text:
\section{Introduction}
\label{sec:1}

\vspace*{1mm}
\noindent
Analytic expressions in perturbative Quantum Chromodynamics, as for partonic sub--system scattering cross
sections and Wilson coefficients, provide an excellent basis for both numerical studies, fits to precision data
and provide analytic insight into the structure of Feynman integrals. Analytic results also allow to derive
important limiting cases and make it easier to incorporate resummations in specific kinematic regions. With a
growing number of scales and loops for the respective processes it becomes more difficult to obtain analytic
results, although there has been significant progress in analytic integration methods recently; for a survey
cf.~\cite{Blumlein:2018cms}. The precise knowledge of these corrections is of importance to measure the
polarized parton densities in high energy collisions and to  determine, related to it, the
strong coupling constant $\alpha_s(M_Z)$ and the heavy quark masses, cf.~\cite{Boer:2011fh}.

The first two--loop QCD heavy flavor corrections to the polarized structure function $g_1(x,Q^2)$ have been
calculated in \cite{Buza:1996xr} in the asymptotic region $Q^2 \gg m^2$, where $Q^2$ denotes the virtuality of
the exchanged photon and $m$ the mass of the heavy quark. The asymptotic two--loop QCD corrections have been
recalculated in \cite{Bierenbaum:2007zz,POL18}. In \cite{Buza:1996xr} the region of low values of $Q^2$ has
been modeled by an ansatz. The leading threshold resummation for the gluonic contributions has been studied
in \cite{Eynck:2000gz}. The complete two--loop polarized heavy flavor Wilson coefficient in the non--singlet case
has been calculated analytically in the tagged flavor case in \cite{Buza:1996xr} and for the complete
contribution to the structure function $g_1(x,Q^2)$ in \cite{Blumlein:2016xcy}, also completing former work on
the polarized Bjorken sum rule in \cite{Blumlein:1998sh}. Numerical results for the polarized two--loop heavy
flavor case have been given in \cite{Hekhorn:2018ywm} recently. Finally, in the non--singlet case the asymptotic
contributions have been calculated to three--loop order analytically in \cite{Behring:2015zaa,Ablinger:2014vwa}.

In the present paper we follow Ref.~\cite{Blumlein:2019qze} in the unpolarized case and calculate the polarized
pure singlet two--loop
heavy flavor corrections for the structure function $g_1(x,Q^2)$ in the whole kinematic range analytically. We
also compute the corresponding massless contributions, which have first been calculated in
\cite{Zijlstra:1993sh} and later in \cite{Vogt:2008yw}.
Since the calculations are carried out using dimensional regularization in $D = 4 + \epsilon$ dimensions one may
work in the Larin--scheme \cite{Larin:1993tq}\footnote{For other $\gamma_5$ schemes see Refs.~\cite{HVBM}.}
and perform a finite renormalization to the {\sf M}--scheme \cite{Matiounine:1998re,Moch:2014sna} afterwards.
We derive
both the result in the asymptotic range $Q^2 \gg m^2$, see also Refs.~\cite{Buza:1996xr,POL18}, and in the
threshold
region. Numerical results are presented. Various technical aspects of the present calculation can be found in
Ref.~\cite{Blumlein:2019qze} already.

The paper is organized as follows. In Section~\ref{sec:2} we summarize basic relations for the polarized
deep--inelastic scattering cross section. In Section~\ref{sec:3} the result for the massless pure singlet
Wilson coefficient $C_{g_1}^{(2),\rm PS}$ is presented. The recalculation of the massless Wilson coefficient
is necessary, since in Ref.~\cite{Zijlstra:1993sh} different schemes have been used in part. The corresponding
massive Wilson coefficient is calculated in Section~\ref{sec:4}. The corresponding results for the twist--2
contributions to the structure function $g_2(x,Q^2)$ can be obtained by using the Wandzura--Wilczek relation
\cite{Wandzura:1977qf}, as has been shown for the massless quarkonic \cite{G2A,G2B}  and gluonic \cite{Blumlein:2003wk}
cases, for diffractive scattering \cite{Blumlein:2002fw}, non--forward scattering \cite{Blumlein:2000cx}, and the
target mass corrections \cite{BT,TM}. Limiting cases are studied in Section~\ref{sec:5} and numerical results
are presented in Section~\ref{sec:6}. The conclusions are given in Section~\ref{sec:7}. Some Mellin convolutions
appearing due to renormalization are listed in the Appendix.

\vspace*{-.3cm}
\section{The Deep-inelastic Scattering Cross Section}
\label{sec:2}

\vspace*{1mm}
\noindent
The scattering cross sections for deep--inelastic charged lepton scattering of polarized nucleons

\newpage
\noindent
are obtained polarizing the incoming lepton longitudinally and the target nucleon either longitudinally or
transversally, resulting into the spin 4-vectors $S_L$ and $S_T$,
\begin{eqnarray}
S_L &=& (0, 0, 0; M) \\
S_T &=& M(0, \cos(\beta), \sin(\beta); 0)
\end{eqnarray}
in the nucleon rest frame. One has $S_L.p = S_T.p =0$, with $p$ the nucleon 4-momentum.
The scattering cross sections are given by, cf.~e.g.~\cite{BT,Lampe:1998eu},
\begin{eqnarray}
\frac{d^2 \sigma(\lambda, \pm S_L)}{dx dy} &=& \pm 2 \pi S \left[-2 \lambda y \left(2 - y - \frac{2xy M^2}{S}\right)
x g_1(x,Q^2)  + 8 \lambda \frac{y x M^2}{S} x g_2(x,Q^2) \right]
\\
\frac{d^3 \sigma(\lambda, \pm S_T)}{dx dy d\phi} &=& \pm S \frac{\alpha^2}{Q^4} 2 \sqrt{\frac{M^2}{S}}
\sqrt{xy \left[1-y - \frac{xyM^2}{S}\right]} \cos(\beta-\phi)
\nonumber\\ && \times
\left[-2 \lambda y x g_1(x,Q^2) - 4 \lambda x
g_2(x,Q^2)\right]
\end{eqnarray}
for pure virtual photon exchange. Here $S$ denotes the cms energy of the process, $M$ is the nucleon mass,
$\lambda$ the degree of lepton polarization, $\alpha = e^2/(4\pi)$ is the fine structure constant,
$Q^2 = -q^2$ denotes the photon virtuality and $x = Q^2/(Sy), y = l.q/p.q$ are the Bjorken variables with
$l$ the incoming charged lepton and proton momenta, $S = (p+l)^2$ and $\phi$ is the
azimuthal angle of the final state lepton, which can
be integrated over in the case of longitudinal polarization.

In the following we will present a series of relations in Mellin--$N$ space for convenience. The respective
quantities
in momentum-fraction $z$--space are related to those in Mellin--space by the transformation
\begin{eqnarray}
\label{eq:MEL}
{\rm\bf M}\left[f(z)\right](N) = \int_0^1 dz z^{N-1} f(z).
\end{eqnarray}
The structure function $g_1(N,Q^2)$ is given in the twist--two approximation using the factorization
theorems \cite{FACT} by
\begin{eqnarray}
g_1(N,Q^2) &=& \frac{1}{2} \Biggl[
\frac{1}{N_F} \sum_{k=0}^{N_F} e_k^2 \left\{\Sigma(N,\mu_F^2) C_q^{\rm PS}\left(N,\frac{Q^2}{\mu_F^2}\right)
+
G(N,\mu_F^2) C_g^{\rm S}\left(N,\frac{Q^2}{\mu_F^2}\right) \right\}
\nonumber\\ &&
+ \Delta\left(N,\mu_F^2\right) C_q^{\rm NS}\left(N,\frac{Q^2}{\mu_F^2}\right) \Biggr].
\end{eqnarray}
Here
\begin{eqnarray}
\Sigma(N) = \sum_{k=1}^{N_F} \left[ q(N) + \bar{q}(N) \right]
\end{eqnarray}
denotes the singlet distribution, $G(N)$ the gluon distribution, $\Delta(N)$ the flavor non--singlet distribution
\begin{eqnarray}
\Delta(N) = \sum_{i=1}^{N_F} \left[e_i^2 - \frac{1}{N_F} \sum_{k=1}^{N_F} e_k^2\right] \left[q_i(N) + \bar{q}_i(N)
\right].
\end{eqnarray}
$e_k$ labels the electric charge of the $k$th light quark.

The Mellin transform of the structure function $g_2$ is related to that of $g_1$ by
\begin{eqnarray}
g_2(N,Q^2) = - \frac{N-1}{N} g_1(N,Q^2)
\end{eqnarray}
or
\begin{eqnarray}
g_2(x,Q^2) = - g_1(x,Q^2) + \int_0^1 \frac{dy}{y} g_1(y,Q^2).
\end{eqnarray}
Note that in the massive pure singlet case the support of both structure functions in limited by $0 < x <
1/(1+4 m^2/Q^2)$ due to the production of two heavy quarks.

The different steps in the renormalization and factorization of the polarized massless Wilson coefficients
have been described in \cite{Zijlstra:1992qd,Zijlstra:1993sh} and for the massive Wilson coefficients in
\cite{POL18} using the Larin--scheme. In the present case the finite renormalization moving to the
{\sf M}--scheme only affects the massless or massive Wilson coefficient by adding the term $-z_{\rm PS}^{(2)}$,
Eq.~(\ref{eq:ZPS2}).
\section{The Massless Wilson Coefficient}
\label{sec:3}

\vspace*{1mm}
\noindent
The Feynman diagrams contributing to the polarized massless two--loop Wilson coefficient are shown in
Figure~\ref{fig:FIG2}.
\begin{figure}[H]
      \centering
      \includegraphics[width=0.5\textwidth]{PLOTS/FIG2.pdf}
      \caption{\small \sf The Feynman diagrams for the massless and massive two--loop pure singlet Wilson
coefficient.}
      \label{fig:FIG2}
\end{figure}
\noindent
forming the amplitude squared $G_{\mu\nu}$, where the indices refer to the coupling of the virtual photon.
Here all quark lines are massless.
The massless resp. massive Wilson coefficients are obtained following Ref.~\cite{Zijlstra:1993sh},
Eqs.~(3.7--3.18).  The corresponding phase space integrals have been given in Ref.~\cite{Blumlein:2019qze}.
We apply the Larin--scheme \cite{Larin:1993tq} in which the contraction of the free indices of the
two appearing Levi--Civita tensors have to be performed in $D$ dimensions.

