Is the ${\Theta }^{+}$ a $K\pi N$ bound state?

Following a recent suggestion that the ${\Theta }^{+}$ could be a $K\pi N$ bound state we perform an investigation under the light of the meson-meson and meson-baryon dynamics provided by the chiral Lagrangians and using methods currently employed to dynamically generate meson and baryon resonances by means of unitary extensions of chiral perturbation theory. We consider two-body and three-body forces and examine the possibility of a bound state below the three-particle pion-kaon-nucleon and above the kaon-nucleon thresholds. Although we find indeed an attractive interaction in the case of isospin $\mathrm{I}=0$ and spin-parity $1∕2^{+}$, the interaction is too weak to bind the system. If we arbitrarily add to the physically motivated potential the needed strength to bind the system and with such strong attraction evaluate the decay width into $KN$, this turns out to be small. A discussion on further work in this direction is done.

Figure 1

Our final result: $K\pi N$ scattering matrix [modulus and denominator of Eq. (28)]. Energy units are $100\text{MeV}$.

Figure 2

We find a resonance with a reasonable width for a potential larger by a factor 6 (see text). Units are $100\text{MeV}$.

Figure 3

We also find a resonance, this time too broad, by increasing the cutoff in the $\kappa N$ loop to $4\text{GeV}$ and about doubling the potential. Units are $100\text{MeV}$.