\zeta ) \right]_+
\biggr\} .\end{aligned}$$ Here $$\begin{aligned}
{^{QQ}\!K^A} (x , x ', \zeta)
\!\!\!&=&\!\!\! K^T (x , x ', \zeta)
+ C_F \, \Theta_{111}^0 ( x , x - \zeta, x - x ' ), \\
{^{Q}\!G} ( x, x', \zeta )
\!\!\!&=&\!\!\! \frac{x' - \zeta}{x - x'}
\ln \left( \frac{x' - x}{x' - \zeta} \right)
\Theta_{11}^0 (x - \zeta, x - x ')
+
\frac{x'}{x - x'}
\ln \left( \frac{x' - x}{x'} \right)
\Theta_{11}^0 (x, x - x '), \\
{^{Q}\!K^b} ( x, x', \zeta )
\!\!\!&=&\!\!\! \frac{x - \zeta}{x - x'}
\Theta_{11}^0 (x - \zeta, x - x ')
+
\frac{x}{x - x'}
\Theta_{11}^0 (x, x - x '),\end{aligned}$$ and a new kernel ${^{Q}\!G} ( x, x', \zeta )$ comes from the expression for ${^{QQ}{\mit\Phi}}$, which is of the form for $\beta = 0$: $${^{QQ}{\mit\Phi}} (x, y, \zeta)
= ({\cal I} - {\cal D})
\left\{
S (x, z) \otimes {^{QQ}\!K^A} ( z, y, \zeta )
- \left[ {^{Q}\!G} ( y, x, \zeta ) \right]_+
\right\}.$$ In the limit $\omega = 1$ and $\zeta = 1$ we obtain from Eq. (\[unpolQCF\]) the results of Refs. [@Cha81; @Bra83; @Rad86] for the pion transition form factor $F_{\pi\gamma\gamma}$.

Unpolarized case.
-----------------

Since 