The unrenormalized two--loop massless pure singlet Wilson coefficient reads  in Mellin--$N$ space
\begin{eqnarray}
\hspace*{-3mm}
\hat{\hat{C}}_{g_1}^{(2),\rm PS} &=& \hat{a}_s^2 S_\epsilon^2 \left(\frac{Q^2}{\mu^2}\right)^\epsilon\Biggl\{
\frac{1}{\epsilon^2} \frac{1}{2} P_{qg}^{(0)} P_{gq}^{(0)}
+ \frac{1}{\epsilon} \left[ \frac{1}{2} P_{qq}^{(1),\rm PS} + P_{gq}^{(0)} \bar{c}^{(1)}_{g_1,g}\right]
+ \bar{c}^{(2), \rm PS}_{g_1,q} + P_{qg}^{(0)} a_{g_1,g}^{(1)} \Biggr\},
\end{eqnarray}
\noindent
where  $\hat{a}_s = \hat{g}_s^2/(4\pi)^2$ denotes the unrenormalized strong coupling constant,
$S_\epsilon = \exp[\tfrac{\epsilon}{2}(\gamma_E - \ln(4\pi))]$ the spherical factor and $\gamma_E$ the
Euler--Mascheroni constant,
$\bar{c}_i^{(k)}$ and $a_{g_1,g}^{(1)}$ are the expansion coefficients of the one-loop Wilson
coefficient with
\begin{eqnarray}
\hat{\hat{C}}_{g_1,g}^{(1)} &=& \hat{a}_s S_\epsilon \left(\frac{Q^2}{\mu^2}\right)^{\epsilon/2}
\left[ \frac{1}{\epsilon} P_{qg} + \bar{c}_{g_1,g}^{(1)} + \epsilon a_{g_1,g}^{(1)}\right],
\end{eqnarray}
given by the Feynman diagrams in Figure~\ref{fig:FIG1},
\begin{figure}[H]
      \centering
      \includegraphics[width=0.5\textwidth]{PLOTS/FIG1.pdf}
      \caption{\small \sf The one--loop Feynman diagrams.}
      \label{fig:FIG1}
\end{figure}
\noindent
where all quark lines are massless.
One obtains
\begin{eqnarray}
\bar{c}^{(1)}_{g_1,g} &=& 4 \textcolor{blue}{T_F N_F} \Bigl[- (2z-1)[{\rm H}_1 + {\rm H}_0] + 3 - 4z\Bigr]
\\
a_{g_1,g}^{(1)} &=& \textcolor{blue}{T_F N_F} \Bigl[
-12 + 16 z + 3 (1 - 2 z) \zeta_2 - (6 - 8 z) \left({\rm H}_0 + {\rm H}_1\right)
- (1 - 2 z) ({\rm H}_0 + {\rm H}_1)^2 \Bigr].
\end{eqnarray}
The contributing splitting functions \cite{Sasaki:1975hk,Ahmed:1976ee,Altarelli:1977zs,Mertig:1995ny,SP_PS1}
are
\begin{eqnarray}
P_{qg}(z) &=& 8 \textcolor{blue}{T_F N_F} \left[z^2 - (1-z)^2\right]
\\
P_{gq}(z) &=& 4 \textcolor{blue}{C_F} \frac{1-(1-z)^2}{z}
\\
P_{qq}^{(1),\rm PS}(z) &=& 16 \textcolor{blue}{C_F T_F N_F} \Bigl[ 1 - z - (1 - 3 z) {\rm H}_0 - (1 + z)
{\rm H}_0^2\Bigr],
\end{eqnarray}
\noindent
where $C_F = (N_c^2-1)/{2 N_c}, T_F = 1/2$ and $N_c = 3$ for the gauge group $SU(3)_c$ of Quantum
Chromodynamics.
In the following we will use the convention
\begin{eqnarray}
\hat{f}(N_F) = f(N_F+1) - f(N_F).
\end{eqnarray}
The harmonic polylogarithms are given by \cite{Remiddi:1999ew}
\begin{equation}
{\rm H}_{b,\vec{a}}(z) = \int_0^z dy f_b(y) {\rm H}_{\vec{a}}(y),~~{\rm H}_\emptyset = 1,~~b, a_i \in \{-1,0,1\},
\end{equation}
and the letters $f_c$ read
\begin{equation}\label{eq:HPL1}
f_0(z) = \frac{1}{z},~~~~
f_1(z) = \frac{1}{1-z},~~~~
f_{-1}(z) = \frac{1}{1+z}.
\end{equation}
We use the shorthand notation ${\rm H}_{\vec{a}}(z) \equiv {\rm H}_{\vec{a}}$ in case of the argument $z$. The harmonic
polylogarithms are
dual to the harmonic sums \cite{Vermaseren:1998uu,Blumlein:1998if} by the Mellin transformation (\ref{eq:MEL}).

\noindent
In the Larin--scheme we obtain
\begin{eqnarray}
\hat{\hat{C}}_{g_1}^{(2),\rm PS,L} &=& - \textcolor{blue}{C_F T_F N_F} \Biggl\{
-\frac{1}{\epsilon^2} \left[80 (1 - z) + 32 (1 + z) {\rm H}_0\right]
+ \frac{1}{\epsilon} [184 (1 - z) - 32 (1 + z) \zeta_2
\nonumber\\ &&
+ 40 (3 - z) {\rm H}_0
+ 24 (1 + z) {\rm H}_0^2 + 80 (1 - z) {\rm H}_1 + 32 (1 + z) {\rm H}_{0,1}]
- \frac{1432}{3} (1-z)
\nonumber\\ &&
- \frac{4}{3} (233-43 z) {\rm H}_0
+\frac{32 (1+z)^3}{3 z} {\rm H}_{-1} {\rm H}_0
-\frac{2}{3} \big(
        129-15 z+8 z^2\big) {\rm H}_0^2
\nonumber\\ &&
-\frac{28}{3} (1+z) {\rm H}_0^3
-(1-z)\big[
        184
        +80 {\rm H}_0
\big] {\rm H}_1
-40 (1-z) {\rm H}_1^2
-(1+z) \big[
        40 +
        32 {\rm H}_0
\big] {\rm H}_{0,1}
\nonumber\\ &&
-\frac{32 (1+z)^3}{3 z} {\rm H}_{0,-1}
+16 (1+z) \left[{\rm H}_{0,0,1}
- 2 {\rm H}_{0,1,1} \right]
\nonumber\\ &&
+\left[
        \frac{4}{3} \big(
                129-45 z+8 z^2\big)
        +56 (1+z) {\rm H}_0
\right] \zeta_2
+16 (1+z) \zeta_3,
\label{eq:C2PSunr}
\end{eqnarray}
performing the phase space integrations as has been outlined in \cite{Blumlein:2019qze,Zijlstra:1992qd}.

At $O(a_s^2)$ neither the renormalization of the heavy quark mass nor the coupling constant contributes
in case of the massive or massless pure singlet Wilson coefficient. The poles in $\epsilon$ in Eq.~(\ref{eq:C2PSunr})
are due to collinear singularities only, which have to be factorized. One may proceed as follows.
The unfactorized quarkonic Wilson coefficients for the structure function $g_1$, $\hat{\hat{C}}_{1,q}^{\rm NS,S}$
in Mellin--space are given by
\begin{eqnarray}
\hat{\hat{C}}_{1,q}^{\rm NS} &=& \Gamma_{qq}^{\rm NS} C_q^{\rm NS}
\\
\hat{\hat{C}}_{1,q}^{\rm S} &=&
  \Gamma_{qq}^{\rm S} C_q^{\rm S}
+ \Gamma_{gq}^{\rm S} C_g^{\rm S}.
\end{eqnarray}
The pure singlet contribution is obtained by
\begin{eqnarray}
\hat{\hat{C}}_{1,q}^{\rm PS} &=& \hat{\hat{C}}_{1,q}^{\rm S} - \hat{\hat{C}}_{1,q}^{\rm NS}
\nonumber\\ &=& \Gamma_{qq}^{\rm S} C_q^{\rm S} - \Gamma_{qq}^{\rm NS} C_q^{\rm NS} + \Gamma_{gq}^{\rm S}
C_g^{\rm S}
\\
&=& \left[\Gamma_{qq}^{\rm NS} + \Gamma_{qq}^{\rm S}\right]\left[C_q^{\rm NS} + C_q^{PS}\right] -
\Gamma_{qq}^{\rm NS} C_q^{\rm NS} + \Gamma_{gq}^{\rm S} C_g^{\rm S}
\end{eqnarray}
with
\begin{eqnarray}
\Gamma_{gq}^{(0)} &=& \hat{a}_s S_\epsilon \left(\frac{\mu_F^2}{\mu^2}\right)^{\epsilon/2}
\frac{1}{\epsilon}
P_{gq}^{(0)}, \\ \Gamma_{qq}^{(1),\rm PS} &=& \hat{a}_s^2 S_\epsilon^2 \left(\frac{\mu_F^2}{\mu^2}\right)^{\epsilon}
\left[
 \frac{1}{\epsilon^2} P_{qg}^{(0)} P_{gq}^{(0)}
+\frac{1}{\epsilon} P_{qq}^{(1),\rm PS} \right].
\end{eqnarray}
and
\begin{eqnarray}
\hat{\hat{C}}_{1,q}^{\rm PS}
&=& a_s^2 \Biggl\{
  \frac{1}{\epsilon^2} \frac{1}{2} P_{qg}^{(0)}   P_{gq}^{(0)}
+ \frac{1}{\epsilon} \Biggl[\frac{1}{2} P_{qq}^{(1), \rm PS}
+ P_{gq}^{(0)} C_g^{(1)} \Biggr] + C_q^{(2, \rm PS} \Biggr\} .
\end{eqnarray}
The factorized massless pure singlet two--loop Wilson coefficient ${C}_{g_1}^{(2),\rm PS,L}$ is given by
\begin{eqnarray}
C_{g_1}^{(2),\rm PS,L}\left(z,\frac{Q^2}{\mu^2}\right) &=& a_s^2 \Biggl\{
\frac{1}{8} P_{qg}^{(0)} P_{gq}^{(0)} L_M^2 + \frac{1}{2}
\left[ P_{qq}^{(1),\rm PS} + P_{gq}^{(0)} \bar{c}_g^{(1)} \right] L_M + \bar{c}_q^{(2), \rm PS}
\Biggr\},
\end{eqnarray}
where
\begin{eqnarray}
L_M = \ln\left(\frac{Q^2}{\mu}\right).
\end{eqnarray}
Here we set $\mu_F = \mu$, and work with a single scale only for the factorization and renormalization scale;
$a_s = g_s^2/(4\pi)$ is the
running coupling constant and the spherical factor $S_\epsilon$, as usually, is set to one at the end of the calculation.
Note that the splitting function $P_{qq}^{(1),\rm PS}$ is correctly obtained, cf.~\cite{Mertig:1995ny,SP_PS1},
despite working in the Larin--scheme, cf. also \cite{Moch:2014sna}.

The massless Wilson coefficient in the {\sf M}--scheme is obtained by the following finite renormalization
\begin{eqnarray}
C_{g_1}^{(2),\rm PS, M} = C_{g_1}^{(2),\rm PS, L} - z_{\rm PS}^{(2)},
\end{eqnarray}
with \cite{Matiounine:1998re}
\begin{eqnarray}
\label{eq:ZPS2}
z_{\rm PS}^{(2)} =
\textcolor{blue}{C_F T_F N_F} \Biggl[16 (1 - z) + 8 (3 - z) {\rm H}_0 + 4 (2 + z) {\rm H}^2_0 \Biggr],
\end{eqnarray}
cf.~\cite{Moch:2014sna,POL18}.
$C_{g_1}^{(2),\rm
PS,M}$ in $z$--space is given by
\begin{eqnarray}
C_{g_1}^{(2),\rm PS,M}\left(z,\frac{Q^2}{\mu^2}\right) &=& a_s^2 \textcolor{blue}{C_F T_F N_F} \Biggl\{
\big[20 (1-z) +8 (1+z) {\rm H}_0 \big] L_M^2
        - \big[
                (1-z) (88 + 40 {\rm H}_1)
\nonumber\\ &&
                +16 (1+z) ({\rm H}_0^2 + {\rm H}_{0,1} - \zeta_2)
                +32 (2-z) {\rm H}_0
           \big] L_M
     +   \frac{760}{3} (1-z)
\nonumber\\ &&
        +\frac{4}{3} (119-13 z) {\rm H}_0
        -\frac{32 (1+z)^3}{3z}  {\rm H}_{-1} {\rm H}_0
        +\frac{2}{3} \big[
                75-15 z+8 z^2\big] {\rm H}_0^2
\nonumber\\ &&
        +\frac{20}{3} (1+z) {\rm H}_0^3
        +(1-z)\big(88 + 40  {\rm H}_0\big)
        {\rm H}_1
        +20 (1-z) {\rm H}_1^2
        +\big[
                8 (3+z)
\nonumber\\ &&
                +16 (1+z) {\rm H}_0
        \big] {\rm H}_{0,1}
        +\frac{32 (1+z)^3}{3z} {\rm H}_{0,-1}
        +16 (1+z) {\rm H}_{0,1,1}
\nonumber\\ &&
        - 32\left[
                \frac{1}{3} \big(
                        9-3 z+z^2\big)
                + (1+z) {\rm H}_0
        \right] \zeta_2
        -16 (1+z) \zeta_3
\Biggr\} .
\end{eqnarray}
\section{The Massive Wilson Coefficient}
\label{sec:4}

