}_s\left[
\left(-4 x^2+5 x-\frac{5}{2}\right) \mathrm{L}_x^2+\left(\frac{8 x^2}{3}-\frac{11
   x}{3}+\frac{1}{3}\right) \mathrm{L}_x+\frac{487 x^2}{36}-\frac{487 x}{36}
\right. \right. \nonumber \\ && \left. \left.
+\pi
   ^2 \left(\frac{x^2}{3}-\frac{x}{3}+\frac{1}{6}\right)+\frac{601}{72}
\right]
+
\left(\frac{x^2}{3}+x-\frac{1}{2}\right) \mathrm{L}_x^3+\left(-\frac{5
   x}{2}+\mathrm{L}_y
   \left(-x^2+\frac{x}{2}-\frac{1}{4}\right)+\frac{3}{4}\right)
   \mathrm{L}_x^2
\right. \nonumber \\ && \left.
+\li2 \left(-2 x^2+x-\frac{1}{2}\right)
   \mathrm{L}_x+\left(\frac{43 x^2}{3}-\frac{151 x}{12}+\pi ^2 \left(\frac{4
   x^2}{3}-\frac{5 x}{6}+\frac{5}{12}\right)+10\right) \mathrm{L}_x
\right. \nonumber \\ && \left.
-\frac{9907
   x^2}{432}+\frac{9907 x}{432}+\pi ^2 \left(-\frac{23 x^2}{72}+\frac{5
   x}{72}+\frac{25}{144}\right)+\li3 \left(2
   x^2-x+\frac{1}{2}\right)+\left(-\frac{23 x^2}{6}
\right. \right. \nonumber \\ && \left. \left.
+\frac{17
   x}{6}-\frac{17}{12}\right) \z3-\frac{10945}{864}
\right\}
+
\mathrm{L}_m^4\left[
\frac{x^2}{4}-\frac{x}{4}+\frac{1}{8}
\right]
+
\mathrm{L}_m^3 \mathrm{L}_s\left[
\frac{2 x^2}{3}-\frac{2 x}{3}+\frac{1}{3}
\right]
+
\mathrm{L}_m \mathrm{L}_s^3\left[
-\frac{4 x^2}{3}
\right. \nonumber \\ && \left.
+\frac{4 x}{3}-\frac{2}{3}
\right]
+
\mathrm{L}_s^4\left[
\frac{x^2}{3}-\frac{x}{3}+\frac{1}{6}
\right]
+
\mathrm{L}_m^3\left[
-\frac{11 x^2}{18}+\frac{11 x}{18}+\mathrm{L}_x
   \left(-\frac{x^2}{3}+\frac{x}{3}-\frac{1}{6}\right)-\frac{5}{36}
\right]
\nonumber \\ &&
+
\mathrm{L}_m^2 \mathrm{L}_s\left[
-\frac{5 x^2}{3}+\frac{5 x}{3}+\mathrm{L}_x \left(-2 x^2+2 x-1\right)-\frac{5}{6}
\right]
+
\mathrm{L}_m \mathrm{L}_s^2\left[
-\frac{4 x^2}{3}+\frac{4 x}{3}+\mathrm{L}_x \left(-4 x^2+4 x-2\right)
\right. \nonumber \\ && \left.
-\frac{5}{3}
\right]
+
\mathrm{L}_s^3\left[
\frac{14 x^2}{9}-\frac{14 x}{9}+\mathrm{L}_x \left(\frac{8 x^2}{3}-\frac{8
   x}{3}+\frac{4}{3}\right)+\frac{10}{9}
\right]
+
\mathrm{L}_m^2\left[
\left(\frac{x}{4}-\frac{1}{8}\right) \mathrm{L}_x^2+\left(\frac{x^2}{2}-\frac{3
   x}{4}\right) \mathrm{L}_x
\right. \nonumber \\ && \left.
+\frac{247 x^2}{36}-\frac{247 x}{36}+\frac{283}{72}
\right]
+
\mathrm{L}_m \mathrm{L}_s\left[
\left(x-\frac{1}{2}\right) \mathrm{L}_x^2+\left(2 x^2-3 x\right)
   \mathrm{L}_x+\frac{23 x^2}{2}-\frac{23 x}{2}+\frac{83}{12}
\right]
\nonumber \\ &&
+
\mathrm{L}_s^2\left[
\left(4 x^2-5 x+\frac{5}{2}\right) \mathrm{L}_x^2+\left(\frac{14 x^2}{3}-\frac{11
   x}{3}+\frac{10}{3}\right) \mathrm{L}_x-\frac{37 x^2}{4}+\frac{37 x}{4}+\pi ^2
   \left(-\frac{x^2}{3}+\frac{x}{3}-\frac{1}{6}\right)
\right. \nonumber \\ && \left.
-\frac{35}{8}
\right]
+
\mathrm{L}_m\left[
\frac{x^2 \mathrm{L}_x^3}{3}+\left(-\frac{x}{2}+\mathrm{L}_y
   \left(-x^2+\frac{x}{2}-\frac{1}{4}\right)+\frac{1}{4}\right)
   \mathrm{L}_x^2+\li2 \left(-2 x^2+x-\frac{1}{2}\right) \mathrm{L}_x
\right. \nonumber \\ && \left.
+\left(-4
   x^2+\frac{19 x}{4}+\pi ^2 \left(x^2-\frac{x}{2}+\frac{1}{4}\right)-2\right)
   \mathrm{L}_x-\frac{781 x^2}{72}+\frac{781 x}{72}+\pi ^2 \left(-\frac{7
   x^2}{12}+\frac{x}{3}-\frac{1}{24}\right)
\right. \nonumber \\ && \left.
+\li3 \left(2
   x^2-x+\frac{1}{2}\right)+\left(\frac{7 x^2}{3}-\frac{10
   x}{3}+\frac{5}{3}\right) \z3-\frac{499}{144}
\right]
+
\mathrm{L}_s\left[
\left(-\frac{2 x^2}{3}-2 x+1\right) \mathrm{L}_x^3
\right. \nonumber \\ && \left.
+\left(\frac{4 x}{3}+\mathrm{L}_y
   \left(2 x^2-x+\frac{1}{2}\right)+\frac{1}{3}\right) \mathrm{L}_x^2+\li2
   \left(4 x^2-2 x+1\right) \mathrm{L}_x+\left(-\frac{86 x^2}{3}+\frac{173
   x}{6}
\right. \right. \nonumber \\ && \left. \left.
+\pi ^2 \left(-\frac{8 x^2}{3}+\frac{5
   x}{3}-\frac{5}{6}\right)-\frac{49}{3}\right) \mathrm{L}_x+\frac{2003
   x^2}{216}-\frac{2003 x}{216}+\li3 \left(-4 x^2+2 x-1\right)
\right. \nonumber \\ && \left.
+\pi ^2
   \left(-\frac{43 x^2}{36}+\frac{61
   x}{36}-\frac{91}{72}\right)+\left(\frac{23 x^2}{3}-\frac{17
   x}{3}+\frac{17}{6}\right) \z3-\frac{919}{432}
\right]
+
\left(-x^2-\frac{x}{24}+\frac{1}{48}\right) \mathrm{L}_x^4
\nonumber \\ &&
+\left(-\frac{7
   x^2}{18}+\frac{13 x}{12}+\mathrm{L}_y \left(\frac{10 x^2}{3}-2
   x+1\right)-\frac{7}{12}\right)
   \mathrm{L}_x^3+\left(\left(-\frac{x^2}{2}-\frac{5 x}{4}+\frac{11}{8}\right)
   \mathrm{L}_y^2+\left(\frac{7 x^2}{6}
\right. \right. \nonumber \\ && \left. \left.
+\frac{13 x}{12}+\frac{13}{12}\right)
   \mathrm{L}_y+\frac{101 x}{72}+\pi ^2 \left(-\frac{25 x^2}{6}+\frac{25
   x}{12}-\frac{25}{24}\right)+\frac{3}{1-x}-\frac{617}{144}\right)
   \mathrm{L}_x^2+\s12 \left(-2 x^2
\right. \nonumber \\ && \left.
-5 x+\frac{11}{2}\right)
   \mathrm{L}_x+\left(\frac{260 x^2}{9}-\frac{139 x}{4}+\pi ^2 \left(-\frac{25
   x^2}{9}+\frac{53 x}{36}-\frac{47}{18}\right)+\mathrm{L}_y \pi ^2
   \left(-\frac{x^2}{3}+\frac{11 x}{6}-\frac{17}{12}\right)
\right. \nonumber \\ && \left.
+\left(\frac{52
   x^2}{3}-\frac{31 x}{3}+\frac{31}{6}\right) \z3+\frac{1027}{72}\right)
   \mathrm{L}_x+\frac{17845 x^2}{2592}-\frac{17845 x}{2592}+\pi ^4
   \left(-\frac{11 x^2}{80}-\frac{x}{240}+\frac{1}{480}\right)
\nonumber \\ &&
+\li4
   \left(2 x^2+3 x-\frac{3}{2}\right)+\s22 \left(2 x^2+5
   x-\frac{11}{2}\right)+\pi ^2 \left(\frac{1009 x^2}{432}-\frac{259
   x}{432}+\frac{403}{864}\right)
\nonumber \\ &&
+\li3 \left(-\frac{7 x^2}{3}-\frac{13
   x}{6}+\mathrm{L}_x \left(-8 x^2+2 x-1\right)+\mathrm{L}_y \left(2 x^2+5
   x-\frac{11}{2}\right)-\frac{13}{6}\right)+\li2 \left(\left(7
   x^2
\right. \right. \nonumber \\ && \left. \left.
-\frac{7 x}{2}+\frac{7}{4}\right) \mathrm{L}_x^2+\left(\frac{7
   x^2}{3}+\frac{13 x}{6}+\mathrm{L}_y \left(-2 x^2-5
   x+\frac{11}{2}\right)+\frac{13}{6}\right) \mathrm{L}_x+\pi ^2
   \left(-\frac{x^2}{3}+\frac{11 x}{6}-\frac{17}{12}\right)\right)
\nonumber \\ &&
+\mathrm{L}_y
   \left(-2 x^2-5 x+\frac{11}{2}\right) \z3+\left(-\frac{14
   x^2}{9}+\frac{55 x}{18}+\frac{149}{36}\right) \z3+\frac{77287}{5184},
\end{eqnarray}

