Recently, many efforts are being put in studying three-hadron systems made of mesons and baryons and interesting results are being found. In this talk, I summarize the main features of the formalism used to study such three hadron systems with strang
eness $S=-1,0$ within a framework built on the basis of unitary chiral theories and solution of the Faddeev equations. In particular, I present the results obtained for the $pibar{K}N$, $Kbar{K}N$ and $KKbar{K}$ systems and their respective coupled channels. In the first case, we find four $Sigma$s and two $Lambda$s with spin-parity $J^P=1/2^+$, in the 1500-1800 MeV region, as two meson-one baryon s-wave resonances. In the second case, a $1/2^+$ $N^*$ around 1900 MeV is found. For the last one a kaon close to 1420 MeV is formed, which can be identified with K(1460).
We investigate the $bar KN$ and coupled channels system in a finite volume and study the properties of the $Lambda(1405)$ resonance. We calculate the energy levels in a finite volume and solve the inverse problem of determining the resonance position
in the infinite volume. We devise the best strategy of analysis to obtain the two poles of the $Lambda(1405)$ in the infinite volume case, with sufficient precision to distinguish them.
In this talk we show recent developments on few body systems involving mesons. We report on an approach to Faddeev equations using chiral unitary dynamics, where an explicit cancellation of the two body off shell amplitude with three body forces st
emming from the same chiral Lagrangians takes place. This removal of the unphysical off shell part of the amplitudes is most welcome and renders the approach unambiguous, showing that only on shell two body amplitudes need to be used. Within this approach, systems of two mesons and one baryon are studied, reproducing properties of the low lying $1/2^+$ states. On the other hand we also report on multirho and $K^*$ multirho states which can be associated to known meson resonances of high spin.
We solve the Faddeev equations for the two meson-one baryon system $pipi N$ and coupled channels using the experimental two-body $t$-matrices for the $pi N$ interaction as input and unitary chiral dynamics to describe the interaction between the rest
of coupled channels. In addition to the $N^*(1710)$ obtained before with the $pipi N$ channel, we obtain, for $J^pi=1/2^+$ and total isospin of the three-body system $I=1/2$, a resonance peak whose mass is around 2080 MeV and width of 54 MeV, while for $I=3/2$ we find a peak around 2126 MeV and 42 MeV of width. These two resonances can be identified with the $N^* (2100)$ and the $Delta (1910)$, respectively. We obtain another peak in the isospin 1/2 configuration, around 1920 MeV which can be interpreted as a resonance in the $N a_0(980)$ and $N f_0(980)$ systems.
We perform a Faddeev calculation for the three mesons system, $phi K bar{K}$, taking the interaction between two pseudoscalar mesons and between a vector and a pseudoscalar meson from the chiral unitary approach. We obtain a neat resonance peak aroun
d a total mass of 2150 MeV and an invariant mass for the $K bar{K}$ system around 970 MeV, very close to the $f_0(980)$ mass. The state appears in I=0 and qualifies as a $phi f_0(980)$ resonance. We enlarge the space of states including $phi pi pi$, since $pi pi$ and $K bar{K}$ build up the $f_0$ (980), and find moderate changes that serve to quantify theoretical uncertainties. No state is seen in I=1. This finding provides a natural explanation for the recent state found at BABAR and BES, the X(2175), which decays into $phi f_0(980)$.
We discuss the dynamical generation of some low-lying $1/2^+$ $Sigma$s and $Lambda$s in two-meson one-baryon systems. These systems have been constructed by adding a pion in $S$-wave to the $bar{K} N$ pair and its coupled channels, where the $1/2^-$
$Lambda$(1405)-resonance gets dynamically generated. We solve Faddeev equations in the coupled-channel approach to calculate the $T$-matrix for these systems as a function of the total energy and the invariant mass of one of the meson-baryon pairs. This squared $T$-matrix shows peaks at the energies very close to the masses of the strangeness -1, $1/2^+$ resonances listed in the particle data book.
We evaluate the $sigma$ exchange contribution to the $bar{K}Ntobar{K}N$ scattering within a chiral unitary approach. We show that the chiral transition potentials for $pi pi to K bar{K}$ in the $t$-channel lead to a $sigma$ contribution that vanishes
in the $bar{K}$ forward direction and, hence, would produce a null $sigma$ exchange contribution to the $K^-$ optical potential in nuclear matter in a simple impulse approximation. This is a consequence of the fact that the leading order chiral Lagrangian gives an I=0 $pipito Kbar{K}$ amplitude proportional to the squared momentum transfer, $q^2$. This finding poses questions on the meaning or the origin of $sigma$ exchange potentials used in relativistic mean field approaches to the $K^-$ nuclear selfenergy. This elementary $sigma$ exchange potential in $bar{K}Ntobar{K}N$ is compared to the Weinberg-Tomozawa term and is found to be smaller than present theoretical uncertainties but will be relevant in the future when aiming at fitting increasingly more accurate data.