\vspace*{1mm}
\noindent
The kinematic domain for the massive Wilson coefficient is given by
\begin{eqnarray}
0 < z <  \frac{Q^2}{4m^2 + Q^2}.
\end{eqnarray}
The unrenormalized two--loop massive pure singlet Wilson coefficient reads in Mellin--$N$ space
\begin{eqnarray}
\hat{\hat{H}}_{g_1}^{(2),\rm PS,L} &=& \hat{a}_s^2 S_\epsilon^2 \left(\frac{Q^2}{\mu^2}\right)^\epsilon\Biggl\{
\frac{1}{\epsilon} P_{gq}^{(0)} h^{(1)}_{g_1,g}
+ h^{(2), \rm PS,L}_{g_1,q} + P_{qg}^{(0)} \bar{b}_{g_1,g}^{(1)} \Biggr\}.
\end{eqnarray}
The contributing Feynman diagrams are shown in Figure~\ref{fig:FIG2}, where now the outgoing quark lines
with momenta $k_1$ and $k_2$ are taken massive, see also Figure~\ref{fig:FIG1}.
Here $\bar{h}^{(1)}_{g_1,g}$ \cite{Watson:1981ce,Gluck:1990in,Vogelsang:1990ug} and $b_{g_1,g}^{(1)}$ are the
expansion coefficients of the one--loop Wilson coefficient
\begin{eqnarray}
\hat{H}_{g_1,g}^{(1)} &=& \hat{a}_s S_\epsilon \left(\frac{Q^2}{\mu^2}\right)^{\epsilon/2}
\left[ h_{g_1,g}^{(1)} + \epsilon \bar{b}_{g_1,g}^{(1)}\right]
\end{eqnarray}
given by the diagrams in Figure~\ref{fig:FIG1} now with massive quark lines.
The expansion coefficients are given by
\begin{eqnarray}
h^{(1)}_{g_1,g} &=& 4 T_F \left[(3-4z) \beta - (1-2z) {\rm H}_0\left(\frac{1+\beta}{1-\beta}\right)\right]
\\
\bar{b}^{(1)}_{g_1,g} &=&
T_F \Biggl\{
        -4 (3-4 z) \beta
        +(1-2 z) {\rm H}_0^2\left(
                \frac{1-\beta }{1+\beta }\right)
        -2 \left[(3-4 z) \beta
                +(1-2 z) {\rm H}_0\left(
                        \frac{1-\beta }{1+\beta }\right)
        \right]
\nonumber\\ && \times
\left[{\rm H}_0 + {\rm H}_1  - 2 \ln(\beta)\right]
        +4 (1-2 z) {\rm H}_{0,1}\left(
                \frac{2 \beta }{1+\beta }\right)
\Biggr\}.
\end{eqnarray}
Here $\beta$ denotes the velocity of the produced heavy quarks,
\begin{eqnarray}
\beta = \sqrt{1 - \frac{4m^2}{Q^2} \frac{z}{1-z}}.
\end{eqnarray}
Since the two heavy quarks do not induce collinear divergences the mass factorization in the massive case reads
\begin{eqnarray}
        \hat{\hat{H}}_{g_1}^{(2),\rm PS,L} &=& H_{g_1}^{(2),\rm PS} + \Gamma_{gq} \otimes H_{g_1,g}^{(1),\rm
PS} ~.
\end{eqnarray}
Here the Mellin convolution of two functions in $z$--space is given by
\begin{eqnarray}
\label{eq:MEL1}
A(z) \otimes B(z) = \int_0^1 dz_1 \int_0^1 dz_2 \delta(z - z_1 z_2) A(z_1) B(z_2).
\end{eqnarray}
We find
\begin{eqnarray}
        H_{g_1}^{(2),\rm PS,L} &=&  \hat{a}_s^2 S_\varepsilon^2 \biggl\{ \left( \frac{Q^2}{\mu^2} \right)^\epsilon
\biggl[ \frac{1}{\epsilon} P_{gq}^{(0)} h_{g_1}^{(1)} + h_{g_1}^{(2),\rm PS} +  P_{gq}^{(0)} \bar{b}_{g_1}^{(1)} \biggr]
\nonumber \\ &&
- \left( \frac{\mu_F^2}{\mu^2} \right)^{\epsilon/2} \left( \frac{Q^2}{\mu^2} \right)^{\epsilon/2}
\biggl[ \frac{1}{\epsilon} P_{gq}^{(0)} h_{g_1}^{(1)} + P_{gq}^{(0)} \bar{b}_{g_1}^{(1)} \biggr] \biggr\}.
\end{eqnarray}
Identifying the renormalization and factorization scale, $\mu = \mu_F$, we finally obtain
\begin{eqnarray}
        H_{g_1}^{(2),\rm PS} &=&  {a}_s^2 \biggl[ \frac{1}{2} P_{gq}^{(0)} h_{g_1}^{(1)} L_M + h_{g_1}^{(2),\rm PS}
\biggr] + O(\epsilon)~.
\end{eqnarray}
Note that in the pure singlet case neither the heavy quark mass nor the coupling constant is  renormalized
at two--loop order.

The massive pure singlet Wilson coefficient is obtained as a four--fold integral over two angular and two
energy variables, cf.~\cite{Blumlein:2019qze} for details of the calculation. These integrals are
systematically turned into iterative integrals. This process leads to a set of letters, through which
these
integrals are defined, see also \cite{Ablinger:2014bra}. We use the code \cite{RAAB1} in {\tt
Mathematica},
which also uses the routine {\tt DSolveRational} of the package {\tt HolonomicFunctions} \cite{KOUTSCHAN};
see \cite{RAAB2, RAAB3} for the general theory underlying \cite{RAAB1}. We also refer to \cite{GUO} for the
simpler case when no singularities are present at the endpoints of integration, which, however, does not
apply here.

