Repulsion and absorption of the $\mathrm{\Sigma }$-nucleus potential for ${\mathrm{\Sigma }}^{-}\text{−}{}^5\mathrm{He}$ in the ${}^6\mathrm{Li}({\pi }^{-},K^{+})$ reaction

We study phenomenologically inclusive spectra of the ${}^6\mathrm{Li}({\pi }^{-},K^{+})$ reaction at 1.2 GeV/$c$ within a distorted-wave impulse approximation with the optimal Fermi-averaging ${\pi }^{-}p\to K^{+}{\mathrm{\Sigma }}^{-}t$ matrix. We attempt to clarify the property of a $\mathrm{\Sigma }$-nucleus potential for ${\mathrm{\Sigma }}^{-}\text{−}{}^5\mathrm{He}$ by comparing the calculated spectra with the data of the J-PARC E10 experiment. The result shows that the repulsive and absorptive components of the ${\mathrm{\Sigma }}^{-}\text{−}{}^5\mathrm{He}$ potential provide the ability to explain the data of the continuum spectra in $\mathrm{\Sigma }$ and $\mathrm{\Lambda }$ regions; the strengths of $V_{\mathrm{\Sigma }}=+30\pm 10$ MeV and $W_{\mathrm{\Sigma }}=-26\pm 2$ MeV are favored within the Woods-Saxon potential, consistent with analyses for heavier nuclei. Effects of the size and potential range for ${\mathrm{\Sigma }}^{-}\text{−}{}^5\mathrm{He}$ in the neutron excess of $(N-Z)/(N+Z)=$ 0.2 are also discussed.

Figure 1

Diagrams for double-charge exchange (${\pi }^{-},K^{+}$) reactions on nuclear targets, leading to production of (a) $\mathrm{\Sigma }$ hypernuclear states by ${\pi }^{-}p\to K^{+}{\mathrm{\Sigma }}^{-}$ processes and (b) $\mathrm{\Lambda }$ hypernuclear states caused by the ${\mathrm{\Sigma }}^{-}p\leftrightarrow \mathrm{\Lambda }n$ coupling.

Figure 2

Calculated angular distributions of the differential cross section for the ${\pi }^{-}p\to K^{+}{\mathrm{\Sigma }}^{-}$ reaction in the c.m. frame at $E_{\mathrm{c}.\mathrm{m}.}=1740$, 1764, 1793, 1818, 1844, 1875, 1930, 1978, and 2025 MeV, together with the data [2, 32].

Figure 3

Energy dependence of the optimal cross section ${(d\sigma /d\mathrm{\Omega })}^{\mathrm{opt}}$ for the ${\pi }^{-}p\to K^{+}{\mathrm{\Sigma }}^{-}$ reaction on the ${}^6\mathrm{Li}$ target at $p_{{\pi }^{-}}=1.2$ GeV/$c$ and ${\theta }_{\mathrm{lab}}=3^{\circ }$, $5^{\circ }$, $7^{\circ }$, $9^{\circ }$, ${11}^{\circ }$, and ${13}^{\circ }$, as a function of $E_{{\mathrm{\Sigma }}^{-}}$. The arrow shows the ${}^5\mathrm{He}+{\mathrm{\Sigma }}^{-}$ threshold.

Figure 4

Radial distributions of the form factors with Gaussian ranges of 0.0, 0.80, and 1.20 fm for the ${}^5\mathrm{He}$(3/$2_{\mathrm{g}.\mathrm{s}.}^{-}$) nucleus, as a function of the radial distance. Solid curves and square symbols denote the distributions with the modified harmonic oscillator (MHO) and the Woods-Saxon (WS) models, respectively.

