  We extend previous work on sum rules for the invariant amplitudes of pion
Compton scattering by deriving a complete lowest order perturbative spectral
function - and its leading non perturbative power corr ections - for a specific
combination of the two helicities $(H_1 + H_2)$ of this process. Using some
properties of a modified version of the Borel transform, we develop a method of
calculation of the gluonic corrections which can be easily extended to other
similar reactions, such as proton Compton scattering. A preliminary comparison
of the new sum rule with the pion form factor sum rule is made.