The result can be expressed in terms of iterative integrals of the following twelve partly square--root
valued letters,
\begin{eqnarray}
        f_{w_1}(t) &=& \frac{1}{1 - k t},
\\
        f_{w_2}(t) &=& \frac{1}{1 + k t},
\\
        f_{w_3}(t) &=& \frac{1}{\beta + t},
\\
        f_{w_4}(t) &=& \frac{1}{\beta - t},
\\
        f_{w_5}(t) &=& \frac{1}{k - z - ( 1 - z ) k t },
\\
        f_{w_6}(t) &=& \frac{1}{k + z - ( 1 - z ) k t },
\\
        f_{w_7}(t) &=& \frac{1}{k - z + ( 1 - z ) k t },
\\
        f_{w_8}(t) &=& \frac{1}{k + z + ( 1 - z ) k t },
\\
\label{eq:A1}
        f_{w_9}(t) &=& \frac{t}{k^2 \left(1 - t^2 \left(1 - z^2\right)\right)-z^2},
\\
        f_{w_{10}}(t) &=& \frac{1}{t \sqrt{1-t^2} \sqrt{1-k^2 t^2}},
\\
        f_{w_{11}}(t) &=& \frac{t}{\sqrt{1-t^2} \sqrt{1-k^2 t^2}},
\\
\label{eq:AA1}
        f_{w_{12}}(t) &=& \frac{t}{\sqrt{1-t^2} \sqrt{1-k^2 t^2} \left(k^2 \left(1 - t^2 \left(1 -
z^2\right)\right)-z^2\right)},
\end{eqnarray}
given before in Ref.~\cite{Blumlein:2019qze}, and the letters spanning the harmonic polylogarithms. These
iterative integrals have maximally
weight {\sf w = 3} and belong to the  Kummer--elliptic integrals, \cite{Blumlein:2019qze}, in general.
The variable $k$ is defined by
\begin{eqnarray}
k = \frac{\sqrt{z}}{\sqrt{1-(1-z)\beta^2}}.
\end{eqnarray}
One obtains for the following analytic result of the massive polarized two--loop Wilson coefficient
\begin{eqnarray}
H_{g_1}^{(2),\rm PS,L} &=& \textcolor{blue}{C_F T_F} \Biggl\{
-\frac{16(1-z) P_1}{3 k^2}
\biggl\{
        {\rm H}_{w_{5},0}
      - {\rm H}_{w_{6},0}
      + {\rm H}_{w_{7},0}
      - {\rm H}_{w_{8},0}
      -\bigl[
              {\rm H}_{w_{5}}
            - {\rm H}_{w_{6}}
            + {\rm H}_{w_{7}}
            - {\rm H}_{w_{8}}
      \bigr]
\nonumber \\ &&
\times {\rm H}_0
\biggr\}
-\frac{8 P_2}{3 k^2} {\rm H}_{w_{2},-1}
-\frac{8 P_3}{3 k^2} {\rm H}_1 {\rm H}_{w_{1}}
+ \frac{4(1-z) P_4}{3 k^2 z}
\biggl\{
        {\rm H}_{w_{6},1}
      - {\rm H}_{w_{8},-1}
      - {\rm H}_{w_{6}} {\rm H}_1
\nonumber \\ &&
      + {\rm H}_{w_{8}} {\rm H}_{-1}
\biggr\}
- \frac{4 (1-z) P_5}{3 k^2 z}
\biggl\{
        {\rm H}_{w_{6},-1}
      - {\rm H}_{w_{8},1}
      + {\rm H}_{w_{8}} {\rm H}_1
      - {\rm H}_{w_{6}} {\rm H}_{-1}
\biggr\}
\nonumber \\ &&
+\frac{4 (1-z) P_6}{3 k^2 z}
\biggl\{
        {\rm H}_{w_{5},-1}
      - {\rm H}_{w_{7},1}
      + {\rm H}_{w_{7}} {\rm H}_1
      - {\rm H}_{w_{5}} {\rm H}_{-1}
\biggr\}
-\frac{4 (1-z) P_7}{3 k^2 z}
\biggl\{
        {\rm H}_{w_{5},1}
\nonumber \\ &&
      - {\rm H}_{w_{7},-1}
      - {\rm H}_{w_{5}} {\rm H}_1
      + {\rm H}_{w_{7}} {\rm H}_{-1}
\biggr\}
+ \frac{16 P_8}{3(1- k \beta)} {\rm H}_{w_{1}}
- \frac{16 P_9}{3(1+k \beta)} {\rm H}_{w_{2}}
\nonumber \\ &&
+ \frac{8 (1-z) P_{10}}{3 (k (2-z)-z) (1-k \beta)} {\rm H}_{w_{5}}
+ \frac{8 (1-z) P_{11}}{3 (k (2-z)+z) (1+k \beta)} {\rm H}_{w_{6}}
\nonumber \\ &&
- \frac{8 (1-z) P_{12}}{3 (k (2-z)-z) (1+k \beta)} {\rm H}_{w_{7}}
- \frac{8 (1-z) P_{13}}{3 (k (2-z)+z) (1-k \beta)} {\rm H}_{w_{8}}
\nonumber \\ &&
+ \frac{32 P_{14}}{3 k^2 \big(k^2 (2-z)^2-z^2\big)} {\rm H}_1
- \frac{32 P_{15}}{3 k^2 \big(k^2 (2-z)^2-z^2\big)} {\rm H}_{-1}
\nonumber \\ &&
+\frac{1216}{3} (1-z) \beta
+ 8 (1-z) (1-2z)
\bigl[
            {\rm H}_1 + {\rm H}_{-1} - 2 \beta
\bigr] \big( \ln(z) + \ln(1-z) \big)
\nonumber \\ &&
+ 16 (1 + 2z)
\biggl\{
        2 \bigl( {\rm H}_{w_{1},w_{4}} + {\rm H}_{w_{2},w_{4}} + {\rm H}_{w_{3},w_{1}} + {\rm H}_{w_{3},w_{2}} \bigr)
\nonumber \\ &&
      + k \bigl( {\rm H}_{w_{1}}^2 - {\rm H}_{w_{2}}^2 \bigr)
      + \bigl[
            - 2 \ln \big(k^2-z\big)
            + 6 \ln (k)
            - \ln \big(1-k^2\big)
            + \ln (k^2-z^2)
            - 2 {\rm H}_{w_{3}}
      \bigr]
\nonumber \\ && \times
      \bigl( {\rm H}_{w_{1}} + {\rm H}_{w_{2}} \bigr)
      + k ( 1 - z ) \bigl[ {\rm H}_{w_{5},w_{1}} + {\rm H}_{w_{6},w_{2}} - {\rm H}_{w_{7},w_{2}} - {\rm H}_{w_{8},w_{1}} \bigr]
\nonumber \\ &&
      - k ( 1 - z ) \bigl[ {\rm H}_{w_{5}} - {\rm H}_{w_{8}} \bigr] {\rm H}_{w_{1}}
      - k ( 1 - z ) \bigl[ {\rm H}_{w_{6}} - {\rm H}_{w_{7}} \bigr] {\rm H}_{w_{2}}
\biggr\}
\nonumber \\ &&
+16 (1-z) (7-2 z) \beta  \ln (k^2-z^2)
+ 8 \left(7-\left(2-\frac{1}{k^2}\right) z\right)
\biggl\{
      2 {\rm H}_1 {\rm H}_0
      - 6 \ln (k) {\rm H}_1
\nonumber \\ &&
      + \ln \big(1-k^2\big) {\rm H}_1
      + 2 \ln \big(k^2-z\big) {\rm H}_1
      + 2 {\rm H}_{w_{3}} {\rm H}_1
      - 6 \ln (k) {\rm H}_{-1}
      + \ln \big(1-k^2\big) {\rm H}_{-1}
\nonumber \\ &&
      + 2 \ln \big(k^2-z\big) {\rm H}_{-1}
      + 2 {\rm H}_{-1} {\rm H}_{w_{3}}
      - 2 {\rm H}_{0,1}
      - 2 {\rm H}_{1,w_{4}}
      - 2  {\rm H}_{w_{3},1}
      - 2 {\rm H}_{w_{3},-1}
      + 2  {\rm H}_{-1,0}
\nonumber \\ &&
      - 2 {\rm H}_{-1,w_{4}}
\biggr\}
+\frac{32}{3} \big(-3 k^2+\frac{z^2}{k^2}-3 \big(1+z^2\big)\big) \bigl[ {\rm H}_{w_{1}} + {\rm H}_{w_{2}} \bigr] {\rm H}_0
+8 \ln (k^2-z^2)
\nonumber \\ && \times
\left(-8+\frac{\big(-1+5 k^2\big) z}{k^2}-2 z^2\right) \bigl[ {\rm H}_1 + {\rm H}_{-1} \bigr]
+\frac{8}{3} \big(6 k^2-\frac{3 z}{k}
-\frac{2 z^2}{k^2}+6 \big(2+2 z
\nonumber \\ &&
+z^2\big)-3 k \big(8-5 z+2 z^2\big)\big) \bigl[ {\rm H}_{w_{1}} {\rm H}_{-1} - {\rm H}_{w_{2}} {\rm H}_1 \bigr]
+\frac{8}{3} \big(6 k^2+\frac{3 z}{k}-\frac{2 z^2}{k^2}+6 \big(2+2 z
\nonumber \\ &&
+z^2\big)+3 k \big(8-5 z+2 z^2\big)\big) {\rm H}_{w_{2}} {\rm H}_{-1}
+ \frac{4}{3} \left(87+\frac{4}{z}-\frac{9 \big(-1+2 k^2\big) z}{k^2}\right) \bigl[ {\rm H}_1^2 - {\rm H}_{-1}^2 \bigr]
\nonumber \\ &&
-\frac{16}{3} k (1-z) \big(3+2 z+3 z^2\big)
\biggl\{
       \frac{1}{\sqrt{z}}
      \bigl[
              {\rm H}_{w_{10},w_{5}}
            - {\rm H}_{w_{10},w_{6}}
            + {\rm H}_{w_{10},w_{7}}
            - {\rm H}_{w_{10},w_{8}}
      \bigr]
\nonumber \\ &&
      - \frac{k}{\sqrt{z}}
      \biggl(
              {\rm H}_{w_{5},w_{11}}
            + {\rm H}_{w_{6},w_{11}}
            + {\rm H}_{w_{7},w_{11}}
            + {\rm H}_{w_{8},w_{11}}
      -\bigl[
              {\rm H}_{w_{5}}
            + {\rm H}_{w_{6}}
            + {\rm H}_{w_{7}}
            + {\rm H}_{w_{8}}
      \bigr] {\rm H}_{w_{11}}
      \biggr)
\nonumber \\ &&
      - 2 k ( 1 - k^2 ) \sqrt{z}
      \biggl(
              {\rm H}_{w_{5},w_{12}}
            + {\rm H}_{w_{6},w_{12}}
            + {\rm H}_{w_{7},w_{12}}
            + {\rm H}_{w_{8},w_{12}}
      - \bigl[
              {\rm H}_{w_{5}}
            + {\rm H}_{w_{6}}
            + {\rm H}_{w_{7}}
\nonumber \\ &&
            + {\rm H}_{w_{8}}
      \bigr] {\rm H}_{w_{12}}
      \biggr)
      - \frac{2}{\sqrt{z}(1-z)} \bigl[ {\rm H}_{w_{10},w_{1}} + {\rm H}_{w_{10},w_{2}} \bigr]
      - \frac{2 (1 - k^2) \sqrt{z}}{1-z} \bigl[ {\rm H}_{w_{12},1}
\nonumber \\ &&
+ {\rm H}_{w_{12},-1} \bigr]
\biggr\}
-\frac{384 \big(k^2-z\big)}{k^2 \beta } {\rm H}_{w_{3}}
- \frac{8}{3} \left(39-\frac{4}{z}+\frac{9 \big(1-2 k^2\big) z}{k^2}\right) {\rm H}_{-1} {\rm H}_1
\nonumber \\ &&
+ 32 \biggl( k^2 - (2-z) z-\frac{z^2}{3 k^2}\biggr) {\rm H}_{w_{1},0}
+\frac{8}{3} \bigl(6 k^2+\frac{3 z}{k}-\frac{2 z^2}{k^2}-3 k z (1-2 z)
\nonumber \\ &&
-6 \big(1+4 z-z^2\big)\bigr) {\rm H}_{w_{1},1}
+\frac{8}{3} \big(6-6 k^2+24 z+\frac{3 z}{k}-6 z^2+\frac{2 z^2}{k^2}
\nonumber \\ &&
-3 k z (1-2 z)\big) {\rm H}_{w_{1},-1}
+32 \biggl( k^2 - (2 - z) z -\frac{z^2}{3 k^2}\biggr) {\rm H}_{w_{2},0}
+\frac{8}{3} \biggl(6 k^2-\frac{3 z}{k}
\nonumber \\ &&
+3 k (1-2 z) z-\frac{2 z^2}{k^2}
-6 \big(1+4 z-z^2\big)\biggr) {\rm H}_{w_{2},1}
+ \biggl( 208-\frac{64}{3 z}+\frac{48 \big(1-2 k^2\big) z}{k^2} \biggr)
\nonumber \\ &&
\times {\rm H}_{-1,1}
- \frac{64 k^2 (1-z^2) \big(1+3 z^2\big)}{3 z}
\biggl\{
      {\rm H}_{w_{9},1}
      + {\rm H}_{w_{9},-1}
      - k (1-z) \bigl[ {\rm H}_{w_{9},w_{5}} + {\rm H}_{w_{9},w_{6}}
\nonumber \\ &&
      + {\rm H}_{w_{9},w_{7}} + {\rm H}_{w_{9},w_{8}} \bigr]
\biggr\}
+ 8 (1+z)
\biggl\{
      - 4 {\rm H}_{0,1,1}
      - 4 {\rm H}_{0,-1,1}
      + 20 {\rm H}_{1,1,1}
      + 4 {\rm H}_{1,1,w_{4}}
\nonumber \\ &&
      + 4 {\rm H}_{1,-1,w_{4}}
      - 4 {\rm H}_{w_{3},1,1}
      + 4 {\rm H}_{w_{3},1,-1}
      - 4 {\rm H}_{w_{3},-1,1}
      + 4 {\rm H}_{w_{3},-1,-1}
      + 4 {\rm H}_{-1,1,0}
\nonumber \\ &&
      + 16 {\rm H}_{-1,1,1}
      - 4 {\rm H}_{-1,1,w_{4}}
      + 4 {\rm H}_{-1,-1,0}
      + 16 {\rm H}_{-1,-1,1}
      - 4 {\rm H}_{-1,-1,w_{4}}
      + 20 {\rm H}_{-1,-1,-1}
\nonumber \\ &&
+ \biggl(
         \ln \big(1-k^2\big)
        - \ln (k^2-z^2)
        +2 \ln \big(k^2-z\big)
        -6  \ln (k)
\biggr) \bigl[ {\rm H}_{-1}^2 - {\rm H}_1^2 - 2 {\rm H}_1 {\rm H}_{-1}
\nonumber \\ &&
      + 4 {\rm H}_{-1,1} \bigr]
+\bigl[
         \bigl(
                10 {\rm H}_{-1}
                -4 {\rm H}_{w_{3}}
        \bigr) {\rm H}_1
        -4 {\rm H}_0 {\rm H}_1
        +2 {\rm H}_1^2
        -2 {\rm H}_{1,1}
        -4 {\rm H}_{w_{3},1}
        -4 {\rm H}_{w_{3},-1}
\nonumber \\ &&
        -12 {\rm H}_{-1,1}
        -10 {\rm H}_{-1,-1}
\bigr] {\rm H}_{-1}
+\bigr[
         4 {\rm H}_{1,1}
        +8 {\rm H}_{-1,1}
        +4 {\rm H}_{-1,-1}
        - 4 {\rm H}_1^2
\bigl] {\rm H}_{w_{3}}
+\bigl[
         4 {\rm H}_{0,1}
\nonumber \\ &&
        +4 {\rm H}_{0,-1}
        -10 {\rm H}_{1,1}
        +4 {\rm H}_{w_{3},1}
        +4 {\rm H}_{w_{3},-1}
        -4 {\rm H}_{-1,1}
        -10 {\rm H}_{-1,-1}
\bigr] {\rm H}_1
\nonumber \\ &&
+\bigl[
        -4 {\rm H}_1^2
        +4 {\rm H}_{1,1}
        +4 {\rm H}_{-1,1}
\bigr] {\rm H}_0
\biggr\}
+ 32 k (1+z)
\biggl\{
         {\rm H}_{w_{1},1,0}
      +  {\rm H}_{w_{1},1,1}
      -  {\rm H}_{w_{1},1,w_{4}}
\nonumber \\ &&
      -  {\rm H}_{w_{1},1,-1}
      +  {\rm H}_{w_{1},-1,0}
      +  {\rm H}_{w_{1},-1,1}
      -  {\rm H}_{w_{1},-1,w_{4}}
      -  {\rm H}_{w_{1},-1,-1}
      -  {\rm H}_{w_{2},1,0}
      -  {\rm H}_{w_{2},1,1}
\nonumber \\ &&
      +  {\rm H}_{w_{2},1,w_{4}}
      +  {\rm H}_{w_{2},1,-1}
      -  {\rm H}_{w_{2},-1,0}
      -  {\rm H}_{w_{2},-1,1}
      + {\rm H}_{w_{2},-1,w_{4}}
      + {\rm H}_{w_{2},-1,-1}
      + {\rm H}_{w_{3},1,w_{1}}
\nonumber \\ &&
      - {\rm H}_{w_{3},1,w_{2}}
      + {\rm H}_{w_{3},-1,w_{1}}
      - {\rm H}_{w_{3},-1,w_{2}}
      + \frac{1}{2}
      \bigl[
             \ln (k^2-z^2)
            - 2 \ln \big(k^2-z\big)
            + 6 \ln (k)
\nonumber \\ &&
            - \ln \big(1-k^2\big)
      \bigr] \bigl[ {\rm H}_{w_{2},-1} - {\rm H}_{w_{1},-1} + {\rm H}_{w_{2},1} - {\rm H}_{w_{1},1} \bigr]
+ \frac{1}{2} \biggl(
        - {\rm H}_{w_{1},1}
        - {\rm H}_{w_{1},-1}
        + {\rm H}_{w_{2},1}
\nonumber \\ &&
        + {\rm H}_{w_{2},-1}
        + \bigl[ {\rm H}_{w_{1}} - {\rm H}_{w_{2}} \bigr] {\rm H}_1
\biggr) {\rm H}_{-1}
+\bigl[
          {\rm H}_{w_{1},1}
        + {\rm H}_{w_{1},-1}
        - {\rm H}_{w_{2},1}
        - {\rm H}_{w_{2},-1}
\bigr] {\rm H}_{w_{3}}
\nonumber \\ &&
- \frac{1}{2} \biggl(
          {\rm H}_{1,1}
        - {\rm H}_1^2
        + 2 {\rm H}_{w_{3},1}
        + 2 {\rm H}_{w_{3},-1}
        + 2 {\rm H}_{-1,1}
        + {\rm H}_{-1,-1}
        + k {\rm H}_{w_{1},1}
        + k {\rm H}_{w_{1},-1}
\nonumber \\ &&
        - k {\rm H}_{w_{2},1}
        - k {\rm H}_{w_{2},-1}
\biggr) \bigl[ {\rm H}_{w_{1}} - {\rm H}_{w_{2}} \bigr]
+\frac{1}{2}\bigl[
          {\rm H}_{w_{1},1}
        + {\rm H}_{w_{1},-1}
        - {\rm H}_{w_{2},1}
        - {\rm H}_{w_{2},-1}
\bigr] {\rm H}_1
\biggr\}
\nonumber \\ &&
- 16 k ( 1 - z^2 )
\biggl\{
- {\rm H}_{1,w_{4},w_{5}}
- {\rm H}_{1,w_{4},w_{6}}
- {\rm H}_{1,w_{4},w_{7}}
- {\rm H}_{1,w_{4},w_{8}}
+ {\rm H}_{w_{5},1,1}
- {\rm H}_{w_{5},1,-1}
\nonumber \\ &&
+ {\rm H}_{w_{5},w_{3},1}
- {\rm H}_{w_{5},w_{3},-1}
+ {\rm H}_{w_{6},1,1}
- {\rm H}_{w_{6},1,-1}
+ {\rm H}_{w_{6},w_{3},1}
- {\rm H}_{w_{6},w_{3},-1}
+ {\rm H}_{w_{7},w_{3},1}
\nonumber \\ &&
- {\rm H}_{w_{7},w_{3},-1}
- {\rm H}_{w_{7},-1,1}
+ {\rm H}_{w_{7},-1,-1}
+ {\rm H}_{w_{8},w_{3},1}
- {\rm H}_{w_{8},w_{3},-1}
- {\rm H}_{w_{8},-1,1}
+ {\rm H}_{w_{8},-1,-1}
\nonumber \\ &&
+ {\rm H}_{-1,w_{4},w_{5}}
+ {\rm H}_{-1,w_{4},w_{6}}
+ {\rm H}_{-1,w_{4},w_{7}}
+ {\rm H}_{-1,w_{4},w_{8}}
+ k \bigl[
 {\rm H}_{w_{1},w_{4},w_{5}}
+{\rm H}_{w_{1},w_{4},w_{6}}
\nonumber \\ &&
+{\rm H}_{w_{1},w_{4},w_{7}}
+{\rm H}_{w_{1},w_{4},w_{8}}
-{\rm H}_{w_{2},w_{4},w_{5}}
-{\rm H}_{w_{2},w_{4},w_{6}}
-{\rm H}_{w_{2},w_{4},w_{7}}
-{\rm H}_{w_{2},w_{4},w_{8}}
-{\rm H}_{w_{5},1,w_{1}}
\nonumber \\ &&
+{\rm H}_{w_{5},1,w_{2}}
-{\rm H}_{w_{5},w_{3},w_{1}}
+{\rm H}_{w_{5},w_{3},w_{2}}
-{\rm H}_{w_{6},1,w_{1}}
+{\rm H}_{w_{6},1,w_{2}}
-{\rm H}_{w_{6},w_{3},w_{1}}
+{\rm H}_{w_{6},w_{3},w_{2}}
\nonumber \\ &&
-{\rm H}_{w_{7},w_{3},w_{1}}
+{\rm H}_{w_{7},w_{3},w_{2}}
+{\rm H}_{w_{7},-1,w_{1}}
-{\rm H}_{w_{7},-1,w_{2}}
-{\rm H}_{w_{8},w_{3},w_{1}}
+{\rm H}_{w_{8},w_{3},w_{2}}
\nonumber \\ &&
+{\rm H}_{w_{8},-1,w_{1}}
-{\rm H}_{w_{8},-1,w_{2}}
\bigr]
+\biggl(
        - {\rm H}_{1,1}
        - {\rm H}_{w_{3},1}
        + {\rm H}_{w_{3},-1}
        - {\rm H}_{-1,1}
        + {\rm H}_1^2
        - k {\rm H}_{w_{1},1}
\nonumber \\ &&
        + k {\rm H}_{w_{2},1}
        + k {\rm H}_{w_{3},w_{1}}
        - k {\rm H}_{w_{3},w_{2}}
\biggr) \bigl[ {\rm H}_{w_{5}} + {\rm H}_{w_{6}} \bigr]
+\biggl(
        - {\rm H}_{w_{3},1}
        + {\rm H}_{w_{3},-1}
        + {\rm H}_{-1,1}
\nonumber \\ &&
        + {\rm H}_{-1,-1}
        - {\rm H}_1 {\rm H}_{-1}
        + k {\rm H}_{w_{1},-1}
        - k {\rm H}_{w_{2},-1}
        + k {\rm H}_{w_{3},w_{1}}
        - k {\rm H}_{w_{3},w_{2}}
\biggr) \bigl[ {\rm H}_{w_{7}} + {\rm H}_{w_{8}} \bigr]
\nonumber \\ &&
+\biggl(
        \bigl(
                  {\rm H}_{w_{5}}
                + {\rm H}_{w_{6}}
                + {\rm H}_{w_{7}}
                + {\rm H}_{w_{8}}
        \bigr) {\rm H}_{w_{3}}
        -  {\rm H}_{w_{5},1}
        -  {\rm H}_{w_{5},w_{3}}
        -  {\rm H}_{w_{6},1}
        -  {\rm H}_{w_{6},w_{3}}
        -  {\rm H}_{w_{7},w_{3}}
\nonumber \\ &&
        +  {\rm H}_{w_{7},-1}
        -  {\rm H}_{w_{8},w_{3}}
        +  {\rm H}_{w_{8},-1}
\biggr)
\nonumber \\ && \times
\bigl[ {\rm H}_1 -  {\rm H}_{-1} - k \bigl( {\rm H}_{w_{1}} - {\rm H}_{w_{2}} \bigr) \bigr]
\biggr\}
+ 576 (1-z) \beta  \ln (k)
- 96 (1-z) \beta  \ln \big(1-k^2\big)
\nonumber \\ &&
- 192 (1-z) \beta {\rm H}_0
- 192 (1-z) \beta  \ln \big(k^2-z\big)
\Biggr\}
\nonumber \\ &&
+ \frac{1}{2} P_{gq}^{(0)} \otimes h_{g_1}^{(1)} L_M
- P_{gq}^{(0)} \otimes \bar{b}_{g_1}^{(1)}.
\label{eq:g1full}
\end{eqnarray}
The remaining convolutions appearing in Eq.~(\ref{eq:g1full}) are given in Appendix~\ref{sec:A1}.
Here and in the Appendix the argument of the iterative integrals ${\rm H}_{\vec{a}}$ is $\beta$.

