In-flight ($K^{-},p$) reactions for the formation of kaonic atoms and kaonic nuclei using the Green function method

We study theoretically the kaonic atom and kaonic nucleus formations in the in-flight ($K^{-},p$) reactions using the Green function method, which is suited to evaluate formation rates both of stable and unstable bound systems. We consider ${}^{12}\mathrm{C}$ and ${}^{16}\mathrm{O}$ as the targets and calculate the spectra of the ($K^{-},p$) reactions. We conclude that no peak structure resulting from kaonic nucleus formation is expected in the reaction spectra calculated with the chiral unitary kaon-nucleus optical potential. In the spectra with the phenomenological deep kaon-nucleus potential, it may be possible to observe some structures because of the formation of the kaonic nucleus states. For all cases, we find clear signals because of the kaonic atom formations in the reaction spectra, which show very interesting structures, such as the RESONANCE DIP instead of the resonance peak for the atomic $1s$ state formation.

Figure 1

The kaon-nucleus optical potential of the chiral unitary approach [18, 37] for ${}^{11}\mathrm{B}$ as a function of the radial coordinate $r$. The upper and lower panels show the real and imaginary parts, respectively. The solid, dotted, and dashed lines indicate the potential strength for the kaon energies $E=0\hspace{1em}\mathrm{MeV},E=-50\hspace{1em}\mathrm{MeV},\hspace{1em}\mathrm{and}\hspace{1em}E=-100\hspace{1em}\mathrm{MeV}$, respectively.

Figure 2

The kaon-nucleus optical potential of the phenomenological fit [12] with the phase-space-factor [36] for ${}^{11}\mathrm{B}$ as a function of the radial coordinate $r$. The upper and lower panels show the real and imaginary parts, respectively. The solid, dotted, and dashed lines indicate the potential strength for the kaon energies $E=0\hspace{1em}\mathrm{MeV},E=-50\hspace{1em}\mathrm{MeV},\hspace{1em}\mathrm{and}\hspace{1em}E=-100\hspace{1em}\mathrm{MeV}$, respectively. The real part does not have the energy dependence.

Figure 3

Subcomponents of the kaonic nucleus formation cross sections in the ${}^{12}\mathrm{C}$($K^{-},p$) reactions at $T_{K^{-}}=600\hspace{1em}\mathrm{MeV}$ are plotted as functions of the emitted proton energy $T_p$ at ${\theta }_p^{\mathrm{lab}}=0$ (deg.) for the phenomenological optical potential Eq. (14). The upper panel shows the results with the reduced imaginary potential $\mathrm{Im}{V}_{\mathrm{opt}}/10$, whereas the lower panel shows those with Eq. (14). The solid lines show the results obtained by the Green function method for the sum of the subcomponents $[(s){}_K\otimes (1p_{3/2}){}_p^{-1}]$ and $[(p){}_K\otimes [(1p_{3/2}){}_p^{-1}]$. The dashed lines show those by the effective number approach for the sum of the subcomponents $[(1s_{\mathrm{nucl}}){}_K\otimes (1p_{3/2}){}_p^{-1}],[(2p_{\mathrm{nucl}}){}_K\otimes (1p_{3/2}){}_p^{-1}]$, and $[(2s_{\mathrm{nucl}}){}_K\otimes (1p_{3/2}){}_p^{-1}]$. The vertical dashed line indicates the kaon production threshold.

