Limits to the Fixed Center Approximation to Faddeev equations: the case of the $\phi(2170)$

TL;DR

This paper critically examines the limitations of the Fixed Center Approximation (FCA) to Faddeev equations in studying the $\

Contribution

It identifies the inaccuracies of FCA for the $\

Findings

FCA can produce inaccurate results for the $\

Limitations of an alternative approach are also discussed.

The study clarifies the applicability boundaries of FCA in three-hadron systems.

Abstract

The Fixed Center Approximation to the Faddeev equations (FCA) has been used lately with success in the study of bound systems of three hadrons. It is also important to set the limits of the approach in those problems to prevent proliferation of inaccurate predictions. In this paper we study the case of the $\phi(2170)$, which has been described by means of Faddeev equations as a resonant state of $\phi$ and $K \bar{K}$, and show the problems derived from the use of the FCA in its study. At the same time we also expose the limitations of an alternative approach recently proposed.