\begin{eqnarray}
B &=&
\frac{1}{\epsilon^4}\left\{
-x^2+x-\frac{1}{2}
\right\}
+
\frac{1}{\epsilon^3}\left\{
\mathrm{L}_m\left[
-2 x^2+2 x-1
\right]
+
\mathrm{L}_s\left[
2 x^2-2 x+1
\right]
-\frac{31 x^2}{4}+\frac{31 x}{4}+\mathrm{L}_y \left(-4 x^2
\right. \right. \nonumber \\ && \left. \left.
+4 x-2\right)+\mathrm{L}_x
   \left(6 x^2-6 x+3\right)-\frac{27}{8}
\right\}
+
\frac{1}{\epsilon^2}\left\{
\mathrm{L}_m \mathrm{L}_s\left[
4 x^2-4 x+2
\right]
+
\mathrm{L}_s^2\left[
-2 x^2+2 x-1
\right]
\right. \nonumber \\ && \left.
+
\mathrm{L}_m\left[
-\frac{47 x^2}{6}+\frac{47 x}{6}+\mathrm{L}_y \left(-4 x^2+4 x-2\right)+\mathrm{L}_x
   \left(6 x^2-6 x+3\right)-\frac{35}{12}
\right]
+
\mathrm{L}_s\left[
\frac{49 x^2}{6}-\frac{49 x}{6}
\right. \right. \nonumber \\ && \left. \left.
+\mathrm{L}_x \left(-12 x^2+12
   x-6\right)+\mathrm{L}_y \left(8 x^2-8 x+4\right)+\frac{37}{12}
\right]
+
\left(-6 x^2+\frac{15 x}{2}-\frac{15}{4}\right) \mathrm{L}_x^2+\left(\frac{67
   x^2}{3}
\right. \right. \nonumber \\ && \left. \left.
-\frac{143 x}{6}+\mathrm{L}_y \left(4 x^2-4
   x+2\right)+\frac{29}{3}\right) \mathrm{L}_x-\frac{415 x^2}{36}+\frac{415
   x}{36}+\mathrm{L}_y \left(-\frac{52 x^2}{3}+\frac{49
   x}{3}-\frac{20}{3}\right)
\right. \nonumber \\ && \left.
+\pi ^2 \left(\frac{17 x^2}{12}-\frac{17
   x}{12}+\frac{17}{24}\right)+\mathrm{L}_y^2 \left(2
   x^2-x+\frac{1}{2}\right)-\frac{17}{9}
\right\}
+
\frac{1}{\epsilon}\left\{
\mathrm{L}_m^3\left[
\frac{2 x^2}{3}-\frac{2 x}{3}+\frac{1}{3}
\right]
\right. \nonumber \\ && \left.
+
\mathrm{L}_m \mathrm{L}_s^2\left[
-4 x^2+4 x-2
\right]
+
\mathrm{L}_s^3\left[
\frac{4 x^2}{3}-\frac{4 x}{3}+\frac{2}{3}
\right]
+
\mathrm{L}_m^2\left[
2 x^2-2 x+\mathrm{L}_x \left(-3 x^2+3 x-\frac{3}{2}\right)
\right. \right. \nonumber \\ && \left. \left.
+\mathrm{L}_y \left(2
   x^2-2 x+1\right)+1
\right]
+
\mathrm{L}_m \mathrm{L}_s\left[
\frac{25 x^2}{3}-\frac{25 x}{3}+\mathrm{L}_x \left(-12 x^2+12
   x-6\right)+\mathrm{L}_y \left(8 x^2-8 x+4\right)
\right. \right. \nonumber \\ && \left. \left.
+\frac{13}{6}
\right]
+
\mathrm{L}_s^2\left[
-\frac{9 x^2}{2}+\frac{9 x}{2}+\mathrm{L}_y \left(-8 x^2+8 x-4\right)+\mathrm{L}_x
   \left(12 x^2-12 x+6\right)-\frac{5}{4}
\right]
+
\mathrm{L}_m\left[
\left(\frac{3 x}{2}
\right. \right. \right. \nonumber \\ && \left. \left. \left.
-\frac{3}{4}\right) \mathrm{L}_x^2+\left(3 x^2-\frac{9
   x}{2}\right) \mathrm{L}_x-\frac{16 x^2}{3}+\mathrm{L}_y^2
   \left(x-\frac{1}{2}\right)+\frac{16 x}{3}+\mathrm{L}_y \left(-2
   x^2+x+1\right)+\pi ^2 \left(\frac{7 x^2}{6}
\right. \right. \right. \nonumber \\ && \left. \left. \left.
-\frac{7
   x}{6}+\frac{7}{12}\right)+\frac{5}{4}
\right]
+
\mathrm{L}_s\left[
\left(12 x^2-15 x+\frac{15}{2}\right) \mathrm{L}_x^2+\left(-\frac{46
   x^2}{3}+\frac{55 x}{3}+\mathrm{L}_y \left(-8 x^2+8
   x-4\right)
\right. \right. \right. \nonumber \\ && \left. \left. \left.
-\frac{14}{3}\right) \mathrm{L}_x+\frac{85 x^2}{18}-\frac{85
   x}{18}+\mathrm{L}_y^2 \left(-4 x^2+2 x-1\right)+\pi ^2 \left(-\frac{17
   x^2}{6}+\frac{17 x}{6}-\frac{17}{12}\right)+\mathrm{L}_y \left(\frac{16
   x^2}{3}
\right. \right. \right. \nonumber \\ && \left. \left. \left.
-\frac{10 x}{3}-\frac{4}{3}\right)-\frac{31}{18}
\right]
+
\left(-x^2-3 x+\frac{3}{2}\right) \mathrm{L}_x^3+\left(\frac{15 x}{2}+\mathrm{L}_y
   \left(3 x^2-\frac{x}{2}+\frac{1}{4}\right)-\frac{9}{4}\right)
   \mathrm{L}_x^2
\right. \nonumber \\ && \left.
+\li2 \left(6 x^2-3 x+\frac{3}{2}\right)
   \mathrm{L}_x+\left(\left(\frac{1}{2}-x\right) \mathrm{L}_y^2-\mathrm{L}_y-\frac{50
   x^2}{3}+\frac{137 x}{12}+\pi ^2 \left(-\frac{31 x^2}{3}+\frac{41
   x}{6}-\frac{41}{12}\right)
\right. \right. \nonumber \\ && \left. \left.
-15\right) \mathrm{L}_x-\frac{419
   x^2}{108}+\frac{419 x}{108}+\mathrm{L}_y^2 \left(4
   x-\frac{5}{2}\right)+\li3 \left(-6 x^2+3
   x-\frac{3}{2}\right)+\s12 \left(-4 x^2+6 x-3\right)
\right. \nonumber \\ && \left.
+\mathrm{L}_y^3
   \left(\frac{2 x^2}{3}-\frac{7 x}{3}+\frac{7}{6}\right)+\pi ^2
   \left(\frac{263 x^2}{72}-\frac{101 x}{72}-\frac{91}{144}\right)+\mathrm{L}_y
   \left(\frac{86 x^2}{3}-\frac{193 x}{6}+\pi ^2 \left(\frac{8
   x^2}{3}-\frac{11
   x}{3}
\right. \right. \right. \nonumber \\ && \left. \left. \left.
+\frac{11}{6}\right)+\frac{47}{2}\right)+\left(\frac{91
   x^2}{6}-\frac{73 x}{6}+\frac{73}{12}\right) \z3+\frac{413}{108}
\right\}
+
\mathrm{L}_m^4\left[
-\frac{x^2}{2}+\frac{x}{2}-\frac{1}{4}
\right]
+
\mathrm{L}_m^3 \mathrm{L}_s\left[
-\frac{4 x^2}{3}+\frac{4 x}{3}
\right. \nonumber \\ && \left.
-\frac{2}{3}
\right]
+
\mathrm{L}_m \mathrm{L}_s^3\left[
\frac{8 x^2}{3}-\frac{8 x}{3}+\frac{4}{3}
\right]
+
\mathrm{L}_s^4\left[
-\frac{2 x^2}{3}+\frac{2 x}{3}-\frac{1}{3}
\right]
+
\mathrm{L}_m^3\left[
\frac{11 x^2}{18}-\frac{11 x}{18}+\mathrm{L}_y \left(-\frac{2 x^2}{3}+\frac{2
   x}{3}
\right. \right. \nonumber \\ && \left. \left.
-\frac{1}{3}\right)+\mathrm{L}_x
   \left(x^2-x+\frac{1}{2}\right)-\frac{1}{36}
\right]
+
\mathrm{L}_m^2 \mathrm{L}_s\left[
-\frac{x^2}{3}+\frac{x}{3}+\mathrm{L}_y \left(-4 x^2+4 x-2\right)+\mathrm{L}_x
   \left(6 x^2-6 x+3\right)
\right. \nonumber \\ && \left.
-\frac{1}{6}
\right]
+
\mathrm{L}_m \mathrm{L}_s^2\left[
-\frac{14 x^2}{3}+\frac{14 x}{3}+\mathrm{L}_y \left(-8 x^2+8 x-4\right)+\mathrm{L}_x
   \left(12 x^2-12 x+6\right)-\frac{1}{3}
\right]
+
\mathrm{L}_s^3\left[
\frac{16 x^2}{9}
\right. \nonumber \\ && \left.
-\frac{16 x}{9}+\mathrm{L}_x \left(-8 x^2+8 x-4\right)+\mathrm{L}_y
   \left(\frac{16 x^2}{3}-\frac{16 x}{3}+\frac{8}{3}\right)+\frac{2}{9}
\right]
+
\mathrm{L}_m^2\left[
\left(\frac{3}{8}-\frac{3 x}{4}\right) \mathrm{L}_x^2+\left(\frac{9 x}{4}
\right. \right. \nonumber \\ && \left. \left.
-\frac{3
   x^2}{2}\right) \mathrm{L}_x-\frac{157 x^2}{36}+\mathrm{L}_y^2
   \left(\frac{1}{4}-\frac{x}{2}\right)+\frac{157 x}{36}+\pi ^2 \left(-\frac{2
   x^2}{3}+\frac{2 x}{3}-\frac{1}{3}\right)+\mathrm{L}_y
   \left(x^2-\frac{x}{2}-\frac{1}{2}\right)
\right. \nonumber \\ && \left.
-\frac{229}{72}
\right]
+
\mathrm{L}_m \mathrm{L}_s\left[
\left(\frac{3}{2}-3 x\right) \mathrm{L}_x^2+\left(9 x-6 x^2\right) \mathrm{L}_x+7
   x^2+\mathrm{L}_y^2 (1-2 x)-7 x+\pi ^2 \left(-\frac{7 x^2}{3}+\frac{7
   x}{3}
\right. \right. \nonumber \\ && \left. \left.
-\frac{7}{6}\right)+\mathrm{L}_y \left(4 x^2-2 x-2\right)-\frac{2}{3}
\right]
+
\mathrm{L}_s^2\left[
\left(-12 x^2+15 x-\frac{15}{2}\right) \mathrm{L}_x^2+\left(\frac{2
   x^2}{3}-\frac{11 x}{3}+\mathrm{L}_y \left(8 x^2-8
   x
\right. \right. \right. \nonumber \\ && \left. \left. \left.
+4\right)-\frac{8}{3}\right) \mathrm{L}_x+\frac{40 x^2}{9}-\frac{40 x}{9}+\pi
   ^2 \left(\frac{17 x^2}{6}-\frac{17 x}{6}+\frac{17}{12}\right)+\mathrm{L}_y^2
   \left(4 x^2-2 x+1\right)+\mathrm{L}_y \left(\frac{28 x^2}{3}-\frac{34
   x}{3}
\right. \right. \nonumber \\ && \left. \left.
+\frac{26}{3}\right)+\frac{161}{36}
\right]
+
\mathrm{L}_m\left[
-x^2 \mathrm{L}_x^3+\left(\frac{3 x}{2}+\mathrm{L}_y \left(3 x^2-\frac{3
   x}{2}+\frac{3}{4}\right)-\frac{3}{4}\right) \mathrm{L}_x^2+\li2 \left(6
   x^2-3 x
\right. \right. \nonumber \\ && \left. \left.
+\frac{3}{2}\right) \mathrm{L}_x+\left(12 x^2-\frac{57 x}{4}+\pi ^2
   \left(-3 x^2+\frac{3 x}{2}-\frac{3}{4}\right)+6\right) \mathrm{L}_x-\frac{115
   x^2}{9}+\mathrm{L}_y^2 \left(x-\frac{1}{2}\right)+\frac{115 x}{9}
\right. \nonumber \\ && \left.
+\li3
   \left(-6 x^2+3 x-\frac{3}{2}\right)+\s12 \left(-4 x^2+6
   x-3\right)+\mathrm{L}_y^3 \left(\frac{2 x^2}{3}-\frac{4
   x}{3}+\frac{2}{3}\right)+\pi ^2 \left(\frac{9
   x^2}{4}-x
\right. \right. \nonumber \\ && \left. \left.
-\frac{5}{24}\right)+\mathrm{L}_y \left(-8 x^2+\frac{13 x}{2}+\pi ^2
   \left(2 x^2-3 x+\frac{3}{2}\right)-\frac{5}{2}\right)+\left(\frac{37
   x^2}{3}-\frac{28 x}{3}+\frac{14}{3}\right) \z3-\frac{67}{18}
\right]
\nonumber \\ &&
+
\mathrm{L}_s\left[
\left(2 x^2+6 x-3\right) \mathrm{L}_x^3+\left(-\frac{23 x}{3}+\mathrm{L}_y \left(-6
   x^2+x-\frac{1}{2}\right)+\frac{5}{6}\right) \mathrm{L}_x^2+\li2
   \left(-12 x^2+6 x-3\right) \mathrm{L}_x
\right. \nonumber \\ && \left.
+\left((2 x-1) \mathrm{L}_y^2+2
   \mathrm{L}_y+\frac{100 x^2}{3}-\frac{181 x}{6}+\pi ^2 \left(\frac{62
   x^2}{3}-\frac{41 x}{3}+\frac{41}{6}\right)+\frac{68}{3}\right)
   \mathrm{L}_x-\frac{1957 x^2}{54}+\mathrm{L}_y^2 \left(\frac{4}{3}
\right. \right. \nonumber \\ && \left. \left.
-\frac{2
   x}{3}\right)+\frac{1957 x}{54}+\mathrm{L}_y^3 \left(-\frac{4 x^2}{3}+\frac{14
   x}{3}-\frac{7}{3}\right)+\pi ^2 \left(\frac{67 x^2}{36}-\frac{229
   x}{36}+\frac{421}{72}\right)+\s12 \left(8 x^2-12
   x
\right. \right. \nonumber \\ && \left. \left.
+6\right)+\li3 \left(12 x^2-6 x+3\right)+\mathrm{L}_y \left(-\frac{172
   x^2}{3}+57 x+\pi ^2 \left(-\frac{16 x^2}{3}+\frac{22
   x}{3}-\frac{11}{3}\right)-\frac{97}{3}\right)
\right. \nonumber \\ && \left.
+\left(-\frac{91
   x^2}{3}+\frac{73 x}{3}-\frac{73}{6}\right) \z3-\frac{553}{27}
\right]
+
\left(3 x^2+\frac{x}{8}-\frac{1}{16}\right)
   \mathrm{L}_x^4+\left(-\frac{x^2}{18}-\frac{35 x}{12}+\mathrm{L}_y \left(-\frac{32
   x^2}{3}
\right. \right. \nonumber \\ && \left. \left.
+\frac{16 x}{3}-\frac{8}{3}\right)+\frac{7}{4}\right)
   \mathrm{L}_x^3+\left(\left(\frac{7 x^2}{2}+\frac{5 x}{4}-\frac{17}{8}\right)
   \mathrm{L}_y^2+\left(\frac{x^2}{6}-\frac{29 x}{12}-\frac{13}{6}\right)
   \mathrm{L}_y+\frac{365 x}{72}+\pi ^2 \left(\frac{95 x^2}{6}
\right. \right. \nonumber \\ && \left. \left.
-\frac{103
   x}{12}+\frac{103}{24}\right)-\frac{7}{1-x}+\frac{1027}{144}\right)
   \mathrm{L}_x^2+\left(\left(-\frac{2 x^2}{3}+\frac{4 x}{3}-\frac{2}{3}\right)
   \mathrm{L}_y^3+\left(\frac{x}{2}-\frac{1}{4}\right) \mathrm{L}_y^2+\left(\pi ^2
   \left(-\frac{43 x^2}{3}
\right. \right. \right. \nonumber \\ && \left. \left. \left.
+\frac{15 x}{2}-\frac{11}{4}\right)+2\right)
   \mathrm{L}_y-\frac{304 x^2}{9}+\frac{153 x}{4}+\pi ^2 \left(-\frac{125
   x^2}{18}+\frac{149 x}{36}+\frac{155}{36}\right)+\left(-54 x^2+45
   x
\right. \right. \nonumber \\ && \left. \left.
-\frac{45}{2}\right) \z3-\frac{1181}{72}\right)
   \mathrm{L}_x+\frac{38959 x^2}{648}-\frac{38959 x}{648}+\s22 \left(-10
   x^2+25 x-\frac{1}{2}\right)+\li4 \left(-6
   x^2
\right. \nonumber \\ && \left.
-x+\frac{1}{2}\right)+\mathrm{L}_y^4 \left(-x^2+\frac{25
   x}{12}-\frac{25}{24}\right)+\mathrm{L}_y^3 \left(-\frac{7 x^2}{9}+\frac{2
   x}{9}-\frac{13}{36}\right)+\pi ^4 \left(\frac{77 x^2}{45}-\frac{29
   x}{24}+\frac{101}{240}\right)
\nonumber \\ &&
+\s13 \left(4 x^2+2
   x-1\right)+\mathrm{L}_y^2 \left(-\frac{281 x}{36}+\pi ^2 \left(-4 x^2+\frac{19
   x}{3}-\frac{19}{6}\right)-\frac{163}{72}+\frac{5}{x}\right)
\nonumber \\ &&
+\s12
   \left(\frac{14 x^2}{3}-\frac{47 x}{3}+\mathrm{L}_y \left(8 x^2-8
   x+4\right)+\mathrm{L}_x \left(10 x^2-9
   x-\frac{3}{2}\right)+\frac{46}{3}\right)+\li3
   \left(-\frac{x^2}{3}
\right. \nonumber \\ && \left.
+\frac{23 x}{6}+\mathrm{L}_y \left(-10 x^2-3
   x+\frac{15}{2}\right)+\mathrm{L}_x \left(24 x^2-10
   x+5\right)+\frac{29}{6}\right)+\li2 \left(\left(-21 x^2+\frac{21
   x}{2}
\right. \right. \nonumber \\ && \left. \left.
-\frac{21}{4}\right) \mathrm{L}_x^2+\left(\frac{x^2}{3}-\frac{23
   x}{6}+\mathrm{L}_y \left(10 x^2+3 x-\frac{15}{2}\right)-\frac{29}{6}\right)
   \mathrm{L}_x+\pi ^2 \left(-5
   x^2-\frac{x}{6}+\frac{25}{12}\right)\right)
\nonumber \\ &&
+\left(\frac{43
   x^2}{18}+\frac{28 x}{9}-\frac{169}{18}\right) \z3+\mathrm{L}_y
   \left(\frac{520 x^2}{9}-\frac{829 x}{18}+\pi ^2 \left(-\frac{50
   x^2}{9}+\frac{28 x}{3}-\frac{25}{3}\right)+\left(\frac{98 x^2}{3}
\right. \right. \nonumber \\ && \left. \left.
-\frac{107
   x}{3}+\frac{71}{6}\right) \z3+\frac{605}{36}\right)+\pi ^2
   \left(-\frac{3797 x^2}{216}+\frac{1585 x}{108}+\left(2 x^2-2 x+1\right)
   \log (2)-\frac{2353}{216}\right)
\nonumber \\ &&
+\frac{41473}{1296},
\end{eqnarray}

