Dispersion relations---a corollary of unitarity and analyticity---are among
the few methods which can be employed to study low energy processes in QCD. In
this work we critically revisit the calculation of decay widths of a
hypothetical light scalar boson into pions and kaons. Strong interactions of
the mesons in the final state affect these decays significantly. We argue that
applications of the dispersion relations to the calculation of these decay
widths, which are present in the literature, rely on an uncontrollable
approximation of the reduced S-matrix. When a limited number of initial and
final decay channels are considered, in general neither a unitary S-matrix nor
its version corrected by an inelasticity factor are justified. The scalar form
factor calculation provides a transparent example to illustrate this statement.
Therefore, the results of this calculation, which are important for
theoretical, cosmological and experimental applications, must be treated with
caution. Our work, therefore, calls for a breakthrough in the understanding of
the dispersion relations of realistic many-channel systems.Comment: 8 pages, 3 figure