The polynomials $P_i$ in Eq.~(\ref{eq:g1full}) read
\begin{eqnarray}
P_1&=&3 k^4+3 k^2 \left(z^2+1\right)-z^2, \\
P_2&=&6 k^4+3 k^3 z (2 z-1)+6 k^2 \left(z^2-4 z-1\right)+3 k z-2 z^2, \\
P_3&=&6 k^4+3 k^3 \left(2 z^2-5 z+8\right)+6 k^2 \left(z^2+2 z+2\right)+3 k z-2
z^2, \\
P_4&=&6 k^4 z+k^3 \left(-16 z^3+33 z^2-24 z+8\right)+6 k^2 z \left(z^2+2 z+2\right)-3 k z^2-2 z^3, \\
P_5&=&6 k^4 z+k^3 \left(-4 z^3+3 z^2+24 z+8\right)+6 k^2 z \left(z^2+2 z+2\right)+3 k z^2-2 z^3, \\
P_6&=&6
k^4 z+k^3 \left(4 z^3-3 z^2-24 z-8\right)+6 k^2 z \left(z^2+2 z+2\right)-3 k z^2-2 z^3, \\ P_7&=&6 k^4 z+k^3
\left(16 z^3-33 z^2+24 z-8\right)+6 k^2 z \left(z^2+2 z+2\right)+3 k z^2-2 z^3, \\ P_8&=&3 \beta k^3 (z-3)+3
k^2 (z-4)+4 \beta k \left(3 z^2-13 z+3\right)-12 z^2+46 z+9, \\ P_9&=&3 \beta k^3 (z-3)-3 k^2 (z-4)+4 \beta k
\left(3 z^2-13 z+3\right)+12 z^2-46 z-9, \\ P_{10}&=&3 \beta k^4 (1-z)^2+k^3 \bigl(-35 \beta +(3-16 \beta )
z^2+3 (17 \beta -9) z+39\bigr)+k^2 \bigl(-52 \beta +12 \beta z^3 \nonumber \\ && +(22-81 \beta ) z^2+(121
\beta -72) z+35\bigr)+k \bigl(12 (\beta -1) z^3+(75-38 \beta ) z^2 \nonumber \\ && +(26 \beta -88)
z+10\bigr)+z \bigl(-12 z^2+32 z-5\bigr), \\ P_{11}&=&3 \beta k^4 (1-z)^2+k^3 \bigl(35 \beta +(16 \beta -3)
z^2+(27-51 \beta ) z-39\bigr)+k^2 \bigl(-52 \beta +12 \beta z^3 \nonumber \\ && +(22-81 \beta ) z^2+(121
\beta -72) z+35\bigr)+k \bigl(-12 (\beta -1) z^3+(38 \beta -75) z^2 \nonumber \\ && +(88-26 \beta )
z-10\bigr)+z \bigl(-12 z^2+32 z-5\bigr), \\ P_{12}&=&3 \beta k^4 (1-z)^2-k^3 \bigl(35 \beta +(16 \beta +3)
z^2-3 (17 \beta +9) z+39\bigr)+k^2 \bigl(-52 \beta +12 \beta z^3 \nonumber \\ && -(81 \beta +22) z^2+(121
\beta +72) z-35\bigr)+k \bigl(12 (\beta +1) z^3-(38 \beta +75) z^2 \nonumber \\ && +(26 \beta +88)
z-10\bigr)+z \bigl(12 z^2-32 z+5\bigr), \\ P_{13}&=&3 \beta k^4 (1-z)^2+k^3 \bigl(35 \beta +(16 \beta +3)
z^2-3 (17 \beta +9) z+39\bigr)+k^2 \bigl(-52 \beta +12 \beta z^3 \nonumber \\ && -(81 \beta +22) z^2+(121
\beta +72) z-35\bigr)+k \bigl(-12 (\beta +1) z^3+(38 \beta +75) z^2 \nonumber \\ && -2 (13 \beta +44)
z+10\bigr)+z \bigl(12 z^2-32 z+5\bigr), \\ P_{14}&=&k^4 \bigl(-3 (36 \beta +1)+(27 \beta -10) z^3+(37-135
\beta ) z^2+(216 \beta -34) z\bigr) \nonumber \\ && +k^2 z \bigl((3-27 \beta ) z^2+(27 \beta +28)
z-28\bigr)+7 z^3, \\ P_{15}&=&k^4 \bigl(-108 \beta +(27 \beta +10) z^3-(135 \beta +37) z^2+(216 \beta +34)
z+3\bigr) \nonumber \\ && +k^2 z \bigl(-3 (9 \beta +1) z^2+(27 \beta -28) z+28\bigr)-7 z^3.
\end{eqnarray}
We finally perform the transformation to the {\sf M}--scheme for the massive two--loop pure singlet Wilson
coefficient. It is given by
\begin{eqnarray}
H_{g_1}^{2,\rm PS,M} = H_{g_1}^{2,\rm PS,L} - \frac{1}{N_F} z_{\rm PS}^{(2)}.
\end{eqnarray}
\section{The Asymptotic and Threshold Expansions}
\label{sec:5}

\vspace*{1mm}
\noindent
The complete expressions calculated in Section~\ref{sec:4} allow now to perform the asymptotic expansion
for $Q^2 \gg m^2$ and the threshold expansion for $\beta \ll 1$.