\begin{eqnarray}
C &=&
\frac{1}{\epsilon^4}\left\{
\frac{x^2}{2}-\frac{x}{2}+\frac{1}{4}
\right\}
+
\frac{1}{\epsilon^3}\left\{
\mathrm{L}_m\left[
x^2-x+\frac{1}{2}
\right]
+
\mathrm{L}_s\left[
-x^2+x-\frac{1}{2}
\right]
+
\frac{5 x^2}{2}-\frac{5 x}{2}+\mathrm{L}_x \left(-4 x^2+4 x
\right. \right. \nonumber \\ && \left. \left.
-2\right)+\mathrm{L}_y
   \left(4 x^2-4 x+2\right)+1
\right\}
+
\frac{1}{\epsilon^2}\left\{
\mathrm{L}_m \mathrm{L}_s\left[
-2 x^2+2 x-1
\right]
+
\mathrm{L}_s^2\left[
x^2-x+\frac{1}{2}
\right]
+
\mathrm{L}_m\left[
3 x^2-3 x
\right. \right. \nonumber \\ && \left. \left.
+\mathrm{L}_x \left(-4 x^2+4 x-2\right)+\mathrm{L}_y \left(4 x^2-4
   x+2\right)+1
\right]
+
\mathrm{L}_s\left[
-5 x^2+5 x+\mathrm{L}_y \left(-8 x^2+8 x-4\right)+\mathrm{L}_x \left(8 x^2
\right. \right. \right. \nonumber \\ && \left. \left. \left.
-8
   x+4\right)-2
\right]
+
\left(6 x^2-7 x+\frac{7}{2}\right) \mathrm{L}_x^2+\left(-10 x^2+11 x+\mathrm{L}_y
   \left(-12 x^2+12 x-6\right)-4\right) \mathrm{L}_x+\frac{73 x^2}{8}
\right. \nonumber \\ && \left.
-\frac{73
   x}{8}+\pi ^2 \left(-\frac{13 x^2}{4}+\frac{13
   x}{4}-\frac{13}{8}\right)+\mathrm{L}_y^2 \left(6 x^2-7
   x+\frac{7}{2}\right)+\mathrm{L}_y \left(10 x^2-9 x+3\right)+\frac{53}{16}
\right\}
\nonumber \\ &&
+
\frac{1}{\epsilon}\left\{
\mathrm{L}_m^3\left[
-\frac{x^2}{3}+\frac{x}{3}-\frac{1}{6}
\right]
+
\mathrm{L}_m \mathrm{L}_s^2\left[
2 x^2-2 x+1
\right]
+
\mathrm{L}_s^3\left[
-\frac{2 x^2}{3}+\frac{2 x}{3}-\frac{1}{3}
\right]
+
\mathrm{L}_m^2\left[
-x^2+x
\right. \right. \nonumber \\ && \left. \left.
+\mathrm{L}_y \left(-2 x^2+2 x-1\right)+\mathrm{L}_x \left(2 x^2-2
   x+1\right)-\frac{1}{2}
\right]
+
\mathrm{L}_m \mathrm{L}_s\left[
-6 x^2+6 x+\mathrm{L}_y \left(-8 x^2+8 x-4\right)
\right. \right. \nonumber \\ && \left. \left.
+\mathrm{L}_x \left(8 x^2-8
   x+4\right)-2
\right]
+
\mathrm{L}_s^2\left[
5 x^2-5 x+\mathrm{L}_x \left(-8 x^2+8 x-4\right)+\mathrm{L}_y \left(8 x^2-8
   x+4\right)+2
\right]
\right. \nonumber \\ && \left.
+
\mathrm{L}_m\left[
\left(\frac{1}{2}-x\right) \mathrm{L}_x^2+\left(3 x-2 x^2\right)
   \mathrm{L}_x+\frac{37 x^2}{4}+\mathrm{L}_y^2 \left(\frac{1}{2}-x\right)-\frac{37
   x}{4}+\pi ^2 \left(-\frac{7 x^2}{6}+\frac{7
   x}{6}-\frac{7}{12}\right)
\right. \right. \nonumber \\ && \left. \left.
+\mathrm{L}_y \left(2 x^2-x-1\right)+\frac{25}{8}
\right]
+
\mathrm{L}_s\left[
\left(-12 x^2+14 x-7\right) \mathrm{L}_x^2+\left(20 x^2-22 x+\mathrm{L}_y \left(24
   x^2-24 x+12\right)
\right. \right. \right. \nonumber \\ && \left. \left. \left.
+8\right) \mathrm{L}_x-\frac{73 x^2}{4}+\frac{73
   x}{4}+\mathrm{L}_y \left(-20 x^2+18 x-6\right)+\mathrm{L}_y^2 \left(-12 x^2+14
   x-7\right)+\pi ^2 \left(\frac{13 x^2}{2}-\frac{13
   x}{2}
\right. \right. \right. \nonumber \\ && \left. \left. \left.
+\frac{13}{4}\right)-\frac{53}{8}
\right]
+
\left(\frac{2 x^2}{3}+3 x-\frac{3}{2}\right) \mathrm{L}_x^3+\left(-6 x+\mathrm{L}_y
   \left(-2 x^2-2 x+1\right)+\frac{3}{2}\right) \mathrm{L}_x^2+\li2
   \left(-4 x^2
\right. \right. \nonumber \\ && \left. \left.
+2 x-1\right) \mathrm{L}_x+\left(\left(3 x-\frac{3}{2}\right)
   \mathrm{L}_y^2+3 \mathrm{L}_y-24 x^2+\frac{55 x}{2}+\pi ^2 \left(\frac{46
   x^2}{3}-\frac{31 x}{3}+\frac{31}{6}\right)-10\right) \mathrm{L}_x
\right. \nonumber \\ && \left.
-\frac{3
   \mathrm{L}_y^2}{2}+\frac{429 x^2}{16}-\frac{429 x}{16}+\pi ^2 \left(-\frac{32
   x^2}{3}+\frac{17 x}{3}-\frac{29}{24}\right)+\mathrm{L}_y^3 \left(-\frac{2
   x^2}{3}-\frac{5 x}{3}+\frac{5}{6}\right)+\s12 \left(4 x^2
\right. \right. \nonumber \\ && \left. \left.
-6
   x+3\right)+\li3 \left(4 x^2-2 x+1\right)+\mathrm{L}_y \left(24
   x^2-\frac{41 x}{2}+\pi ^2 \left(-\frac{46 x^2}{3}+\frac{61
   x}{3}-\frac{61}{6}\right)+\frac{13}{2}\right)
\right. \nonumber \\ && \left.
+\left(-\frac{34
   x^2}{3}+\frac{28 x}{3}-\frac{14}{3}\right) \z3+\frac{283}{32}
\right\}
+
\mathrm{L}_m^4\left[
\frac{x^2}{4}-\frac{x}{4}+\frac{1}{8}
\right]
+
\mathrm{L}_m^3 \mathrm{L}_s\left[
\frac{2 x^2}{3}-\frac{2 x}{3}+\frac{1}{3}
\right]
\nonumber \\ &&
+
\mathrm{L}_m \mathrm{L}_s^3\left[
-\frac{4 x^2}{3}+\frac{4 x}{3}-\frac{2}{3}
\right]
+
\mathrm{L}_s^4\left[
\frac{x^2}{3}-\frac{x}{3}+\frac{1}{6}
\right]
+
\mathrm{L}_m^3\left[
\mathrm{L}_x \left(-\frac{2 x^2}{3}+\frac{2 x}{3}-\frac{1}{3}\right)+\mathrm{L}_y
   \left(\frac{2 x^2}{3}-\frac{2 x}{3}
\right. \right. \nonumber \\ && \left. \left.
+\frac{1}{3}\right)+\frac{1}{6}
\right]
+
\mathrm{L}_m^2 \mathrm{L}_s\left[
2 x^2-2 x+\mathrm{L}_x \left(-4 x^2+4 x-2\right)+\mathrm{L}_y \left(4 x^2-4
   x+2\right)+1
\right]
+
\mathrm{L}_m \mathrm{L}_s^2\left[
6 x^2
\right. \nonumber \\ && \left.
-6 x+\mathrm{L}_x \left(-8 x^2+8 x-4\right)+\mathrm{L}_y \left(8 x^2-8
   x+4\right)+2
\right]
+
\mathrm{L}_s^3\left[
-\frac{10 x^2}{3}+\frac{10 x}{3}+\mathrm{L}_y \left(-\frac{16 x^2}{3}+\frac{16
   x}{3}
\right. \right. \nonumber \\ && \left. \left.
-\frac{8}{3}\right)+\mathrm{L}_x \left(\frac{16 x^2}{3}-\frac{16
   x}{3}+\frac{8}{3}\right)-\frac{4}{3}
\right]
+
\mathrm{L}_m^2\left[
\left(\frac{x}{2}-\frac{1}{4}\right) \mathrm{L}_x^2+\left(x^2-\frac{3
   x}{2}\right) \mathrm{L}_x-\frac{5 x^2}{2}+\mathrm{L}_y^2
   \left(\frac{x}{2}
\right. \right. \nonumber \\ && \left. \left.
-\frac{1}{4}\right)+\frac{5 x}{2}+\mathrm{L}_y
   \left(-x^2+\frac{x}{2}+\frac{1}{2}\right)+\pi ^2 \left(\frac{2
   x^2}{3}-\frac{2 x}{3}+\frac{1}{3}\right)-\frac{3}{4}
\right]
+
\mathrm{L}_m \mathrm{L}_s\left[
(2 x-1) \mathrm{L}_x^2+\left(4 x^2
\right. \right. \nonumber \\ && \left. \left.
-6 x\right) \mathrm{L}_x-\frac{37 x^2}{2}+\frac{37
   x}{2}+\mathrm{L}_y^2 (2 x-1)+\mathrm{L}_y \left(-4 x^2+2 x+2\right)+\pi ^2
   \left(\frac{7 x^2}{3}-\frac{7 x}{3}+\frac{7}{6}\right)-\frac{25}{4}
\right]
\nonumber \\ &&
+
\mathrm{L}_s^2\left[
\left(12 x^2-14 x+7\right) \mathrm{L}_x^2+\left(-20 x^2+22 x+\mathrm{L}_y \left(-24
   x^2+24 x-12\right)-8\right) \mathrm{L}_x+\frac{73 x^2}{4}-\frac{73 x}{4}
\right. \nonumber \\ && \left.
+\pi
   ^2 \left(-\frac{13 x^2}{2}+\frac{13 x}{2}-\frac{13}{4}\right)+\mathrm{L}_y^2
   \left(12 x^2-14 x+7\right)+\mathrm{L}_y \left(20 x^2-18
   x+6\right)+\frac{53}{8}
\right]
+
\mathrm{L}_m\left[
\frac{2 x^2 \mathrm{L}_x^3}{3}
\right. \nonumber \\ && \left.
+\left(-x+\mathrm{L}_y \left(-2
   x^2+x-\frac{1}{2}\right)+\frac{1}{2}\right) \mathrm{L}_x^2+\li2 \left(-4
   x^2+2 x-1\right) \mathrm{L}_x+\left(-8 x^2+\frac{19 x}{2}+\pi ^2 \left(2
   x^2
\right. \right. \right. \nonumber \\ && \left. \left. \left.
-x+\frac{1}{2}\right)-4\right) \mathrm{L}_x+\frac{189 x^2}{8}+\mathrm{L}_y^2
   \left(\frac{1}{2}-x\right)-\frac{189 x}{8}+\pi ^2 \left(-\frac{5
   x^2}{3}+\frac{2 x}{3}+\frac{1}{4}\right)+\mathrm{L}_y^3 \left(-\frac{2
   x^2}{3}+\frac{4 x}{3}
\right. \right. \nonumber \\ && \left. \left.
-\frac{2}{3}\right)+\s12 \left(4 x^2-6
   x+3\right)+\li3 \left(4 x^2-2 x+1\right)+\mathrm{L}_y \left(8
   x^2-\frac{13 x}{2}+\pi ^2 \left(-2 x^2+3
   x
\right. \right. \right. \nonumber \\ && \left. \left. \left.
-\frac{3}{2}\right)+\frac{5}{2}\right)+\left(-\frac{44 x^2}{3}+\frac{38
   x}{3}-\frac{19}{3}\right) \z3+\frac{115}{16}
\right]
+
\mathrm{L}_s\left[
\left(-\frac{4 x^2}{3}-6 x+3\right) \mathrm{L}_x^3+\left(12 x+\mathrm{L}_y \left(4
   x^2
\right. \right. \right. \nonumber \\ && \left. \left. \left.
+4 x-2\right)-3\right) \mathrm{L}_x^2+\li2 \left(8 x^2-4 x+2\right)
   \mathrm{L}_x+\left((3-6 x) \mathrm{L}_y^2-6 \mathrm{L}_y+48 x^2-55 x+\pi ^2
   \left(-\frac{92 x^2}{3}
\right. \right. \right. \nonumber \\ && \left. \left. \left.
+\frac{62 x}{3}-\frac{31}{3}\right)+20\right)
   \mathrm{L}_x+3 \mathrm{L}_y^2-\frac{429 x^2}{8}+\frac{429 x}{8}+\li3
   \left(-8 x^2+4 x-2\right)+\s12 \left(-8 x^2
\right. \right. \nonumber \\ && \left. \left.
+12 x-6\right)+\mathrm{L}_y^3
   \left(\frac{4 x^2}{3}+\frac{10 x}{3}-\frac{5}{3}\right)+\pi ^2
   \left(\frac{64 x^2}{3}-\frac{34 x}{3}+\frac{29}{12}\right)+\mathrm{L}_y
   \left(-48 x^2+41 x+\pi ^2 \left(\frac{92 x^2}{3}
\right. \right. \right. \nonumber \\ && \left. \left. \left.
-\frac{122
   x}{3}+\frac{61}{3}\right)-13\right)+\left(\frac{68 x^2}{3}-\frac{56
   x}{3}+\frac{28}{3}\right) \z3-\frac{283}{16}
\right]
+
\left(-3 x^2-\frac{x}{12}+\frac{1}{24}\right) \mathrm{L}_x^4+\left(\frac{5
   x^2}{3}
\right. \nonumber \\ && \left.
+\frac{7 x}{3}+\mathrm{L}_y \left(12 x^2-\frac{13
   x}{3}+\frac{13}{6}\right)-\frac{17}{12}\right) \mathrm{L}_x^3+\left(\left(-7
   x^2+\frac{7 x}{2}-\frac{7}{4}\right) \mathrm{L}_y^2+\left(-5
   x^2-\frac{1}{4}\right) \mathrm{L}_y
\right. \nonumber \\ && \left.
-\frac{83 x}{4}+\pi ^2 \left(-\frac{58
   x^2}{3}+\frac{34
   x}{3}-\frac{17}{3}\right)+\frac{3}{1-x}+\frac{35}{8}\right)
   \mathrm{L}_x^2+\left(\left(2 x^2-4 x+2\right)
   \mathrm{L}_y^3+\left(\frac{3}{4}-\frac{3 x}{2}\right) \mathrm{L}_y^2
\right. \nonumber \\ && \left.
+\left(\pi ^2
   \left(\frac{116 x^2}{3}-\frac{89 x}{3}+\frac{89}{6}\right)-6\right)
   \mathrm{L}_y-48 x^2+\frac{125 x}{2}+\pi ^2 \left(25 x^2-13
   x+\frac{7}{6}\right)+\left(\frac{128 x^2}{3}
\right. \right. \nonumber \\ && \left. \left.
-\frac{164
   x}{3}+\frac{82}{3}\right) \z3-\frac{97}{4}\right)
   \mathrm{L}_x+\frac{2479 x^2}{32}-\frac{2479 x}{32}+\s13 \left(-36 x^2+62
   x-31\right)+\mathrm{L}_y^4 \left(-3 x^2
\right. \nonumber \\ && \left.
+\frac{71
   x}{12}-\frac{71}{24}\right)+\mathrm{L}_y^3 \left(-\frac{5 x^2}{3}+\frac{2
   x}{3}+\frac{11}{12}\right)+\pi ^4 \left(-\frac{829 x^2}{720}-\frac{191
   x}{720}+\frac{191}{1440}\right)+\s22 \left(8 x^2
\right. \nonumber \\ && \left.
-56
   x+28\right)+\li4 \left(12 x^2-14 x+7\right)+\mathrm{L}_y^2
   \left(\frac{37 x}{4}+\pi ^2 \left(-\frac{20 x^2}{3}+\frac{29
   x}{3}-\frac{29}{6}\right)-\frac{37}{8}+\frac{3}{x}\right)
\nonumber \\ &&
+\s12
   \left(10 x^2-11 x+\mathrm{L}_x \left(-16 x^2+56 x-28\right)+\mathrm{L}_y \left(8
   x^2-8 x+4\right)\right)+\li3 \left(10 x^2+3 x
\right. \nonumber \\ && \left.
+\mathrm{L}_x \left(-28
   x^2+18 x-9\right)+\mathrm{L}_y \left(16 x^2-20 x+10\right)-1\right)+\li2
   \left(\left(22 x^2-11 x+\frac{11}{2}\right) \mathrm{L}_x^2
\right. \nonumber \\ && \left.
+\left(-10 x^2-3
   x+\mathrm{L}_y \left(-16 x^2+20 x-10\right)+1\right) \mathrm{L}_x+\pi ^2
   \left(\frac{76 x^2}{3}-\frac{52
   x}{3}+\frac{26}{3}\right)\right)+\left(-\frac{265 x^2}{6}
\right. \nonumber \\ && \left.
+\frac{259
   x}{6}-\frac{293}{12}\right) \z3+\mathrm{L}_y \left(48 x^2-\frac{67
   x}{2}+\pi ^2 \left(-25 x^2+30 x-\frac{61}{6}\right)+\left(-\frac{128
   x^2}{3}+\frac{140 x}{3}
\right. \right. \nonumber \\ && \left. \left.
-\frac{70}{3}\right)
   \z3+\frac{39}{4}\right)+\pi ^2 \left(-\frac{1631 x^2}{48}+\frac{285
   x}{16}+\left(-2 x^2+2 x-1\right) \log
   (2)-\frac{247}{96}\right)+\frac{1621}{64},
\end{eqnarray}