Figure 4

Subcomponents of the kaonic atom formation cross sections in the ${}^{12}\mathrm{C}$($K^{-},p$) reactions at $T_{K^{-}}=600$ MeV are plotted as functions of the emitted proton energy $T_p$ at ${\theta }_p^{\mathrm{lab}}=0$ (deg.) for the chiral unitary optical potential [18, 37]. Left and right panel shows the calculated results at energies of kaonic atom $1s$ and $2p$ states formation, respectively. The upper and middle panels show the results with the reduced imaginary potential $\mathrm{Im}{V}_{\mathrm{opt}}/8$ and $\mathrm{Im}{V}_{\mathrm{opt}}/4$, whereas the lower panels show those with the full chiral unitary potential. The solid lines show the results obtained by the Green function method for the sum of the subcomponents $[(s){}_K\otimes (1p_{3/2}){}_p^{-1}]$ and $[(p){}_K\otimes (1p_{3/2}){}_p^{-1}]$. The dashed lines show those by the effective number approach for the sum of subcomponents $[(1s_{\mathrm{atom}}){}_K\otimes (1p_{3/2}){}_p^{-1}]$ and $[(2p_{\mathrm{atom}}){}_K\otimes (1p_{3/2}){}_p^{-1}]$. The vertical dashed lines indicate the binding energies of kaonic atom $1s$ state (left panels) and $2p$ state (right panels).

Figure 5

Kaonic atom formation cross sections in ${}^{12}\mathrm{C}$($K^{-},p$) reactions at $T_{K^{-}}=600$ MeV plotted as functions of the emitted proton energy $T_p$ at ${\theta }_p^{\mathrm{lab}}=0$ (deg.) for the chiral unitary optical potential [18, 37]. Solid line indicates the result of the Green function method and dashed line that of the effective number approach.

Figure 6

Calculated spectra of the ${}^{12}\mathrm{C}$($K^{-},p$) reaction at $T_{K^{-}}=600\hspace{1em}\mathrm{MeV}$ plotted as functions of the emitted proton energy $T_p$ at ${\theta }_p^{\mathrm{lab}}=0$ (deg.). The energy-independent phenomenological potential evaluated at $E=0$ in Eq. (14) is used. Each subcomponent is indicated in the figure. The vertical dashed line indicates the kaon production threshold.

Figure 7

Calculated spectra of the ${}^{12}\mathrm{C}$($K^{-},p$) (upper panels) and the ${}^{16}\mathrm{O}$($K^{-},p$) (lower panels) reactions at $T_{K^{-}}=600$ MeV plotted as functions of the emitted proton energy $T_p$ at ${\theta }_p^{\mathrm{lab}}=0$ (deg.) for the chiral unitary optical potential (left panels) and the phenomenological optical potential (right panels). Solid lines show the results with the energy-dependent potentials and dashed lines those with the energy-independent potentials fixed at $E=0$ MeV. The solid arrows indicate the calculated binding energies of the kaonic nuclear states coupled to the ground states of the final state nucleus, namely $(1p_{3/2}){}_p^{-1}$ state for C target case and $(1p_{1/2}){}_p^{-1}$ state for O target case, respectively. The binding energies are calculated with the energy-dependent potential. The dashed arrows in lower panels indicate the binding energies of the kaonic nuclear states suggested by the experimental group [1, 2]. The vertical dashed line indicates the kaon production threshold.

Figure 8

Calculated spectra of the ${}^{12}\mathrm{C}$($K^{-},p$) reaction at $T_{K^{-}}=600$ MeV plotted as functions of the emitted proton energy $T_p$ at ${\theta }_p^{\mathrm{lab}}=0$ (deg.) for the energy-dependent phenomenological optical potential with the different branching ratio of the decay channels defined in Eq. (20) (see text in detail). Each subcomponent is indicated in the figure. The vertical dashed line indicates the kaon production threshold.

Figure 9

Calculated spectra of the ${}^{12}\mathrm{C}$($K^{-},p$) and ${}^{16}\mathrm{O}$($K^{-},p$) reactions at $T_{K^{-}}=600$ MeV plotted as functions of the emitted proton energy $T_p$ at ${\theta }_p^{\mathrm{lab}}=0$ (deg.) for the phenomenological optical potential with different energy dependence defined in Eq. (21) (see text in detail). Each subcomponent is indicated in the figure. The vertical dashed lines indicate the kaon production threshold.