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

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 $\varphi (2170)$, which has been described by means of Faddeev equations as a resonant state of $\varphi$ 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.

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• $K^{\ast }$ mesons with hidden charm arising from $KX(3872)$ and $K{Z}_c(3900)$ dynamics
• Few-body systems consisting of mesons
• $\bar{K}NN$ Absorption within the Framework of the Fixed Center Approximation to Faddeev equations
• Theoretical support for the $\pi (1300)$ and the recently claimed $f_0(1790)$ as molecular resonances
• Vector mesons and electromagnetic form factor of the $\Lambda$ hyperon
• A prediction of $D^{\ast }$-multi-$\rho$ states
• The $\varphi (2170)$ production in the process $\gamma p\to \eta \varphi p$
• A study of $\eta K\bar{K}$ and ${\eta }^{\prime }K\bar{K}$ with the fixed center approximation to Faddeev equations
• Three-body model for $K(1460)$ resonance
• Prediction of possible exotic states in the $\eta \bar{K}{K}^{\ast }$ system
• States of $\rho B^{\ast }{\bar{B}}^{\ast }$ with $J=3$ within the Fixed Center Approximation to Faddeev equations
• Possible bound states with hidden bottom from ${\bar{K}}^{(\ast )}{B}^{(\ast )}{\bar{B}}^{(\ast )}$ systems
• Pseudotensor mesons as three-body resonances
• Partial decay widths of $\varphi (2170)$ to kaonic resonances
• Tri-meson bound state $BB{B}^{\ast }$ via delocalized $\pi$~Bond
• Prediction of new states from $D^{(\ast )}{B}^{(\ast )}{\bar{B}}^{(\ast )}$ three-body interactions
• Description of $\rho (1700)$ as a $\rho K\bar{K}$ system with the fixed center approximation
• On the two-body decay processes of the predicted three-body $K^{\ast }(4307)$ resonance
• Predictions of super-exotic heavy mesons from $K^{\ast }{B}^{\ast }{B}^{\ast }$ interactions
• Investigations of the $D$-multi-$\rho$ interactions
• Looking for bound states and resonances in the ${\eta }^{\prime }K\bar{K}$ system
• $D_1(2420)$ and its interactions with a kaon: open charm states with strangeness
• States generated in the $K$-multi-$\rho$ interactions
• Dynamically generated states from the $\eta K^{\ast }{\bar{K}}^{\ast }$, $\pi K^{\ast }{\bar{K}}^{\ast }$, and $K{K}^{\ast }{\bar{K}}^{\ast }$ systems within the fixed-center approximation
• Three-body scattering problem in the fixed center approximation: the case of attraction
• $N^{\ast }$ states with hidden charm and a three-body nature
• Correlation function for the $n{\bar{D}}_{s0}^{\ast }(2317)$ interaction and the issue of elastic unitarity