\begin{eqnarray}
D_l &=&
\frac{1}{\epsilon^3}\left\{
-\frac{x^2}{2}+\frac{x}{2}-\frac{1}{4}
\right\}
+
\frac{1}{\epsilon^2}\left\{
\mathrm{L}_m\left[
-\frac{x^2}{3}+\frac{x}{3}-\frac{1}{6}
\right]
+
\mathrm{L}_s\left[
-\frac{x^2}{3}+\frac{x}{3}-\frac{1}{6}
\right]
+
\frac{5 x^2}{9}-\frac{5 x}{9}+\mathrm{L}_x \left(\frac{2 x^2}{3}
\right. \right. \nonumber \\ && \left. \left.
-\frac{2
   x}{3}+\frac{1}{3}\right)+\frac{19}{36}
\right\}
+
\frac{1}{\epsilon}\left\{
\mathrm{L}_m \mathrm{L}_s\left[
-\frac{2 x^2}{3}+\frac{2 x}{3}-\frac{1}{3}
\right]
+
\mathrm{L}_s^2\left[
x^2-x+\frac{1}{2}
\right]
+
\mathrm{L}_m\left[
\frac{5 x^2}{3}-\frac{5 x}{3}+1
\right]
\right. \nonumber \\ && \left.
+
\mathrm{L}_s\left[
-\frac{40 x^2}{9}+\frac{40 x}{9}+\mathrm{L}_x \left(\frac{4 x^2}{3}-\frac{4
   x}{3}+\frac{2}{3}\right)-\frac{37}{18}
\right]
+
\frac{649 x^2}{108}-\frac{649 x}{108}+\mathrm{L}_x \left(-\frac{10
   x^2}{3}+\frac{10 x}{3}-2\right)
\right. \nonumber \\ && \left.
+\pi ^2 \left(-\frac{11 x^2}{36}+\frac{11
   x}{36}-\frac{11}{72}\right)+\frac{589}{216}
\right\}
+
\mathrm{L}_m^3\left[
\frac{x^2}{9}-\frac{x}{9}+\frac{1}{18}
\right]
+
\mathrm{L}_m^2 \mathrm{L}_s\left[
\frac{2 x^2}{3}-\frac{2 x}{3}+\frac{1}{3}
\right]
\nonumber \\ &&
+
\mathrm{L}_m \mathrm{L}_s^2\left[
\frac{4 x^2}{3}-\frac{4 x}{3}+\frac{2}{3}
\right]
+
\mathrm{L}_s^3\left[
-\frac{8 x^2}{9}+\frac{8 x}{9}-\frac{4}{9}
\right]
+
\mathrm{L}_m^2\left[
-\frac{23 x^2}{18}+\frac{23 x}{18}-\frac{23}{36}
\right]
+
\mathrm{L}_m \mathrm{L}_s\left[
-4 x^2
\right. \nonumber \\ && \left.
+4 x-\frac{5}{3}
\right]
+
\mathrm{L}_s^2\left[
\frac{11 x^2}{9}-\frac{11 x}{9}+\mathrm{L}_x \left(-\frac{8 x^2}{3}+\frac{8
   x}{3}-\frac{4}{3}\right)+\frac{1}{9}
\right]
+
\mathrm{L}_m\left[
\frac{73 x^2}{18}-\frac{73 x}{18}+\pi ^2
   \left(-\frac{x^2}{6}
\right. \right. \nonumber \\ && \left. \left.
+\frac{x}{6}-\frac{1}{12}\right)+\frac{43}{36}
\right]
+
\mathrm{L}_s\left[
\left(\frac{2 x}{3}-\frac{1}{3}\right) \mathrm{L}_x^2+\left(\frac{20
   x^2}{3}-\frac{22 x}{3}+\frac{10}{3}\right) \mathrm{L}_x+\frac{91
   x^2}{18}-\frac{91 x}{18}+\pi ^2 \left(\frac{17 x^2}{18}
\right. \right. \nonumber \\ && \left. \left.
-\frac{17
   x}{18}+\frac{17}{36}\right)+\frac{19}{4}
\right]
+
\frac{2 x^2 \mathrm{L}_x^3}{9}+\left(-\frac{13 x}{9}+\mathrm{L}_y \left(-\frac{2
   x^2}{3}+\frac{2 x}{3}-\frac{1}{3}\right)+\frac{5}{9}\right)
   \mathrm{L}_x^2+\li2 \left(-\frac{4 x^2}{3}
\right. \nonumber \\ && \left.
+\frac{4
   x}{3}-\frac{2}{3}\right) \mathrm{L}_x+\left(-\frac{56 x^2}{9}+8 x+\pi ^2
   \left(\frac{25 x^2}{9}-\frac{17
   x}{9}+\frac{17}{18}\right)-\frac{29}{9}\right) \mathrm{L}_x-\frac{2639
   x^2}{216}+\frac{2639 x}{216}
\nonumber \\ &&
+\pi ^2 \left(\frac{2 x^2}{27}-\frac{14
   x}{27}+\frac{137}{216}\right)+\li3 \left(\frac{4 x^2}{3}-\frac{4
   x}{3}+\frac{2}{3}\right)+\left(-\frac{73 x^2}{9}+\frac{73
   x}{9}-\frac{73}{18}\right) \z3-\frac{3937}{432}, \nonumber \\ &&
\end{eqnarray}