In the asymptotic limit $Q^2 \gg m^2$ the first expansion coefficients of the polarized massive pure singlet
Wilson coefficient read setting $\mu^2 = Q^2$
\begin{eqnarray}
        H_{g_1}^{(2),\rm PS,M} &=& \textcolor{blue}{C_F T_F} \Biggl\{
-\big(
        20 (1-z)
        + 8 (1+z) {\rm H}_0
\big) \ln^2 \left( \frac{\textcolor{blue}{m^2}}{\textcolor{blue}{Q^2}} \right)
-\big(
        8 (1-z)
        -8 (1-3 z) {\rm H}_0
\nonumber \\ &&
        -8 (1+z) {\rm H}_0^2
\big)  \ln \left( \frac{\textcolor{blue}{m^2}}{\textcolor{blue}{Q^2}} \right)
+\frac{592}{3} (1-z)
+\biggl(
         \frac{256}{3} (2-z)
        -\frac{32 (1+z)^3}{3 z} {\rm H}_{-1}
\biggr) {\rm H}_0
\nonumber \\ &&
+\frac{8}{3} \big(21+2 z^2\big) {\rm H}_0^2
+\frac{16}{3} (1+z) {\rm H}_0^3
+\biggl(
         88 (1-z)
        +80 (1-z) {\rm H}_0
\biggr) {\rm H}_1
+20 (1-z) {\rm H}_1^2
\nonumber \\ &&
-\biggl(
        16 (1-3 z)
        -32 (1+z) {\rm H}_0
\biggr) {\rm H}_{0,1}
+\frac{32 (1+z)^3}{3 z} {\rm H}_{0,-1}
-32 (1+z) {\rm H}_{0,0,1}
\nonumber \\ &&
+16 (1+z) {\rm H}_{0,1,1}
-\big(
         \frac{32}{3} \big(9-3 z+z^2\big)
        +32 (1+z) {\rm H}_0
\big) \zeta_2
+16 (1+z) \zeta_3
\nonumber \\ &&
- \Biggl[16 (1 - z) + 8 (3 - z) {\rm H}_0 + 4 (2 + z) {\rm H}^2_0 \Biggr]
\nonumber \\ &&
+ \frac{\textcolor{blue}{m^2}}{\textcolor{blue}{Q^2}}
\Biggl[
\bigl(
        16 (1-z) (1-3 z)
        -32 z {\rm H}_0
\bigr) \ln \left( \frac{\textcolor{blue}{m^2}}{\textcolor{blue}{Q^2}} \right)
+8 \bigl(18-12 z-7 z^2\bigr)
\nonumber \\ &&
+16 \bigl(6+z+6 z^2\bigr) {\rm H}_0
+16 z {\rm H}_0^2
+16 \bigl(3-7 z+3 z^2\bigr) {\rm H}_1
\Biggr]
\nonumber \\ &&
+ \left(\frac{\textcolor{blue}{m^2}}{\textcolor{blue}{Q^2}}\right)^2
\Biggl[
-4 (1-z) (3+4 z) \ln^2 \left( \frac{\textcolor{blue}{m^2}}{\textcolor{blue}{Q^2}} \right)
+\biggl(
         \frac{4 P_{18}}{1-z}
        -16 (1-z) (5+4 z) {\rm H}_0
\nonumber \\ &&
        -8 (1-z) (5+4 z) {\rm H}_1
\biggr) \ln \left( \frac{\textcolor{blue}{m^2}}{\textcolor{blue}{Q^2}} \right)
+\frac{2 P_{19}}{3 (1-z)^2}
+\biggl(
         \frac{16 P_{16}}{1-z}
        -64 (1-z^2) {\rm H}_{-1}
\biggr) {\rm H}_0
\nonumber \\ &&
+\biggl(
         \frac{4 P_{17}}{1-z}
        -32 (1-z) {\rm H}_0
\biggr) {\rm H}_1
-4 (1-z) (7+4 z) {\rm H}_1^2
-16 (1-z) (3+4 z) {\rm H}_{0,1}
\nonumber \\ &&
+64 (1-z^2) {\rm H}_{0,-1}
+16 (1-z) \zeta_2
\Biggr]
+ O \left( \kappa^3 \ln^2(\kappa) \right)
\Biggr\},
\label{eq:ASY}
\end{eqnarray}
with the polynomials
\begin{eqnarray}
P_{16} &=& 3 z^4+z^3-11 z^2+13 z-7,
\\
P_{17} &=& 6 z^4+2 z^3-63 z^2+84 z-32,
\\
P_{18} &=& 6 z^4+2 z^3-57 z^2+76 z-28,
\\
P_{19} &=& 15 z^5-27 z^4+393 z^3-1079 z^2+1069 z-339.
\end{eqnarray}
In this expansion the Kummer--elliptic integrals turn into harmonic polylogarithms.
The leading term, which is free of power corrections of $O((m^2/Q^2)^k), k \in \mathbb{N}, k \geq 1$,
can be predicted using the representation of the massive Wilson coefficient by massive operator matrix
elements (OMEs), cf.~\cite{Buza:1995ie,Buza:1996xr,Bierenbaum:2007qe,POL18}, and massless Wilson
coefficients,
\begin{eqnarray}
H_{g_1}^{(2),\rm PS}\left(z,\frac{Q^2}{m^2}\right) &=&  A_{Qq}^{(2),\rm PS}(N_F+1) + \hat{C}_{g_1}^{(2),\rm
PS}(N_F+1).
\end{eqnarray}
Here the massless Wilson coefficient $\tilde{C}_{g_1}^{(2),\rm PS}(N_F+1)$ is the one given in
Section~\ref{sec:3} normalized by $N_F+1$. The massive two--loop operator matrix element
$A_{Qq}^{(2),\rm PS}$ in Mellin space reads
\begin{eqnarray}
A_{Qq}^{(2),\rm PS} &=& -\frac{1}{8} \hat{P}_{qg}^{(0)} P_{gq}^{(0)} \ln^2\left(\frac{m^2}{\mu^2}\right)
- \frac{1}{2} \hat{P}_{qq}^{(1), \rm PS} \ln\left(\frac{m^2}{\mu^2}\right)
+ \frac{1}{8} \hat{P}_{qg}^{(0)} P_{gq}^{(0)} \zeta_2 + a_{Qq}^{(2),\rm PS},
\label{eq:asymp}
\end{eqnarray}
cf.~\cite{Buza:1995ie,Buza:1996xr,Bierenbaum:2007qe,POL18}; for its renormalization see
Ref.~\cite{Bierenbaum:2009mv}. The constant part of the unrenormalized polarized OME $a_{Qq}^{(2),\rm PS}$
is given by \cite{Buza:1996xr,POL18}
\begin{eqnarray}
a_{Qq}^{(2),\rm PS}(z) &=& \textcolor{blue}{C_F T_F} \Biggl\{
- 72 (1-z)
-12 (1+5 z) {\rm H}_0
-2 (1-3 z) {\rm H}_0^2
-\frac{4}{3} (1+z) {\rm H}_0^3
+40 (1-z) {\rm H}_0 {\rm H}_1
\nonumber\\ &&
-\big(
        40 (1-z)
       -16 (1+z) {\rm H}_0
\big) {\rm H}_{0,1}
-32 (1+z) {\rm H}_{0,0,1}
-\big(
        20 (1-z) -
        8 (1+z) {\rm H}_0
\big) \zeta_2
\nonumber\\ &&
+32 (1+z) \zeta_3
\Biggr\}
\end{eqnarray}
in $z$-space. The calculation of $A_{Qq}^{(2),\rm PS}$ is performed in the Larin--scheme. One has to apply
the tensor decomposition method, however, to obtain the correct result. These aspects are discussed in
Ref.~\cite{POL18} in detail.

The threshold expansion of the Wilson coefficients for $\beta \ll 1$ is given
by
\begin{eqnarray}
H_{g_1}^{(1)}\left(z,\frac{Q^2}{m^2}\right) &=& 4 \textcolor{blue}{T_F} \beta
\biggl\{ 1
-\frac{2}{3} (1-2 z) \beta ^2
-\frac{2}{5} (1-2 z) \beta ^4
-\frac{2}{7} (1-2 z) \beta ^6
\\ \nonumber &&
-\frac{2}{9} (1-2 z) \beta ^8
+ O(\beta^{10})
\biggr\},
\\
H_{g_1}^{(2),\rm PS,L}\left(z,\frac{Q^2}{m^2}\right) &=& \textcolor{blue}{C_F T_F} (1-z) \beta^3
\Biggl\{
-\frac{256}{9}
+\frac{16}{3} \bigl[ \ln (1-z) - \ln (z) + 4 \ln (2 \beta ) \bigr]
\\ \nonumber &&
+\beta ^2 \biggl(
        -\frac{32}{75} (41+20 z)
        +\frac{16}{5} \bigl[ \ln (1-z) - \ln (z) + 4 \ln (2 \beta ) \bigr]
\biggr)
\\ \nonumber &&
-\beta ^4 \biggl(
        \frac{16 \big(2723+20504 z-12352 z^2\big)}{11025}
        -\frac{16}{105} \big(7+16 z-8 z^2\big) \bigl[ \ln (1-z)
\\ \nonumber &&
- \ln (z) + 4 \ln (2 \beta ) \bigr]
\big)
+\beta ^6 \biggl(
         \frac{16 \big(47203-909904 z+950864 z^2-345728 z^3\big)}{297675}
\\ \nonumber &&
        +\frac{16}{945} \big(1+272 z-232 z^2+64 z^3\big) \bigl[ \ln (1-z) - \ln (z) + 4 \ln (2 \beta \biggr)
\biggr]
\biggr)
\\ \nonumber &&
+ O ( \beta^8 )
\Biggr\}.
\end{eqnarray}

\section{Numerical Results}
\label{sec:6}

\vspace*{1mm}
\noindent
Let us now illustrate the analytic results numerically. In Figure~\ref{fig:F1} the two--loop heavy flavor
Wilson coefficient $H_{g_1}^{(2),\rm PS,M}$ is shown  as a function of $z$ for different values of $Q^2
\in [10, 10^4]~\mathrm{GeV}^2$, setting the charm quark mass to $m_c = 1.59~\mathrm{GeV}$, cf.~\cite{Ablinger:2014vwa}.
\begin{figure}[h!]
\centering
\includegraphics[width=0.7 \linewidth]{PLOTS/H_g1-overview.pdf}
\caption{\small \sf The Wilson coefficient $H_{g_{1}}^{(2),\rm PS}$
as a function of $z$ for
different values of $Q^2$ and the scale choice $\mu^2 = \mu_F^2 = Q^2$.
Upper full line (Blue):      $Q^2=10^4~\mathrm{GeV}^2$;
upper dashed line (Orange):  $Q^2=10^3~\mathrm{GeV}^2$;
upper dotted line (Magenta): $Q^2=500~\mathrm{GeV}^2$;
dash-dotted line (Blue):     $Q^2=100~\mathrm{GeV}^2$;
lower full line (Red):       $Q^2=50~\mathrm{GeV}^2$;
lower dashed line (Gray):    $Q^2=25~\mathrm{GeV}^2$;
lower dotted line (Brown):   $Q^2=10~\mathrm{GeV}^2$.}
\label{fig:F1}
\end{figure}
\begin{figure}[H]\centering
\includegraphics[width=0.7\linewidth]{PLOTS/R-H_g1-1.pdf}
\caption{\sf \small The ratio $R_{g_1}^{(1)}$, Eq.~(\ref{eq:RAT1}),  as a
function
of $\chi=Q^2/m^2$. Solid line: $z=10^{-4}$; dotted line: $z=10^{-2}$; dashed line: $z=1/2$.}
\label{fig:RAT1}
\end{figure}
\noindent
For large values of $Q^2$ these results approach the asymptotic result for $H_{g_1,q}^{2,\rm PS}$.
In the small $x$ region this Wilson coefficient is negative.

Next we study the ratios
\begin{eqnarray}
\label{eq:RAT1}
        R_{g_1}^{(1)} &=& \frac{H_{g_1,q}^{2,\rm PS}}{\tilde{H}_{g_1,q}^{2,\rm PS}} (\mu =\mu_F =m)~,
\end{eqnarray}
comparing the full (\ref{eq:g1full}) and the asymptotic results, $\tilde{H}$,
(\ref{eq:ASY}) for the leading term in Figure~\ref{fig:RAT1}.

For $H_{g_1,q}^{2,\rm PS}$ the asymptotic expansion agrees with the full calculation
up to $Q^2/m^2 \equiv \chi = 10$ to about $2\%$ for $z=10^{-4}$, $\chi = 40$ for $z = 10^{-2}$ and $\chi =
200$ for $z = 1/2$. However, the Wilson coefficients are very small already for the last value.
Similar to the ratio of the full and asymptotic Wilson coefficient we define the ratio
\begin{eqnarray}
\label{eq:RAT2}
        R_{g_{1}} &=& \frac{ g_{1,q}^{(2), \rm PS} }{ \tilde{g}_{1,q}^{(2), \rm PS} },
\end{eqnarray}
where $\tilde{g}_{1,q}^{(2), \rm PS}$ is the structure function obtained by using the expansion of the
respective Wilson coefficient up to the desired level. The corresponding results are depicted in
Figure~\ref{fig:RLandR2b}. We use the parameterization of the parton distributions Ref.~\cite{Blumlein:2010rn} at
NLO with the corresponding values of $\alpha_s(Q^2)$ at NNLO \cite{Alekhin:2017kpj} to compare to previous
non--singlet results in \cite{Behring:2015zaa}. Demanding an agreement within $\pm 2\%$ for $g_1^{\rm PS}$ in the
range $z \in [10^{-4}, 10^{-2}, 1/2]$ leads to values $Q_0^2/m^2 \in [5, 5, 13]$ of the $O((m^2/Q^2)^2)$
improved result, $Q_0^2/m^2 \in [10, 12, 30]$ of the $O(m^2/Q^2)$ improved
result, and $Q_0^2/m^2 \in [12, 100, 170]$ for the asymptotic result.

