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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>W</a:mi></a:math>- and <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>Z</c:mi></c:math>-bosons into fermion pairs in the double-logarithmic approximation.