\begin{eqnarray}
D_h &=&
\frac{1}{\epsilon^2}\left\{
\mathrm{L}_m\left[
-\frac{2 x^2}{3}+\frac{2 x}{3}-\frac{1}{3}
\right]
+
\mathrm{L}_s\left[
-\frac{4 x^2}{3}+\frac{4 x}{3}-\frac{2}{3}
\right]
+
\frac{10 x^2}{9}-\frac{10 x}{9}+\frac{5}{9}
\right\}
+
\frac{1}{\epsilon}\left\{
\mathrm{L}_m^2\left[
-\frac{x^2}{3}+\frac{x}{3}
\right. \right. \nonumber \\ && \left. \left.
-\frac{1}{6}
\right]
+
\mathrm{L}_s^2\left[
2 x^2-2 x+1
\right]
+
\mathrm{L}_m\left[
-\frac{5 x^2}{9}+\frac{5 x}{9}+\mathrm{L}_x \left(\frac{4 x^2}{3}-\frac{4
   x}{3}+\frac{2}{3}\right)+\frac{1}{18}
\right]
+
\mathrm{L}_s\left[
-\frac{50 x^2}{9}
\right. \right. \nonumber \\ && \left. \left.
+\frac{50 x}{9}+\mathrm{L}_x \left(\frac{8 x^2}{3}-\frac{8
   x}{3}+\frac{4}{3}\right)-\frac{19}{9}
\right]
+
\frac{131 x^2}{27}-\frac{131 x}{27}+\mathrm{L}_x \left(-\frac{20 x^2}{9}+\frac{20
   x}{9}-\frac{10}{9}\right)
\right. \nonumber \\ && \left.
+\pi ^2
   \left(-\frac{x^2}{3}+\frac{x}{3}-\frac{1}{6}\right)+\frac{101}{54}
\right\}
+
\mathrm{L}_m^3\left[
\frac{7 x^2}{9}-\frac{7 x}{9}+\frac{7}{18}
\right]
+
\mathrm{L}_m^2 \mathrm{L}_s\left[
\frac{4 x^2}{3}-\frac{4 x}{3}+\frac{2}{3}
\right]
+
\mathrm{L}_m \mathrm{L}_s^2\left[
\frac{2 x^2}{3}
\right. \nonumber \\ && \left.
-\frac{2 x}{3}+\frac{1}{3}
\right]
+
\mathrm{L}_s^3\left[
-\frac{14 x^2}{9}+\frac{14 x}{9}-\frac{7}{9}
\right]
+
\mathrm{L}_m^2\left[
\frac{37 x^2}{18}-\frac{37 x}{18}+\mathrm{L}_x \left(-\frac{4 x^2}{3}+\frac{4
   x}{3}-\frac{2}{3}\right)+\frac{43}{36}
\right]
\nonumber \\ &&
+
\mathrm{L}_m \mathrm{L}_s\left[
\frac{4 x^2}{9}-\frac{4 x}{9}+\mathrm{L}_x \left(-\frac{8 x^2}{3}+\frac{8
   x}{3}-\frac{4}{3}\right)+\frac{2}{9}
\right]
+
\mathrm{L}_s^2\left[
\frac{7 x^2}{3}-\frac{7 x}{3}+\mathrm{L}_x \left(-4 x^2+4 x-2\right)+\frac{1}{6}
\right]
\nonumber \\ &&
+
\mathrm{L}_m\left[
\frac{76 x^2}{9}-\frac{76 x}{9}+\mathrm{L}_x \left(-\frac{20 x^2}{9}+\frac{20
   x}{9}-\frac{16}{9}\right)+\frac{9}{2}
\right]
+
\mathrm{L}_s\left[
\left(\frac{2 x}{3}-\frac{1}{3}\right) \mathrm{L}_x^2+\left(\frac{40
   x^2}{9}-\frac{46 x}{9}
\right. \right. \nonumber \\ && \left. \left.
+\frac{14}{9}\right) \mathrm{L}_x+\frac{199
   x^2}{27}-\frac{199 x}{27}+\pi ^2
   \left(x^2-x+\frac{1}{2}\right)+\frac{349}{54}
\right]
+
\frac{2 x^2 \mathrm{L}_x^3}{9}+\left(-\frac{13 x}{9}+\mathrm{L}_y \left(-\frac{2
   x^2}{3}+\frac{2 x}{3}
\right. \right. \nonumber \\ && \left. \left.
-\frac{1}{3}\right)+\frac{5}{9}\right)
   \mathrm{L}_x^2+\li2 \left(-\frac{4 x^2}{3}+\frac{4
   x}{3}-\frac{2}{3}\right) \mathrm{L}_x+\left(-\frac{224 x^2}{27}+\frac{272
   x}{27}+\pi ^2 \left(\frac{8 x^2}{3}-\frac{16
   x}{9}+\frac{8}{9}\right)
\right. \nonumber \\ && \left.
-\frac{130}{27}\right) \mathrm{L}_x+\frac{113
   x^2}{54}-\frac{113 x}{54}+\pi ^2 \left(\frac{17 x^2}{54}-\frac{41
   x}{54}+\frac{83}{108}\right)+\li3 \left(\frac{4 x^2}{3}-\frac{4
   x}{3}+\frac{2}{3}\right)
\nonumber \\ &&
+\left(-\frac{76 x^2}{9}+\frac{76
   x}{9}-\frac{38}{9}\right) \z3-\frac{149}{108},
\end{eqnarray}

