Remarks on applying dispersion relations to amplitudes with infrared singularities

Dispersion relations (DR) are known to be a powerful instrument for studying scattering amplitudes. In particular, they often apply to calculations in perturbative QCD and the Standard Model. We argue that applying DR to amplitudes with double-logarithmic (DL) contributions should be done with proper caution because DL terms are often infrared divergent. Ignoring this circumstance leads to incorrect results. As an example of such a situation, we consider applying DR to decays of on-shell $W$- and $Z$-bosons into fermion pairs in the double-logarithmic approximation.

Figure 1

First-loop contribution to $Z\to f\overline{f}$. The waved line is for the $Z$-boson, straight lines denote fermions, and the dotted line is for the virtual photon. The dashed line denotes the cut.

Figure 2

First-loop contributions to the amplitude of $W$-decay into fermions. The waved lines are for the $W$-boson, straight lines denote the fermions, and the dotted lines are for the virtual photons. The dashed lines denote the cuts.