The method of cluster decomposition, well known in statistical mechanics and recently applied to ${\varphi }^3$ field theory by Chang, Yan, and Yao and by Campbell and Chang, is here extended to the calculation of the electromagnetic form factor in a renormalizable neutral pseudoscalar theory. In particular the set of so-called uncrossed "rainbow" diagrams are analyzed in the limit of spacelike momentum transfer squared, $q^2$, much larger than the pion or proton mass squared, ${\mu }^2$ or $M^2$. In this limit the Dirac form factor behaves as $F_1\left(q^2\right)=B(g^2,\frac{{\mu }^2}{m^2},\frac{M^2}{m^2}){(-\frac{q^2}{m^2})}^{-A\left(g^2\right)}$ where $g$ is the ${\pi }^{0}p$ coupling and $m$ is an arbitrary scale factor. The functions $A$ and $B$ are shown to arise in momentum space from "volume" and "surface" effects, respectively. They are given as a power series in $g^2$. The first two terms in $A$ are calculated explicitly, giving $A=\left(\frac{g^2}{32{\pi }^2}\right)+\frac{5}{2}{\left(\frac{g^2}{32{\pi }^2}\right)}^2+\cdots$.

• Received 31 January 1972

DOI:https://doi.org/10.1103/PhysRevD.5.2999