In Figures~\ref{fig:FIG4} we show the complete results for the two--loop pure singlet contributions to $xg_1$
and $xg_2$ as a function of $x$ for a series of $Q^2$--values. Both functions show an oscillatory
behaviour,
which is enlarged for $xg_2$ due to the Wandzura--Wilczek relation.
\begin{figure}[H]
\includegraphics[width=0.49 \linewidth]{PLOTS/R-H_g1-e-4.pdf}
\includegraphics[width=0.49 \linewidth]{PLOTS/R-H_g1-e-2.pdf}
\includegraphics[width=0.49 \linewidth]{PLOTS/R-H_g1-half.pdf}
\caption{\small \sf The ratio $R_{g_1}^{(1)}$, Eq.~(\ref{eq:RAT1}),
as a function of $\chi=Q^2/m^2$ for different values of $z$ gradually improved with $\kappa$ suppressed terms.
Dotted lines: asymptotic result; dashed lines: $O(m^2/Q^2)$ improved; solid lines : $O((m^2/Q^2)^2)$ improved.}
\label{fig:RLandR2b}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=0.49 \linewidth]{PLOTS/g1-overview.pdf}
\includegraphics[width=0.49 \linewidth]{PLOTS/g2-overview.pdf}
\caption{\small \sf The pure singlet contributions
$xg_{1}^{(2),\rm PS}$
and
$xg_{2}^{(2),\rm PS}$
for different values of $Q^2$ and the scale choice $\mu^2 = \mu_F^2 = Q^2$.
Full line (Blue):        $Q^2=10^4~\mathrm{GeV}^2$;
dashed line (Orange):    $Q^2=10^3~\mathrm{GeV}^2$;
dotted line (Magenta):   $Q^2=500~\mathrm{GeV}^2$;
dash-dotted line (Blue): $Q^2=100~\mathrm{GeV}^2$;
full line (Red):         $Q^2=50~\mathrm{GeV}^2$;
dashed line (Gray):      $Q^2=25~\mathrm{GeV}^2$;
dotted line (Brown):     $Q^2=10~\mathrm{GeV}^2$,
using the parameterization of the parton distribution \cite{Blumlein:2010rn}.}
\label{fig:FIG4}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=0.49 \linewidth]{PLOTS/R-g1-asymp-lowx.pdf}
\includegraphics[width=0.49 \linewidth]{PLOTS/R-g1-asymp-highx.pdf}
\caption{\small \sf The ratio of the structure function  $g_1^{(2),\rm PS}$
in the full calculation over the asymptotic approximation for different values of $Q^2$ and the scale choice $\mu^2 = \mu_F^2 =
Q^2$.
Full line (Black):        $Q^2=10^4~\mathrm{GeV}^2$;
dashed line (Gray):       $Q^2=10^3~\mathrm{GeV}^2$;
dotted line (Brown):      $Q^2=500~\mathrm{GeV}^2$;
lower dashed line (Blue): $Q^2=100~\mathrm{GeV}^2$,
using the parameterization of the parton distribution \cite{Blumlein:2010rn}}.
\label{fig:FIG5}
\end{figure}
In Figure~\ref{fig:FIG5} we illustrate the ratios Eq.~(\ref{eq:RAT2}) as a function of $x$ for different values
of $Q^2$ for $g_1^{\rm PS}$ comparing the asymptotic result to the full result. For a better visibility and to avoid
to depict zero transitions in the denominator we separate the small $x$ and large $x$ part into two plots.
The corrections behave widely flat in $x$ for larger values of $Q^2$ and develop some profile for $Q^2 <
100~\mathrm{GeV}^2$.

In Figure~\ref{fig:FIG6} we depict the ratio of the full result over the $O((m^2/Q^2)^2)$ improved asymptotic
results for $g_1^{\rm PS}$ as a function of $x$ for a series of $Q^2$-values, again separating the small $x$ and
the large $x$ ranges because of zero transitions for this ratio.
For $Q^2 \raisebox{-0.07cm 100 \mathrm{GeV}^2$ the ratios are rather flat and are close to one. The line for $Q^2 = 100
\mathrm{GeV}^2$ for $x > 0.5$ deviates from one by more than 5\%. Larger deviations are found for $Q^2 = 50
\mathrm{GeV}^2$, where the 5\% marging is only met for $x < 3 \cdot 10^{-3}$.
We limited the expansion to terms of $\sim O((m^2/Q^2)^2)$ in the present paper, but higher
order terms can
be given straigtforwardly. The expanded expressions do also allow direct Mellin transforms and provide
a suitable analytic basis for Mellin--space programmes.
\begin{figure}[H]
\centering
\includegraphics[width=0.49 \linewidth]{PLOTS/R-g1-lowx.pdf}
\includegraphics[width=0.49 \linewidth]{PLOTS/R-g1-highx.pdf}
\caption{\small \sf The ratio of the pure singlet structure function $g_1^{(2),\rm PS}$
in the full calculation over the $O((m^2/Q^2)^2)$ improved approximation for different values of $Q^2$ and the
scale choice $\mu^2 = \mu_F^2 = Q^2$.
Full lines (Black):         $Q^2=10^4~\mathrm{GeV}^2$;
dashed lines (Gray):        $Q^2=10^3~\mathrm{GeV}^2$;
dotted lines (Brown):       $Q^2=500~\mathrm{GeV}^2$;
dashed lines (Blue):  $Q^2=100~\mathrm{GeV}^2$;
dash-dotted lines (Red):    $Q^2=50~\mathrm{GeV}^2$,
using the parameterization of the parton distribution \cite{Blumlein:2010rn}.}
\label{fig:FIG6}
\end{figure}

\section{Conclusions}
\label{sec:7}

\vspace*{1mm}
\noindent
We have calculated the massless and massive polarized two--loop pure singlet Wilson coefficients for
deep-inelastic scattering in analytic form. The calculation has been performed in the Larin--scheme, with a
final
finite renormalization to the {\sf M}--scheme, which has been introduced in
Refs.~\cite{Matiounine:1998re,Moch:2014sna}. The massless Wilson coefficient can be expressed by the
harmonic
polylogarithms in $z$--space and harmonic sums in Mellin--$N$ space. In the massive case the polarized
two--loop pure singlet Wilson coefficient is also given by iterative integrals, however, of a more general kind,
the Kummer--elliptic integrals, here based on an alphabet of 12 letters, cf.~\cite{Blumlein:2019qze}. From the
expansion of the massive Wilson coefficient in the region $Q^2 \gg m^2$ one obtains the asymptotic result, which
can be given in terms of a massive OME and the massless Wilson coefficient, cf.~\cite{POL18}.
In the region of lower values of $Q^2$ and larger values of $x$, the power corrections to the massive
two--loop Wilson coefficient are essential. From the available analytic result one can construct
the series in $m^2/Q^2$ analytically. Since the deep--inelastic process is usually considered only for
virtualities $Q^2 \raisebox{-0.07cm 5~\mathrm{GeV}^2$, this series gives the proper numerical representation in case of the
charm-quark corrections retaining a relatively small number of terms. The latter representation has the advantage
that it can be transformed into Mellin space directly, since the expansion coefficients are given in terms of
harmonic polylogarithms in $z$--space.

redpajama_set_name: RedPajamaArXiv


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The choice for the papers is to be made while applying for the exam.
Only the eligible candidates can fill the application form of Maharashtra TET 2015. The candidates can their eligibility status from the eligibility criteria of the exam. The selection of the aspirants, who will have been participated in the exam, will be on the basis of their written test and merit list.
The aspirants should have 10+2/diploma/Graduate under the Government of Maharashtra Revolution.
The candidates who want to apply for the exam, their age should not exceed 28 years under the application form of MahaTET 2015. The age limit will be 20-28 years.
महाटीईटी २०१५ ऑनलाईन नोंदणी सुरु झाली आहे.


Continuing to add to the FAQ page on my website. Check it out or make suggestions in the various comment sections here.
Some planes have mechanical adjusters to help with the alignment of a blade in the plane's body. Some planes have none and rely upon slight taps with a hammer. Laterally adjusting an iron can be a tricky task for the newly initiated. Efficiency with this skill is acquired only with time and attempts.
Every time, with every plane, the depth of cut must be addressed each time the plane is disassembled for sharpening. Additionally, each time you take the iron out of any plane you must laterally adjust the plane's iron to be in line with the plane's sole, right?
No, actually. I sharpen the profile of my side escapement irons so that there is no lateral adjustment necessary, only depth of cut. How is this possible? I grind my irons so that when the plane's iron is pressed firmly against the blind side of the plane's body (the side of the plane that the shaving is not ejected from) the iron's profile is directly in line with the profile of the plane's body.
My purpose of spending the time to profile the planes in this manner may not be apparent if you only have one or two planes that you use. After all, you will only need to remember that plane A's iron is centered at the top while touching the blind side at the bottom while Plane B's iron is very different and if you get it wrong you'll figure it out.
Though every plane you will own from various manufacturers has it's own feel, I go out of my way to make certain that each plane works in the same manner as its mate and every other plane in its set. The irons all set the exact same way.
Finally, due to the graduating widths of the irons; the mortise, wedge and tang increase in size similarly. This ensures that the same strike of an adjusting hammer will advance the iron a similar amount, regardless of the plane's size.
Notes: I have never had a left-handed student choose to use right-handed planes. I have never had right-handed students choose to use left-handed planes, unless posing for a joke.
I am teaching a class at CVSW this weekend. There are two spots open. Come learn how to use hollows and rounds, left or right. Come learn what appropriately tuned planes feel like, left and right. Come listen to Bob Van Dyke chastise me for listening to the soundtrack to "Hamilton", neither left nor right.
I continue to add pages to my FAQ page, this is one.
Finding the perfect plane for you lefties may be difficult, regardless of where you fall on the political spectrum.
(1) You will want a specific ogee so you watch the various antique dealers hoping they will post it and you will be the first to buy it…now. It is possible. It is unlikely. You will think that "there should be a law."
(2) You want a missing pair of hollows and rounds to match your set so you watch eBay. With enough time, your desired pair will likely show up. Best case scenario you will get into a bidding war, you buy it, it is not exact, you sell it on eBay to your bidding partner at 40%. You will think that "there should be a law."
(3) You want any left handed plane. You wait, wait, read, and wait. It is not going to happen so you wish for a law to fix the left vs. right conundrum and realize there once was an unwritten law to fix exactly this.
Left handed antique planes essentially do not exist. Left handed planes are desirable for those that can not conform and may only exist because a few have not conformed.
I had a long series of paragraphs, pictures and illustrations typed out to explain the advantages of having the correctly handed planes. Having done that and now erased it, I think I can make this subject quick and apparent. You need to be able to see your progress, as measured by the shavings your plane is producing.
This is a right handed person using a right handed plane. He is working in the normal direction, right to left. Do you see how he is able see inside of the escapement?
When using hollows and rounds we want predictable, desirable results. To get these results we need to be able to gauge our progress. We are able to measure our progress by watching the shavings being ejected from the mouth. If you cannot see the shavings then you cannot see the results. Additionally, if you cannot manipulate the plane to change the shavings because you cannot see the shavings or you're not steering with your dominate hand then you will not get predictable results.
Left handed planes are best for left handed people, as evidenced by every single left handed student I have had and given the choice. You may not choose to purchase all of your planes new and left handed. However, if you are left handed and you own some lefties while still owning some righties, the new lefties won't make you worse with your old righties. You will be better overall.
While there is no shadow between the ramp and the wedge (above), there is also no shadow or space between the wedge's tip and the body. No matter how I try to force the wedge, I cannot create a void here. There is no room for play between the wedge and body of the plane. This means that there is no possible void in which the shavings may be trapped.
I sell planes that work correctly and are, in the parlance of our industry, "ready to go." They have wedges that fit appropriately, irons that are bedded properly, cutting edges that are sharp and match flat soles, and soles that match each other. There are other advantages to new versus old, but those are better demonstrated than pictured.
I have added another common question to my FAQ page. Take note that I will keep the page on my site regarding plane availability updated. This blog post, however, may not be accurate in the future. Please check my page for current options.
Planes are generally made to order. However, on occasion I will make an extra few planes or pairs due to popularity and work flow. I can have planes available for immediate sale and delivery. All are listed below.
Please know that I do not sharpen the planes until they have officially been sold. This ensures that the plane's iron exactly matches the sole on the day that it is sent. As a result, a plane that is available for delivery will still take a day or two to leave the shop.
The planes listed on this page include domestic shipping, which may account for any discrepancies compared to other pages. Ordering multiple planes will slightly reduce the total price.
To order a plane off of this page or for answers to any other questions please send an email to matt@msbickford.com. Payments are best done through PayPal.
Note: the plane you receive may not be the exact one pictured. CT residents will also be charged sales tax.
If you do not know what sizes you want but there is a certain profile you need to execute, send it to me and I'll tell you what planes are used. Or you can find the radii of the included arcs with a circle template. Otherwise, I often recommend getting a pair of #4s and 8s (they cut a radius of 4/16ths and 8/16ths, respectively) if the scale you work to is waist height. Consider 6s and 10s (6/16" Radius and 10/16" Radius) if the scale of your furniture is shoulder height. 4s and 6s are a good place to start for piece that will stand upon another surface, i.e. mantle clock.


