the $QQ$-evolution kernels are the same in the polarized and non-polarized cases, the only difference will arise from the forward coefficient functions [@ZVN94]. Straightforward calculation gives for the quark as well as for gluon non-polarized coefficient functions: $$\begin{aligned}
{^Q T_{(1)}^V}( \omega, x, \zeta, Q^2 ) \!\!\!&=&\!\!\!
{^Q T_{(0)}} ( \omega, y ) \otimes
\biggl\{
{^{QQ}\!K^V} ( y, x, \zeta )\,
\ln \frac{Q^2}{\mu^2}
- \frac{5}{2} \left[ {^{Q}\!K^b} ( y, x, \zeta ) \right]_+
+ \delta (y - x) \nonumber \\
&+& \left[
\ln \left( 1 - y \omega \right) - 1
\right]
{^{QQ}\!K^V} ( y, x, \zeta )
- \left[ {^{Q}\!G} ( y, x, \zeta ) \right]_+
\biggr\} , \\
{^G T_{(1)}^V}( \omega, x, \zeta, Q^2 ) \!\!\!&=&\!\!\!
{^Q T_{(0)}} ( \omega, y ) \otimes
\biggl\{
{^{QG}\!K^V} ( y, x, \zeta )\,
\ln \frac{Q^2}{\mu^2}
- \frac{1}{2}
\left[
{^{QG}\!K^V} ( y, x, \zeta ) + {^{QG}\!K^A} ( y, x, \zeta )
\right]
\nonumber \\
&+& \ln \left( 1 - y \omega \right) {^{QG}\!K^V} ( y, x, \zeta )
\biggr\} ,\end{aligned}$$ the momentum space evolution kernels ${^{Q(Q,G)}\!K^V}$ read $$\begin{aligned}
{^{QQ}\!K^V} (x , x ', \zeta) &=& {^{QQ}\!K^A} (x , x ', \zeta),\\
{^{QG}\!K^V} (x , x ', \zeta)
&=& {^{QG}\!K^A} (x , x 