the two fermion propagators with a Feynman parameter $x$, $$\frac{1}{AB}=\int_{0}^{1}\frac{1}{[Ax+B(1-x)]^2},$$ and after integration over the four–momenta $k$ in the loop \[[I am assuming that a gauge invariant regularization of this integral, like e.g. dimensional regularization, has been made.]{}\], the one–particle irreducible part of this diagram, when subtracted at $q^2=0$ if we choose on–shell renormalization, leads to the renormalized self–energy function: $$\label{eq:selffeynman}
\Pi_{\mbox{\tiny\rm R}}(q^2)=-\frac{\alpha}{\pi}\int_{0}^{1}
dx\,2x(1-x)\log\left(1-\frac{q^2}{m^2-i\epsilon}x(1-x)\right),$$ With the change of variables $y=1-2x$, and using the fact that the resulting integral is symmetric when $y{\rightarrow}-y$, we get $$\Pi_{\mbox{\tiny\rm R}}(q^2)=\frac{\alpha}{\pi}\int_{0}^{1}
dy(1-y^2)\log\left[1-\frac{q^2}{4m^2 -i\epsilon}(1-y^2)\right].$$ Integrating by parts this equation using the identity: $1-y^2=\frac{\partial}{\partial y}(y-\frac{1}{3}y^3)$ results in a rational integral: $$\Pi_{\mbox{\tiny\rm R}}(q^2)=\frac{\alpha}{\pi}\int_{0}^{1}dy\,
2y(y-\frac{1}{3}y^3)\frac{q^2}{4m^2-q^2(1-y^2)-i\epsilon},$$ and with a new change of variables: $t=\frac{4m^2}{1-y^2}$ we finally get a rather interesting representation of the renormalized self–energy $$\label{eq:fdisprel}
\frac{\Pi_{\mbox{\tiny\rm
R}}(q^2)}{q^2}=\frac{\alpha}{\pi}\int_{4m^2}^{\infty}
\frac{dt}{t}\,\frac{1}{t-q^2-i\epsilon}\,
\frac{1}{3}\left(1+\frac{2m^2}{t}\right)\sqrt{1-\frac{4m^2}{t}}.$$ The reason why this representation is interesting is that it is in fact a dispersion relation! We have succeeded in rewriting the initial Feynman parametric representation in eq. (\[eq:selffeynman\]) as a dispersion relation by simple changes of variables. Using the identity $$\label{eq:ppartid}
\frac{1}{t-q^2-i\epsilon}=\mbox{\rm PP}\frac{1}{t-q^2} +i\pi\delta(t-q^2),$$ we immediately see that $$\frac{1}{\pi}{\mbox{\rm Im}}\Pi(t)=\frac{\alpha}{\pi}
\frac{1}{3}\left(1+\frac{2m^2}{t}\right)\sqrt{1-\frac{4m^2}{t}}
\theta(t-4m^2).$$ Equation (\[eq:fdisprel\]) is a particular case of the general dispersion relation written in eq. (\[eq:selfdisp\]), when the 