not mix with each other up to terms proportional to $\beta$. The transformation matrix $B$, determined completely by the special conformal anomaly: $$\begin{aligned}
\label{blefdt-1}
\hat{B} = {\hat{1} \over \hat{1}+{\cal J}\hat{\gamma}^c}
= \hat{1}-{\cal J}\hat{\gamma}^c + {\cal J}(\hat{\gamma}^c
{\cal J}\hat{\gamma}^c) - \dots,\end{aligned}$$ with ${\cal J}\hat{A} = \theta_{jj' + 1} A_{jj'} / a( j, j' )$, specifies the partial conformal eigenfunctions of the evolution kernels. This matrix was calculated in Ref. [@Mue94] for the non-singlet parity-odd channel and used for a consistency check between the available NLO result for the coefficient function of the $\gamma^*\gamma \to \pi$ transition form factor and the forward NLO Wilson coefficients for polarized DIS [@Mue97].

Recently, the $\alpha_s$-correction to the non-forward coefficient function in the singlet channel has been calculated for unpolarized DVCS [@JiOs97]. In this section we evaluate the one-loop coefficient functions for the polarized $(T^A_{(1)})$ and non-polarized $(T^V_{(1)})$ cases as well: $$\begin{aligned}
{^i T}^{\mit\Gamma}( \omega, x, \zeta, Q^2 | \alpha_s )
= {^i T_{(0)}} ( \omega, x ) - \frac{\alpha_s}{2\pi}
{^i T_{(1)}^{\mit\Gamma}}( \omega, x, \zeta, Q^2 )
+ {\cal O} (\alpha_s^2).\end{aligned}$$ The predictions for the coefficient function can be obtained by employing the conformal OPE from the one side and the conformal constraints from the other. 