Uncertainties of hadronic scalar decay calculations

TL;DR

This paper critically examines the use of dispersion relations in calculating decay widths of hypothetical light scalar bosons into mesons, highlighting limitations and calling for improved understanding of multi-channel systems.

Contribution

It identifies fundamental issues with current dispersion relation applications in multi-channel meson decays and emphasizes the need for a more rigorous approach.

Findings

Dispersion relation methods rely on uncontrollable approximations.

Limited channels invalidate unitarity assumptions.

Results of current calculations should be treated with caution.

Abstract

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.