\begin{eqnarray}
E_l &=&
\frac{1}{\epsilon^3}\left\{
\frac{x^2}{2}-\frac{x}{2}+\frac{1}{4}
\right\}
+
\frac{1}{\epsilon^2}\left\{
\mathrm{L}_m\left[
\frac{x^2}{3}-\frac{x}{3}+\frac{1}{6}
\right]
+
\mathrm{L}_s\left[
\frac{x^2}{3}-\frac{x}{3}+\frac{1}{6}
\right]
-\frac{5 x^2}{9}+\frac{5 x}{9}+\mathrm{L}_x \left(-\frac{4 x^2}{3}+\frac{4
   x}{3}
\right. \right. \nonumber \\ && \left. \left.
-\frac{2}{3}\right)+\mathrm{L}_y \left(\frac{4 x^2}{3}-\frac{4
   x}{3}+\frac{2}{3}\right)-\frac{19}{36}
\right\}
+
\frac{1}{\epsilon}\left\{
\mathrm{L}_m \mathrm{L}_s\left[
\frac{2 x^2}{3}-\frac{2 x}{3}+\frac{1}{3}
\right]
+
\mathrm{L}_s^2\left[
-x^2+x-\frac{1}{2}
\right]
\right. \nonumber \\ && \left.
+
\mathrm{L}_m\left[
-\frac{5 x^2}{3}+\frac{5 x}{3}-1
\right]
+
\mathrm{L}_s\left[
\frac{40 x^2}{9}-\frac{40 x}{9}+\mathrm{L}_x \left(-\frac{8 x^2}{3}+\frac{8
   x}{3}-\frac{4}{3}\right)+\mathrm{L}_y \left(\frac{8 x^2}{3}-\frac{8
   x}{3}+\frac{4}{3}\right)
\right. \right. \nonumber \\ && \left. \left.
+\frac{37}{18}
\right]
-\frac{649 x^2}{108}+\frac{649 x}{108}+\mathrm{L}_y \left(-\frac{20
   x^2}{3}+\frac{20 x}{3}-4\right)+\pi ^2 \left(\frac{59 x^2}{36}-\frac{59
   x}{36}+\frac{59}{72}\right)+\mathrm{L}_x \left(\frac{20 x^2}{3}
\right. \right. \nonumber \\ && \left. \left.
-\frac{20
   x}{3}+4\right)-\frac{589}{216}
\right\}
+
\mathrm{L}_m^3\left[
-\frac{x^2}{9}+\frac{x}{9}-\frac{1}{18}
\right]
+
\mathrm{L}_m^2 \mathrm{L}_s\left[
-\frac{2 x^2}{3}+\frac{2 x}{3}-\frac{1}{3}
\right]
+
\mathrm{L}_m \mathrm{L}_s^2\left[
-\frac{4 x^2}{3}+\frac{4 x}{3}
\right. \nonumber \\ && \left.
-\frac{2}{3}
\right]
+
\mathrm{L}_s^3\left[
\frac{8 x^2}{9}-\frac{8 x}{9}+\frac{4}{9}
\right]
+
\mathrm{L}_m^2\left[
\frac{23 x^2}{18}-\frac{23 x}{18}+\frac{23}{36}
\right]
+
\mathrm{L}_m \mathrm{L}_s\left[
4 x^2-4 x+\frac{5}{3}
\right]
+
\mathrm{L}_s^2\left[
-\frac{55 x^2}{9}
\right. \nonumber \\ && \left.
+\frac{55 x}{9}+\mathrm{L}_y \left(-\frac{16 x^2}{3}+\frac{16
   x}{3}-\frac{8}{3}\right)+\mathrm{L}_x \left(\frac{16 x^2}{3}-\frac{16
   x}{3}+\frac{8}{3}\right)-\frac{23}{9}
\right]
+
\mathrm{L}_m\left[
-\frac{73 x^2}{18}+\frac{73 x}{18}
\right. \nonumber \\ && \left.
+\pi ^2
   \left(\frac{x^2}{6}-\frac{x}{6}+\frac{1}{12}\right)-\frac{43}{36}
\right]
+
\mathrm{L}_s\left[
\left(\frac{2}{3}-\frac{4 x}{3}\right) \mathrm{L}_x^2+\left(-\frac{40
   x^2}{3}+\frac{44 x}{3}-\frac{20}{3}\right) \mathrm{L}_x+\frac{1027
   x^2}{54}
\right. \nonumber \\ && \left.
+\mathrm{L}_y^2 \left(\frac{2}{3}-\frac{4 x}{3}\right)-\frac{1027
   x}{54}+\pi ^2 \left(-\frac{89 x^2}{18}+\frac{89
   x}{18}-\frac{89}{36}\right)+\mathrm{L}_y \left(\frac{40 x^2}{3}-12
   x+\frac{16}{3}\right)+\frac{787}{108}
\right]
\nonumber \\ &&
-\frac{4}{9} x^2 \mathrm{L}_x^3+\left(\frac{26 x}{9}+\mathrm{L}_y \left(\frac{4
   x^2}{3}-\frac{4 x}{3}+\frac{2}{3}\right)-\frac{10}{9}\right)
   \mathrm{L}_x^2+\li2 \left(\frac{8 x^2}{3}-\frac{8
   x}{3}+\frac{4}{3}\right) \mathrm{L}_x+\left(\frac{112 x^2}{9}
\right. \nonumber \\ && \left.
-16 x+\pi ^2
   \left(-\frac{50 x^2}{9}+\frac{34
   x}{9}-\frac{17}{9}\right)+\frac{58}{9}\right) \mathrm{L}_x-\frac{13963
   x^2}{648}+\frac{13963 x}{648}+\mathrm{L}_y^2 \left(\frac{26
   x}{9}-\frac{16}{9}\right)
\nonumber \\ &&
+\li3 \left(-\frac{8 x^2}{3}+\frac{8
   x}{3}-\frac{4}{3}\right)+\s12 \left(-\frac{8 x^2}{3}+\frac{8
   x}{3}-\frac{4}{3}\right)+\mathrm{L}_y^3 \left(\frac{4 x^2}{9}-\frac{8
   x}{9}+\frac{4}{9}\right)
\nonumber \\ &&
+\pi ^2 \left(\frac{250 x^2}{27}-\frac{202
   x}{27}+\frac{631}{216}\right)+\mathrm{L}_y \left(-\frac{112 x^2}{9}+\frac{80
   x}{9}+\pi ^2 \left(\frac{50 x^2}{9}-\frac{22
   x}{3}+\frac{11}{3}\right)-\frac{26}{9}\right)
\nonumber \\ &&
+\left(\frac{13
   x^2}{9}-\frac{13 x}{9}+\frac{13}{18}\right) \z3-\frac{10069}{1296},
\end{eqnarray}

