In this report we will calculate the amplitude for a scalar-to-scalars (φ3 <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Crightarrow" /> φ2φ2) decay which involves a triangle loop. We compute the real and imaginary part of the amplitude separately and will argue that this is much more straightforward and practical in this case rather than having to deal with or worry about branch cuts of logarithms. We will derive simple cutting rules closely related to the imaginary part of the amplitude. In doing this, we derive a formula that deals with expressions of the form δ[f(x,y)]δ[g(x,y)], containing two Dirac delta functions.

We give a rather self-contained introduction to the rare pion to dielectron decay which in nontrivial leading order is given by a QED triangle loop. We work within the dispersive framework where the imaginary part of the amplitude is obtained via the Cutkosky rules. We derive these rules in detail. Using the twofold Mellin-Barnes representation for the pion transition form factor, we derive a simple expression for the branching ratio B(π0 <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Crightarrow" /> e+e-) which we then test for various models. In particular a more recent form factor derived from a Lagrangian for light pseudoscalars and vector mesons inspired by effective field theories. Comparison with the KTeV experiment at Fermilab is made and we find that we are more than 3σ below the KTeV experiment for some of the form factors. This is in agreement with other theoretical models, such as the Vector Meson Dominance model and the quark-loop model within the constituent-quark framework. But we also find that we can be in agreement with KTeV if we explore some freedom of the form factor not fixed by the low-energy Lagrangian.