Does the existence of $f_0$ meson enlarge the $K^+ \to \pi^0\pi^0\pi^0 e^+ \nu$ decay probability?

TL;DR

This paper investigates whether the broad $f_0(500)$ resonance significantly increases the rare decay rate of $K^+ o 3ar{ u} e^+$, finding its contribution is proportional to its width but remains negligible.

Contribution

It provides a quantitative estimate of the $f_0(500)$ resonance's impact on the $K^+ o 3ar{ u} e^+$ decay probability across a wide range of widths.

Findings

Resonance contribution scales with its width.

Contribution never exceeds the decay probability without resonance.

Narrow resonance effects are negligible in this decay.

Abstract

We estimate a possible contribution of the two-pion system $f_0(500)$ wide resonance to the decay probability of the super rare decay $K^+ \to 3\pi^0 e^+ \nu$. We vary the width of the resonance from 0.01 MeV to 1000 MeV. We found that the contribution of the narrow resonance in a system of several final state particles into the decay probability is proportional to its width. However it never exceeds the value of the decay probability calculated on free final state particles neglecting the resonance contribution.