\begin{eqnarray}
E_h &=&
\frac{1}{\epsilon^2}\left\{
\mathrm{L}_m\left[
\frac{2 x^2}{3}-\frac{2 x}{3}+\frac{1}{3}
\right]
+
\mathrm{L}_s\left[
\frac{4 x^2}{3}-\frac{4 x}{3}+\frac{2}{3}
\right]
+
-\frac{10 x^2}{9}+\frac{10 x}{9}-\frac{5}{9}
\right\}
+
\frac{1}{\epsilon}\left\{
\mathrm{L}_m^2\left[
\frac{x^2}{3}-\frac{x}{3}+\frac{1}{6}
\right]
\right. \nonumber \\ && \left.
+
\mathrm{L}_s^2\left[
-2 x^2+2 x-1
\right]
+
\mathrm{L}_m\left[
\frac{5 x^2}{9}-\frac{5 x}{9}+\mathrm{L}_x \left(-\frac{8 x^2}{3}+\frac{8
   x}{3}-\frac{4}{3}\right)+\mathrm{L}_y \left(\frac{8 x^2}{3}-\frac{8
   x}{3}+\frac{4}{3}\right)-\frac{1}{18}
\right]
\right. \nonumber \\ && \left.
+
\mathrm{L}_s\left[
\frac{50 x^2}{9}-\frac{50 x}{9}+\mathrm{L}_x \left(-\frac{16 x^2}{3}+\frac{16
   x}{3}-\frac{8}{3}\right)+\mathrm{L}_y \left(\frac{16 x^2}{3}-\frac{16
   x}{3}+\frac{8}{3}\right)+\frac{19}{9}
\right]
-\frac{131 x^2}{27}
\right. \nonumber \\ && \left.
+\frac{131 x}{27}+\mathrm{L}_y \left(-\frac{40
   x^2}{9}+\frac{40 x}{9}-\frac{20}{9}\right)+\pi ^2 \left(\frac{5
   x^2}{3}-\frac{5 x}{3}+\frac{5}{6}\right)+\mathrm{L}_x \left(\frac{40
   x^2}{9}-\frac{40 x}{9}+\frac{20}{9}\right)
\right. \nonumber \\ && \left.
-\frac{101}{54}
\right\}
+
\mathrm{L}_m^3\left[
-\frac{7 x^2}{9}+\frac{7 x}{9}-\frac{7}{18}
\right]
+
\mathrm{L}_m^2 \mathrm{L}_s\left[
-\frac{4 x^2}{3}+\frac{4 x}{3}-\frac{2}{3}
\right]
+
\mathrm{L}_m \mathrm{L}_s^2\left[
-\frac{2 x^2}{3}+\frac{2 x}{3}-\frac{1}{3}
\right]
\nonumber \\ &&
+
\mathrm{L}_s^3\left[
\frac{14 x^2}{9}-\frac{14 x}{9}+\frac{7}{9}
\right]
+
\mathrm{L}_m^2\left[
-\frac{37 x^2}{18}+\frac{37 x}{18}+\mathrm{L}_y \left(-\frac{8 x^2}{3}+\frac{8
   x}{3}-\frac{4}{3}\right)+\mathrm{L}_x \left(\frac{8 x^2}{3}-\frac{8
   x}{3}+\frac{4}{3}\right)
\right. \nonumber \\ && \left.
-\frac{43}{36}
\right]
+
\mathrm{L}_m \mathrm{L}_s\left[
-\frac{4 x^2}{9}+\frac{4 x}{9}+\mathrm{L}_y \left(-\frac{16 x^2}{3}+\frac{16
   x}{3}-\frac{8}{3}\right)+\mathrm{L}_x \left(\frac{16 x^2}{3}-\frac{16
   x}{3}+\frac{8}{3}\right)-\frac{2}{9}
\right]
\nonumber \\ &&
+
\mathrm{L}_s^2\left[
-\frac{65 x^2}{9}+\frac{65 x}{9}+\mathrm{L}_y \left(-8 x^2+8 x-4\right)+\mathrm{L}_x
   \left(8 x^2-8 x+4\right)-\frac{47}{18}
\right]
+
\mathrm{L}_m\left[
-\frac{76 x^2}{9}+\frac{76 x}{9}
\right. \nonumber \\ && \left.
+\mathrm{L}_y \left(-\frac{40 x^2}{9}+\frac{40
   x}{9}-\frac{32}{9}\right)+\mathrm{L}_x \left(\frac{40 x^2}{9}-\frac{40
   x}{9}+\frac{32}{9}\right)-\frac{9}{2}
\right]
+
\mathrm{L}_s\left[
\left(\frac{2}{3}-\frac{4 x}{3}\right) \mathrm{L}_x^2+\left(-\frac{80
   x^2}{9}
\right. \right. \nonumber \\ && \left. \left.
+\frac{92 x}{9}-\frac{28}{9}\right) \mathrm{L}_x+\frac{451
   x^2}{27}+\mathrm{L}_y^2 \left(\frac{2}{3}-\frac{4 x}{3}\right)-\frac{451
   x}{27}+\pi ^2 \left(-5 x^2+5 x-\frac{5}{2}\right)+\mathrm{L}_y \left(\frac{80
   x^2}{9}
\right. \right. \nonumber \\ && \left. \left.
-\frac{68 x}{9}+\frac{16}{9}\right)+\frac{301}{54}
\right]
+
-\frac{4}{9} x^2 \mathrm{L}_x^3+\left(\frac{26 x}{9}+\mathrm{L}_y \left(\frac{4
   x^2}{3}-\frac{4 x}{3}+\frac{2}{3}\right)-\frac{10}{9}\right)
   \mathrm{L}_x^2+\li2 \left(\frac{8 x^2}{3}
\right. \nonumber \\ && \left.
-\frac{8
   x}{3}+\frac{4}{3}\right) \mathrm{L}_x+\left(\frac{448 x^2}{27}-\frac{544
   x}{27}+\pi ^2 \left(-\frac{16 x^2}{3}+\frac{32
   x}{9}-\frac{16}{9}\right)+\frac{260}{27}\right) \mathrm{L}_x-\frac{5809
   x^2}{162}
\nonumber \\ &&
+\frac{5809 x}{162}+\mathrm{L}_y^2 \left(\frac{26
   x}{9}-\frac{16}{9}\right)+\li3 \left(-\frac{8 x^2}{3}+\frac{8
   x}{3}-\frac{4}{3}\right)+\s12 \left(-\frac{8 x^2}{3}+\frac{8
   x}{3}-\frac{4}{3}\right)
\nonumber \\ &&
+\mathrm{L}_y^3 \left(\frac{4 x^2}{9}-\frac{8
   x}{9}+\frac{4}{9}\right)+\pi ^2 \left(\frac{487 x^2}{54}-\frac{391
   x}{54}+\frac{301}{108}\right)+\mathrm{L}_y \left(-\frac{448 x^2}{27}+\frac{352
   x}{27}+\pi ^2 \left(\frac{16 x^2}{3}
\right. \right. \nonumber \\ && \left. \left.
-\frac{64
   x}{9}+\frac{32}{9}\right)-\frac{164}{27}\right)+\left(\frac{16
   x^2}{9}-\frac{16 x}{9}+\frac{8}{9}\right) \z3-\frac{5023}{324},
\end{eqnarray}