text:
Seo Tai-ji Tops Album Sales
Seo Tai-ji
South Korea's music icon Seo Tai-ji ranked first in album sales so far this year.
According to Hanteo Chart, an online site measuring local music record sales, Friday, Seo sold 198,215 copies between Jan. 1 and Sept. 17 this year to stand atop other pop idols.
Seo, 37, released two single albums, one regular album and Seo Tai-ji and Boys vol. 1-4 since making his comeback in July last year.
Seo, a famous rock singer, musician and songwriter, emerged on the Korean music scene in the 1990s and was often called the ``Culture President of Korea.''
Girls' Generation, one of South Korea's leading girl bands, took second place by selling 190,475 copies thanks to the success of ``Gee'' and ``Genie.''
Given album sales of girl groups usually remain weak compared with those of boy bands, Girls' Generation's result is noteworthy, music experts said.
Big Bang, the five-member boy band, and Super Junior, the 13-member boy group, placed third and fourth with 161,460 and 131,480 album sales, respectively. G-Dragon, Big Bang's leader, ranked fifth with 113,100 copies sold.

redpajama_set_name: RedPajamaCommonCrawl


The estate was built by the Peabody trustees on a site acquired from the Metropolitan Board of Works in one of London's first slum clearance schemes. The blocks were designed by Peabody's architect, Henry Astley Darbishire, and the estate was opened in 1881. On 8 September 1940, the second night of the London Blitz, one block took a direct hit, and nearly 80 people were killed. In 1995 a war memorial listing their names was placed on the external wall of Block L.


Way back in February of this year, we featured a new Teen Daze track called "Brooklyn Summer" that was the lead single from the album All of Us, Together, which was released towards the end of May via Lefse Records. The album was a fantastic interweaving of electronic beats but ultimately left me wanting a bit more of the style from his Beach Dreams EP with the incorporation of vocals into the tracks. While Teen Daze is certainly always atmospheric in the best way, the extra layer of fuzzy vocals he has featured from time to time brings an added element and meaning to his songs. Thankfully, I might not have to wait long for my wish as I apparently forgot that Teen Daze is the Ty Segall of the electronic indie scene and never stops releasing new music.
Two days ago, "Union" was released along with the details for his second full-length album of the year, The Inner Mansions, which will hit the shelves on 11-6-12. And to my great surprise and joy, he enlisted the stellar vocal services of Frankie Rose for the track. The song itself is fantastic, with the two singing on top of each other to create an interesting blend of voice to put effects to. Beginning with a chaotic clamor of sound, it eventually settles out into a melodic, lush, pool of tranquility to sink into. Dive in and give it a listen below.
Thought on Tracks is a forum for musical thought and discussion. The blog will focus primarily on Indiana artists, record labels and performances. The content consists of album reviews, concert reviews, music videos and much more. We are based out of Indianapolis but have a few writers contributing form afar. We hope that you will take an active interest in the pieces we publish and are always happy to hear feedback–positive or negative. Thank you for finding us, we hope that you will stay a while.


Posted by siteadmin on Friday 7th of December 2018.
For National Stress Awareness, we thought it would be best to shine a light on what some of the people at Chiltern do to de-stress. After all, we all know what it's like to feel stressed, but when it comes to winding down, it's important to take the time to relax.
Stress is our body's response to pressures from a situation or life event. It is inevitable, there will be stressful periods in your life and the causes of stress can vary. Stress tends to originate from experiencing something new or unexpected, something that threatens your feeling of self, or feeling you have little control over a situation – all of these can be at work or in your personal life.
Becoming overwhelmed by stress can lead to mental health problems or make existing problems worse. Stress plays a big part in absenteeism.Figures released this month by the Health and Safety Executive (HSE) state that work-related stress anxiety or depression accounts for over half of all working days lost due to ill health in Great Britain.
Here at Chiltern, we encourage staff to look after their wellbeing at work. Maintaining a work-life balance, is incredibly important and is used to combat stress and the pressures of work. In addition to upholding a balance between work and your personal life, it is just as important to de-stress at home as it is at work.
There is no right or wrong way to manage stress. Ways of coping with stress is all down to the individual.


text:
Qa-34, 26 April 2018, Statement Of The Spokesperson Of The Ministry Of Foreign Affairs, Mr. Hami Aksoy, In Response To A Question Regarding The Statements On Turkey By Jean-claude Juncker, President Of The European Commission At The Greek Parliament
We have observed that Jean-Claude Juncker, President of the European Commission, during his speech at the Greek Parliament today (26 April), continued the habit of making judgements, as if he were an expert, under the so-called "Union solidarity". With such statements based on the Greek Cypriot and Greek claims, EU officials and institutions lose more of their credibility and reliability each passing day. We reject these frivolous statements that lack any legal foundation.
We are also astonished by Mr Juncker's call for the release of the two Greek soldiers who are in custody in Turkey. The Greek soldiers who have violated the Turkish law do not enjoy any immunity before the independent judiciary. We condemn this hypocritical approach, which reminds the rule of law whenever it suits, and yet doesn't refrain from making statements amounting to intervening in the ongoing judicial process when it befits its interests.


text:
Q: Pexpect.before out of syncc I collect some data from users' routers, connecting via SSH, reading from pexpect. before.
Having trouble on only one router, that his Pexpect output is out of sync. I have 16 parameters that need collecting data from the router.
Solution from @user145078:
session.send(cmd)
index = session.expect([cmd, pexpect.TIMEOUT], 1)
session.send('\n')
index = session.expect([whatever you expect])

Worked a little bit, from 16 parameters, 14 are now in sync, and 2 of them aren't.
I don't know why those 2 parameters are still out of sync. Data is gathered in alphabetical order, meaning if I have commands for 'router_speed', 'signal_to_noise_ratio', 'serial', 'device_uptime','attn' they would be gathered in order:
'attn'
'device_uptime'
'router_speed'
'serial'
'SNR'
With the solution I used for this, only 'SNR' and 'attn' parameters are out of sync, with 'SNR' .before outputting last commands output and 'attn' .before outputing welcoming message to router:

Speedport Plus181\r\n \r\n\r\n\r\n\r\nBusyBox v1.15.3 () built-in
shell (ash)\r\nEnter 'help' for a list of built-in commands.

Don't know what to do, all those other solutions are a bit complicated to get into my running code.
Sorry for the bad grammar or bad explanation but I need help :D

A: Well I fixed it, it was a syntax bug/error in the commands for those two parameters.

missing ), unterminated subpattern at position 37

All is fine now.

redpajama_set_name: RedPajamaStackExchange


A new level of ease and convenience awaits CRC users thanks to a new host of features available on the organization's new and improved website. The system includes features allowing users to sign up for programs, check the availability of places such as the pool or the Roe Stamps field, and check when rentals are due, all from any location where users have access to the site.
The changes have come from a need to simplify many services for CRC users so that they no longer have to call or come in person to sign up for programs. Additionally, students will be able to sign up for programs after hours, which previously would have been impossible. With a comprehensive list of all CRC programs now available online there is also the hope that less well known programs will gain more usage, such as massage therapy. That also depends on awareness of software itself disseminating. As Walter Kolis, Assistant Director of Member Services and Staff Development at the CRC, explained, "I don't think it's as well known as it could be, because we just started with online in the spring." He expects, however, that that will soon change now that most programs are online.
These online resources might be particularly useful for those who frequently need to check information available on the website, such as Victoria Contreras, a fourth year biology and psychology major. Because of her frequent involvement with ORGT, she frequently needs to know the number of slots open for a given trip, and can now do so through her computer.
After around a year of adding new features to the system, most of the features slated to become available are online and running, with the exception of racquetball court reservations. The system also has the capability to make physical purchases possible, so users may at some point be able to buy goggles, locks, and t-shirts online.
Unlike previous years, where the CRC required students to sign up for classes and events in person at the CRC, students will now be able to not only access the CRC's revamped calendar, but also create a member profile and register and pay for courses including G.I.T. FIT, instructional programs, ORGT activities, swim lessons, and more. Enjoy the CRC's multitude of classes without the hassle of signing up in person at the CRC at their website www.mycrc.gatech.edu.


text:
Domenico Rivarola (* 1575 in Genua; † 3. Januar 1627 in Rom) war ein italienischer Erzbischof und Kardinal.

Leben
Er wurde 1575 in Genua als zweiter Sohn von Ottavio Rivarola und Nicoletta Lomellini geboren. Rivarola entstammte einer Adelsfamilie und war vermutlich ein Neffe von Giulio Rivarola, dem Apostolischen Nuntius in Portugal, und von Domenico Rivarola, dem designierten Bischof von Fermo. Sein Studium in Genua schloss er in utroque iure ab und wurde als junger Mann Kanoniker des Domkapitels von San Lorenzo in Genua.

Er zog nach Rom an den Hof von Kardinal Scipione Caffarelli Borghese. Am 10. Dezember 1608 wurde er zum Bischof von Aléria auf Korsika ernannt. Die Bischofsweihe spendete ihm am 28. desselben Monats Kardinal Michelangelo Tonti; Mitkonsekratoren waren Metello Bichi, ehemaliger Bischof von Sovana, und Valeriano Muti, Bischof von Città di Castello. Domenico Rivarola blieb jedoch in Rom, wo ihm Kardinal Caffarelli eine Befreiung von der Residenzpflicht für das Amt eines Uditore gewährte. Am 30. März 1609 erhielt er, nachdem er auf den Sitz von Aleria verzichtet hatte, den Titularsitz von Nazareth.

Er wurde zum außerordentlichen Legaten bei Heinrich IV. von Frankreich ernannt und mit der Aufgabe betraut, den König aufzufordern, von seinen Plänen zur Invasion Italiens Abstand zu nehmen. Der König starb jedoch vor der Ankunft von Rivarola, der dem apostolischen Nuntius in Frankreich für eine gewisse Zeit zur Seite stand und dann nach Italien zurückkehrte. Hier wurde er mit der Aufgabe betraut, einen Streit zwischen der Stadt Rieti und dem Schloss Cantalice zu schlichten.

Im Konsistorium vom 17. August 1611 ernannte ihn Papst Paul V. zum Kardinal. Am 12. September desselben Jahres wurde ihm mit dem Rang eines Kardinalpriesters die Titelkirche Santi Silvestro e Martino ai Monti verliehen. 1612 erhielt er eine Legation in der Romagna, wo er gegen Banditen und Räuber kämpfte. Da er jedoch keine bedeutenden Ergebnisse erzielte, stellte ihm der Pontifex in den Jahren 1617–1619 den seinerzeitigen Bischof von Forlì, Cesare Bartolelli, als apostolischen Delegaten zur Seite.

Rivarola nahm an den Konklaven von 1621 und 1623 teil, bei denen Gregor XV. und Urban VIII. gewählt wurden.

Er starb in Rom und wurde zunächst in der Kirche Santa Maria della Scala beigesetzt und später nach Santa Maria della Vittoria überführt.

Literatur

Einzelnachweise

Weblinks

 Eintrag zum Grabmal im REQUIEM-Projekt
 Eintrag zur Prosopographie im REQUIEM-Projekt, von Wolfgang Reinhard bearbeitet.

Kardinal (17. Jahrhundert)
Titularerzbischof
Römisch-katholischer Bischof (17. Jahrhundert)
Historische Person (Italien)
Geboren 1575
Gestorben 1627
Mann

redpajama_set_name: RedPajamaWikipedia


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If you're also interested in "asking questions", would you like to add it to your profile?
Journal Title: I met you in the midst of stardust.
Journal Title: Why did I open the window that night?
Journal Title: I'm so thankful that I met you.