Variational calculation of the $\mathit{pp}{K}^{-}$ system based on chiral SU(3) dynamics

The ${\mathit{ppK}}^{-}$ system, as a prototype for possible quasibound $\overline{K}$ nuclei, is investigated using a variational approach. Several versions of energy-dependent effective $\overline{K}N$ interactions derived from chiral SU(3) dynamics are employed as input, together with a realistic $\mathit{NN}$ potential (Av18). Taking into account theoretical uncertainties in the extrapolations below the $\overline{K}N$ threshold, we find that the antikaonic dibaryon ${\mathit{ppK}}^{-}$ is not deeply bound. With the driving $s$-wave $\overline{K}N$ interaction the resulting total binding energy is $B({\mathit{ppK}}^{-})=20\pm 3$ MeV and the mesonic decay width involving $\overline{K}N\to \pi Y$ is expected to be in the range 40–70 MeV. Properties of this quasibound ${\mathit{ppK}}^{-}$ system (such as density distributions of nucleons and antikaon) are discussed. The $\Lambda (1405)$, as an $I=0$ quasibound state of $\overline{K}$ and a nucleon, appears to survive in the ${\mathit{ppK}}^{-}$ cluster. Estimates are given for the influence of $p$-wave $\overline{K}N$ interactions and for the width from two-nucleon absorption $(\overline{K}\mathit{NN}\to \mathit{YN})$ processes. With inclusion of these effects and dispersive corrections from absorption, the ${\mathit{ppK}}^{-}$ binding energy is expected to be in the range 20–40 MeV, whereas the total decay width can reach 100 MeV but with large theoretical uncertainties.

Figure 1

Strength of the “corrected” (solid lines) and “uncorrected” (dotted lines) potentials at $r=0$ with HNJH model [37]. The real parts are shown as solid/dotted lines. Imaginary parts are given as dashed lines. (a) $I=0$ channel; (b) $I=1$ channel.

Figure 2

Distribution of total binding energy and mesonic decay width. The models ORB, HNJH, BNW, and BMN are shown with symbols square, circle, triangle, and diamond, respectively. Closed (open) symbols indicate the Type I (Type II) ansatz.

Figure 3

(a) $\mathit{NN}$ density in ${\mathit{ppK}}^{-}$, as function of $\mathit{NN}$ relative distance, calculated with the Type I ansatz. ORB, HNJH, and BNW results are shown as crossed, solid, and dashed lines, respectively. For comparison, the deuteron density calculated with Av18 potential is depicted as a dashed line with filled circles. (b) Display of $r^2\rho (r)$ for the same densities.

Figure 4

$\overline{K}N$ density in ${\mathit{ppK}}^{-}$, as function of $\overline{K}N$ relative distance, calculated with the Type I ansatz. (a) Separate display of each isospin component ($I=0$ and $I=1$). Lines without (with) symbols show the $I=0$ ($I=1$) component of the $\overline{K}N$ density. Models ORB, HNJH, and BNW are depicted with crossed, solid, and dashed lines, respectively. (b) Normalized $I=0\hspace{0.5em}\overline{K}N$ relative density in ${\mathit{ppK}}^{-}$ for the model HNJH with the Type I ansatz (solid line). Solid line with diamond shows $\overline{K}N$ density in $\Lambda (1405)$ calculated with the same model.

Figure 5

Energy dependence of $K^{-}$-proton scattering volume, $C_p$. Its real (imaginary) part is drawn with a solid line (dotted line with diamond).

Figure 6

Feynman diagrams for typical microscopic mechanisms of the two-nucleon absorption process. (a) One pion exchange that is forbidden in the present system; (b) and (c) two-meson exchange.

Figure 7

Absorptive width $\Delta {\Gamma }_{\mathrm{abs}}$ for ${\mathit{ppK}}^{-}\to \mathit{YN}$ as function of Gaussian range parameter $a$. Results are shown for the models ORB, HNJH, BNW, and BMN discussed in the text.

Figure 8

Central plus spin-spin $\mathit{NN}$ potentials in singlet-even (a) and singlet-odd (b) channels. (Solid line) Original Argonne v18 potential (Original). Points represent fits to the original potential with four Gaussians (Fitted).

Figure 9

$L^2$ potential in $\mathit{NN}$ potential. (a) and (b) ${}^{1}E$ and ${}^{1}O$ channels, respectively. Solid line is the original Argonne v18 potential (Original). Points represent the potential used in the present study, which fit the original potential with three Gaussians (Fitted).

Figure 10

Comparison of the loop functions: dimensional regularization (solid lines), three-momentum sharp cutoff (dashed lines), and smooth monopole form factor (dotted lines). (a) $\overline{K}N$ channel; (b) $\pi \Sigma$ channel.

Figure 11

Comparison of Minnesota potential with Av18 potential. ${}^{1}E$ channel of both potentials are shown. Minnesota potential is depicted with a solid line with diamond. Av18 potential is depicted with a solid line.

Figure 12

$\mathit{NN}$ density for Av18 potential (solid line) and Minnesota potential (solid line with diamond). A chiral-based $\overline{K}N$ potential (HNJH) is used.