Studying the bound state of the $K^{-}p$ system in the Bethe-Salpeter formalism

We study the $s$-wave kaon-nucleon bound state with the strangeness $S=-1$ in the Bethe-Salpeter formalism in the ladder and instantaneous approximations. We solve the Bethe-Salpeter equation of the bound state and obtain the Bethe-Salpeter amplitude. It is shown that the $K^{-}p$ bound state exists in this formalism. We also study the decay width of the bound state based on the Bethe-Salpeter techniques. The mass of this bound state is 1422 MeV, and its decay width is obviously smaller than that of $\mathrm{\Lambda }(1405)$. These results indicate that there may be some other structures in the observed resonance.

Figure 1

Feynman diagram for the Weinberg-Tomozawa interaction in ChPT.

Figure 2

Feynman diagram for the decay ${\mathrm{\Lambda }}^{*}\to {\mathrm{\Sigma }}^{+}{\pi }^{-}$.

Figure 3

The truncated Bethe-Salpeter irreducible part $T(x_2,y_2;x_1,y_1)$.

Figure 4

Numerical result for the normalized Bethe-Salpeter amplitude of the bound state. The units of $\widetilde{f_1}$ and $\widetilde{f_2}$ are 1 and ${\mathrm{MeV}}^{-1}$, respectively.