Inclusive annihilation in Yukawa theory

In the model of Yukawa field theory we consider the inclusive process $e^{+}+e^{-}\to \gamma \mathrm{}\left(p\right)\mathrm{}\to \pi \mathrm{}\left(k\right)+\mathrm{anything}$. We find, by summing up the logarithms, that certain transforms $H^{\mathrm{}\left(n\right)\mathrm{}}$ of the amplitude, which in the high-energy limit $u\equiv p^2\to \infty$ may be identified with the moments $\int 0^{1}d\omega {\omega }^{n+1\mathrm{}}\overline{F}(\omega ,p^2)$ of the structure functions $\overline{F}(\omega ,p^2)$, have simple leading behavior. At a fixed point of the renormalization group the $H^{\mathrm{}\left(n\right)\mathrm{}}$ would grow like a fractional power ${\delta }_n$ of $p^2. {\delta }_n$ is a property of the underlying theory (Yukawa) rather than the structure of the pion, treated as a bound state, and may be calculated in perturbation theory.