\begin{eqnarray}
F &=&
\mathrm{L}_s^2\left[
\frac{4 x^2}{9}-\frac{4 x}{9}+\frac{2}{9}
\right]
+
\mathrm{L}_s\left[
-\frac{40 x^2}{27}+\frac{40 x}{27}-\frac{20}{27}
\right]
+
\frac{100 x^2}{81}-\frac{100 x}{81}+\pi ^2 \left(-\frac{4 x^2}{9}+\frac{4
   x}{9}-\frac{2}{9}\right)+\frac{50}{81}. \nonumber \\ &&
\end{eqnarray}
}}
Notice that our results are expressed not only in terms of the classic
polylogarithms up to weight four, but also in terms of Nielsen polylogarithms
\begin{equation}
\mathrm{S}_{n,p}(x) = \frac{(-1)^{n+p-1}}{(n-1)!p!} \int_0^1 dy
\frac{\log^{n-1}(y)\log^p(1-x y)}{y}.
\end{equation}

\subsection*{Conclusions}

In this letter, we have presented the two-loop virtual QCD corrections to the
production of heavy-quarks in the light quark annihilation channel in the
ultra-relativistic limit. Our results form a crucial part of the NNLO predictions for
heavy-quark production in hadron-hadron collisions. However, depending on the
mass of the heavy quark (bottom or top) and the kinematics of the process
under consideration power corrections in the heavy-quark mass may have to be
considered as well.
Our results have been derived by combining two completely different methods which have substantial overlap.
This provides direct and highly non-trivial checks of the QCD
factorization approach of Ref.~\cite{Mitov:2006xs},
of the direct calculation of massive Feynman diagrams and, last but not least,
also on the available masseless results of Ref.~\cite{Anastasiou:2000kg}.

In order to yield physical cross sections, our result for $\langle {\cal M}^{(0)} | {\cal M}^{(2)} \rangle $
still has to be combined with the tree-level $2 \to 4$, the one-loop $2 \to 3$
as well as the square of the one-loop $2 \to 2$ processes.
While some of the matrix elements (including the full mass dependence)
can be easily generated, others became available in the literature only rather recently,
see e.g. Refs.\cite{Korner:2005rg,Dittmaier:2007wz}.
The combination of all these contributions enables the analytic cancellation of the remaining
infrared divergences as well as the isolation of the initial state singularities
which need to be absorbed into parton distribution functions.
This is a necessary prerequisite e.g. to the construction of numerical programs
which provide NNLO QCD estimates of observable scattering cross sections.

Finally, while we have chosen to work in this letter with squared matrix elements,
it should, of course, be clear that with the help of Refs.~\cite{DeFreitas:2004tk,Glover:2004si}
analogous results can be derived for the massive two-loop amplitude
$|{\cal M}^{(2)} \rangle$ itself.

A {\sc Mathematica} file with our results can be obtained by downloading the
source from the preprint server {\tt http://arXiv.org}.
The results are also available from the authors upon request.

{\bf{Acknowledgments:}}
We acknowledge useful discussions with
N.~Glover, J.~Gluza, Z.~Merebashvili and T.~Riemann.
The work of M.C. was supported by the Sofja Kovalevskaja Award of the
Alexander von Humboldt Foundation
and by the ToK Program ``ALGOTOOLS'' (MTKD-CD-2004-014319).
A.M. thanks the Alexander von Humboldt Foundation for support and
S.M. acknowledges contract VH-NG-105 of the Helmholtz Gemeinschaft.
This work was supported in part by the Deutsche Forschungsgemeinschaft in
Sonderforschungs\-be\-reich/Transregio~9.

{\footnotesize

redpajama_set_name: RedPajamaArXiv


MAUI MALL. Friday: Whole Foods Aloha Friday live music, 10 a.m. Sunday: Zumba by Keoni's Hot Lava Dance Fit Studio, 9 a.m.; Whole Foods Arts & Crafts Fair, 10 a.m. to 2 p.m.; "Endless Summer Cruise" fourth annual custom car cruise, show and shine featuring food trucks, DJ music and raffles for prizes, 10 a.m. to 5 p.m.
OUTLETS OF MAUI. Today, Friday, Saturday and Monday: Tahitian dance show, 6 p.m. Friday: Aloha Friday Sunset Sale, 5 to 7 p.m. Tuesday: Free ukulele lessons, 11 a.m. Wednesday: Tahitian dance show, 5:30 p.m.
THE SHOPS AT WAILEA. Today: Coconut frond weaving, 2:30 p.m.; Polynesian show, 5:30 p.m. Friday: Island wood carving, 12:30 p.m.; Ukulele lesson, 3 p.m. Monday: Papa Hula, 1 p.m.; Lei po'o making, 2:30 p.m.; Ukulele lessons, 5:30 p.m. Tuesday: Coconut husking, 2:30 p.m.; Polynesian show, 5:30 p.m. Wednesday: Island wood carving, 10:30 a.m.; Lei po'o making, 1:30 p.m. Concerts at The Shops featuring Maui-born musician and multiple Na Hoku Hanohano Award-winner Anuhea, 5:30 to 7 p.m. at the main Fountain Courtyard.
WHALERS VILLAGE. Tonight: Free movie: "Smurfs: The Lost Village" screening on lower level courtyard, 15 minutes after sunset. Today and Tuesday: Lei making class, 11 a.m. to 1 p.m. Friday: Hula lessons, 3 p.m. to 4 p.m. Saturday and Wednesday: Hula shows, 7 p.m. to 7:25 p.m. and 7:30 p.m. to 8 p.m. Sunday: Live music, 6 to 8 p.m. Monday: Ukulele lessons, 2 to 3 p.m. and 3 to 4 p.m. Tuesday: Free movie: "Cars 3" screening on lower level courtyard, 15 minutes after sunset.


text:
Colchagua is a Chilean village located in San Rosendo, Bío Bío Province, Bío Bío Region. It is located close to the historic town of Rere and the Estación Buenuraqui railway station.

References

Populated places in Bío Bío Province

redpajama_set_name: RedPajamaWikipedia


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The team was formed for this competition, motivated by a strong friendship and the will to share ideas to develop together a project in a challenging way. Our different backgrounds, origins and experiences helped us to keep the mind open.
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text:
Q: Align 2 columns equally in excel I have 2 columns in excel col1 and col2. Both have same data. I want to align col2 as per col1 as col1 is master data.
For ex:
    col1        col2      col3
    abc          ghi       jjj
    def          abc       222
    ghi          def       666
    mmm
    lll

I want to organize the col2 data as col1 and also col1 will have data that is not present in col2. I want to retain all data in col1. I also want to retain the mapped data from col3 w.r.t col2
My result should look like
    col1        col2      col3
    abc          abc      222
    def          def      666
    ghi          ghi      jjj
    mmm
    lll

Appreciate your help

A: Try Index/Match combination like-
=IFERROR(INDEX($B$2:$B$6,MATCH(D2,$B$2:$B$6,0)),"")

VLOOKUP() will also work.
=IFERROR(VLOOKUP(D2,$B$2:$B$6,1,FALSE),"")

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Study suggests impact of concussions reduced in children with more years of sport experience
April 8, 2019 Newsmuellerm
More years in sports – not less – may help protect the brains of children who have had a concussion against future concussions, a new study says.
Research from York University's Faculty of Health found that children who have played in a performance sport for at least seven years, and have a history of concussion, recover better from concussions than children who have fewer years in the sport. Their years in the sport may give them more skill-related motor "reserve" that helps them to get back to the level they were playing at previously.
"Our results suggest there's an advantage to staying with skilled activity to the point where your brain can maintain performance even when it's still being affected in subtle ways by a past injury," said senior author Lauren Sergio, a professor in the School of Kinesiology & Health Science and the Centre for Vision Research at York University. "This performance may be protective and would reduce vulnerability to another concussion when playing with non-concussed peers."
Alison Macpherson
Lauren Sergio
Sergio and her former postdoctoral student and lead author Marc Dalecki, now an assistant professor in the School of Kinesiology at Louisiana State University, along with co-author Alison Macpherson, a professor in York's School of Kinesiology & Health Science, examined factors that may influence performance recovery after a concussion.
"The findings of this study are important because it demonstrates that athletes with more years of experience return to pre-concussion levels more quickly than inexperienced athletes," said Macpherson. "This can be one factor to help guide decisions about returning athletes to play."
Their study looked at 126 youth, aged eight to 17 years old, 64 with a history of concussion and 62 without, over a two-year period from 2013 to 2015. Participants, parents, team managers and coaches were interviewed to obtain detailed information about the concussion history. All concussion history participants were defined as asymptomatic in accordance with current return-to-play protocol guidelines at the time of testing. Participants were asked to perform two visuomotor tasks over 20 trials that required sliding the index finger of the dominant hand along a dual touchscreen laptop, with touchscreens in the vertical and horizontal planes.
In the first condition, they moved their hand on the vertical screen in the same direction as the target on the screen. In the second condition, participants had to slide their finger along the horizontal touchscreen in the opposite direction of a presented target on the screen. In other words, to move the cursor to the left, they had to slide their finger to the right. This type of condition is similar to passing a hockey puck to a teammate on the left while skating to the right.
Researchers found youth with a concussion history with seven or more years of sport experience and higher levels of eye-limb co-ordination-related sport experience had quicker motor skill recovery times (around 12 months) compared to their peers with less than six years of sport experience (around 30 months or two seasons later).They found no differences in these results when factoring in sex or age.
The study is published in the European Journal of Sport Science.
Courtesy of YFile.
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The Office of the Vice President Research and Innovation will launch the 2019-2020 York Research Chairs (YRC) competition. The YRC program mirrors and complements the Canada Research Chairs program, and recognizes researchers who have demonstrated[...]
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