Basic $\overline{K}$ nuclear cluster, $K^{-}\mathit{pp}$, and its enhanced formation in the $p+p\to K^{+}+X$ reaction

We have studied the structure of $K^{-}\mathit{pp}$ nuclear cluster comprehensively by solving this three-body system exactly in a variational method starting from the Ansatz that the $\Lambda (1405)$ resonance $(\equiv {\Lambda }^{*})$ is a $K^{-}p$ bound state. We have found that our original prediction for the presence of $K^{-}\mathit{pp}$ as a compact bound system with $M=2322$ MeV/$c^2,B_K=48$ MeV, and $\Gamma =60$ MeV remains unchanged by varying the $\overline{K}N$ and $\mathit{NN}$ interactions widely as far as they reproduce $\Lambda (1405)$. The structure of $K^{-}\mathit{pp}$ reveals a molecular feature, namely the $K^{-}$ in ${\Lambda }^{*}$ as an “atomic center” plays a key role in producing strong covalent bonding with the other proton. We have shown that the elementary process, $p+p\to K^{+}+{\Lambda }^{*}+p$, which occurs in a short impact parameter and with a large momentum transfer ($Q~1.6$ GeV/$c$), leads to unusually large self-trapping of ${\Lambda }^{*}$ by the participating proton, because the ${\Lambda }^{*}\text{−}p$ system exists as a compact doorway state propagating to $K^{-}\mathit{pp}$ ($R_{{\Lambda }^{*}p}~1.67$ fm).

Figure 1

Parametric presentation of the $K^{-}p$ energy, $E_{K^{-}p}$, and the scattering length, $a_{K^{-}p}$, in the plane of the $\overline{K}N$ interaction strength ($V_0$) and the range parameter ($b$) in the expression $v_{\overline{K}N}^{I=0}(r)=(V_0+i{W}_0)\exp [-(r/b){}^2]$. The imaginary part is adjusted to reproduce $\Gamma =40$ MeV. The experimental values, $E_{K^{-}p}=-27$ MeV (blue squares) and $a_{K^{-}p}=1.7$ fm (red circles), determine $V_0$ and $b$.

Figure 2

Dependence of the $\overline{K}N$ scattering amplitude on the range parameter $b$. (Upper) The real part with $b=0.7$ fm reproduces the known chiral dynamics result [34]. (Lower) The imaginary part with $b=0.7$ fm accounts for the observed total cross section of $\Lambda (1405)$.

Figure 3

Energy dependence of the $\overline{K}N$ interaction strength, where $s^{\mathrm{cmp}}=s_{\mathrm{R}}^{\mathrm{cmp}}+i{s}_{\mathrm{I}}^{\mathrm{cmp}}$.

Figure 4

(Upper) Structure of $K^{-}p$ and $K^{-}\mathit{pp}$, as calculated in Ref. [2]. (Middle) The effective potentials for relative motions of $N\text{−}(\overline{K}N)$ and $\overline{K}\text{−}(\mathit{NN})$, deduced from the exact variational wave function for $K^{-}\mathit{pp}$. The $K^{-}\text{−}p$ potential for $\Lambda (1405)$ is also shown. (Lower) Density distributions of various coordinates in $K^{-}\mathit{pp}$ as well as in $\Lambda (1405)=K^{-}p$ together with its density reduced by a factor of 3/4 (brown dots). The values of the rms distances and momenta are also given.

Figure 5

Comparison of the density distributions, $r^2\rho (r_{\mathit{KN}})$, of the $\overline{K}\text{−}N$ distance in the $\overline{K}N$ pair in $\Lambda (1405)$ and in $K^{-}\mathit{pp}$. The latter is decomposed into the $I=0$ and $I=1$ pairs. The density distribution in $\Lambda (1405)$ after multiplication of a factor 0.625 is also shown.

Figure 6

The adiabatic potential when a proton approaches a $\Lambda (1405)$ as a function of the $p\text{−}p$ distance. For comparison the Tamagaki potential $v_{\mathit{NN}}$ [40] is shown.

Figure 7

The molecular structure of $K^{-}\mathit{pp}$. (Middle) The projected density distributions of $K^{-}$ in $K^{-}\mathit{pp}$ with a fixed $p\text{−}p$ distance ($=2.0$ fm). (Lower) The corresponding $K^{-}$ contour distribution.

Figure 8

The artificially increased hard core in the $\mathit{NN}$ potential, while keeping the $\mathit{NN}$ scattering lengths to empirical values. The calculated energy and width of $K^{-}\mathit{pp}$ with varied hard core values $V_{\mathrm{core}}$ are listed in inset.

Figure 9

Dependence of the $K^{-}\mathit{pp}$ energy and width on the range parameter $b$ in the present treatment (red curves), where the interaction strength is varied so as to reproduce the energy and width of $\Lambda (1405)$. For comparison is shown the case of the chiral dynamics treatment with a fixed “LO” term (black curves).

Figure 10

The $\overline{K}N$ potentials with artificially introduced hard core values, $V_{\mathrm{RC}}=0,500,1000,1500$, and 2000 MeV, while keeping the energy and width of $K^{-}p$ to the ${\Lambda }^{*}$ values. The calculated potential parameters and energy and width of $K^{-}\mathit{pp}$ with varied hard core values $V_{\mathrm{RC}}$ are listed in inset.

Figure 11

(a) Diagram for the $d({\pi }^{+},K^{+})K^{-}\mathit{pp}$ reaction. (b) Calculated spectral shape of the $d({\pi }^{+},K^{+})K^{-}\mathit{pp}$ reaction. (c) Diagram for the $p(p,K^{+})K^{-}\mathit{pp}$ reaction. (d) Forward cross sections of the $p(p,K^{+})K^{-}\mathit{pp}$ reaction for different rms distances $R({\Lambda }^{*}p)$. The cases (A) and (B) of the $\overline{K}N$ interaction correspond to 1.67 and 1.44 fm, respectively.

Figure 12

Effects of various parameters on each factor of the transition amplitude in the $p+p\to K^{+}+K^{-}\mathit{pp}$ reaction. Columns (I) and (III) for the $\overline{K}N$ interaction case (B) and with ρ and π exchange cases, respectively. Column (II) for the less bound case. The solid (red) and broken (blue) curves represent the bound [$E_{\mathrm{B}}({\Lambda }^{*}p)=-60$ MeV] and the quasifree [$E_{\mathrm{QF}}({\Lambda }^{*}p)=100$ MeV] regions.

Figure 13

Kinematic relations in $\mathit{pp}\to K^{+}+X$ reaction at $T_p=3.0$ GeV for $M_X=2280$ MeV/$c^2$. The momentum vectors for a typical backward $K^{+}$ event are also shown.

Figure 14

Predicted differential cross sections of $p+p\to K^{+}+X$ at $T_p=3.0$ GeV for the $\overline{K}N$ interaction case (A). (a) $M_X$ spectra at various $K^{+}$ laboratory angles. (b) $K^{+}$ energy spectra at various $K^{+}$ laboratory angles.

Figure 15

Predicted differential cross sections of $p+p\to K^{+}+X$ at $T_p=3.0$ GeV at forward angle in the three cases of the $\overline{K}N$ interaction: cases (A), (B), and (C), as given in Table . The $E({\Lambda }^{*}p)=0$ value corresponds to the ${\Lambda }^{*}$ emission threshold.