Figure 5

(a) Real and (b) imaginary parts of the $\mathrm{\Sigma }$-nucleus potentials $U_{\mathrm{\Sigma }}$ for ${\mathrm{\Sigma }}^{-}\text{−}{}^5\mathrm{He}$. Solid curves denote the WS potential which has the strength of $V_{\mathrm{\Sigma }}=+60,+30$, and $+10$ MeV and $W_{\mathrm{\Sigma }}=-13$, $-26$, and $-39$ MeV with $R=r_0{A}^{1/3}=1.43$ fm where $r_0=0.835$ fm and $a=0.706$ fm.

Figure 6

Contour plots of the ${\chi }^2$-value distribution in the {$V_{\mathrm{\Sigma }},-W_{\mathrm{\Sigma }}$} plane from fitting to the average cross section of ${\overline{\sigma }}_{2^{\circ }\text{−}{14}^{\circ }}$ in the ${}^6\mathrm{Li}$(${\pi }^{-},K^{+}$) reaction at $p_{{\pi }^{-}}=1.2$ GeV/$c$. A solid circle denotes the minimum position of ${\chi }_{\min }^2=84.5$ at $(V_{\mathrm{\Sigma }},-W_{\mathrm{\Sigma }})=$ (30 MeV, 26 MeV) with $f_s=1.23$. Thick curves indicate $\mathrm{\Delta }{\chi }^2=$ 2.30, 4.61, and 9.21 which correspond to 68%, 90%, and 99% confidence levels for two parameters, respectively.

Figure 7

Comparison of the calculated spectra for ${\overline{\sigma }}_{2^{\circ }\text{−}{14}^{\circ }}$ with the data of the ${}^6\mathrm{Li}$(${\pi }^{-},K^{+}$) reaction at $p_{{\pi }^{-}}=1.2$ GeV/$c$ [2]. Solid curves denote the spectrum for the WS potential with ($V_{\mathrm{\Sigma }},W_{\mathrm{\Sigma }}$) listed in Table 1, together with the value of ${\chi }^2/N$ and the renormalization factor $f_s$. The spectra are folded with a detector resolution of 3 MeV FWHM.

Figure 8

Calculated spectrum for ${\overline{\sigma }}_{2^{\circ }\text{−}{14}^{\circ }}$ in the WS potential with $V_{\mathrm{\Sigma }}=+30$ MeV, $W_{\mathrm{\Sigma }}=-26$ MeV, $r_0=$ 0.835 fm, and $a=$ 0.706 fm, together with the data of the ${}^6\mathrm{Li}$(${\pi }^{-},K^{+}$) reaction at $p_{{\pi }^{-}}=1.2$ GeV/$c$ [2]. Solid, dot-dashed, and dashed curves denote total, $s$-hole, and $p$-hole contributions, respectively. The spectra are folded with a detector resolution of 3 MeV FWHM.

Figure 9

Shapes and magnitudes of the calculated spectra for ${\overline{\sigma }}_{2^{\circ }\text{−}{14}^{\circ }}$ in the ${}^6\mathrm{Li}$(${\pi }^{-},K^{+}$) reaction at $p_{{\pi }^{-}}=1.2$ GeV/$c$, depending on (a) the strengths of $V_{\mathrm{\Sigma }}$ when $W_{\mathrm{\Sigma }}=-26$ MeV is chosen and (b) the strengths of $W_{\mathrm{\Sigma }}$ when $V_{\mathrm{\Sigma }}=+30$ MeV is chosen. Solid curves denote the spectrum for the WS potential with $(V_{\mathrm{\Sigma }},W_{\mathrm{\Sigma }})=$ ($+30$ MeV, $-26$ MeV) as a guide. The spectra are folded with a detector resolution of 3 MeV FWHM.

Figure 10

Comparison among the shapes of the calculated spectra for ${\overline{\sigma }}_{2^{\circ }\text{−}{14}^{\circ }}$ in the ${}^6\mathrm{Li}$(${\pi }^{-},K^{+}$) reaction at $p_{{\pi }^{-}}=1.2$ GeV/$c$, together with the data [2]. The solid curve denotes the spectrum with $(V_{\mathrm{\Sigma }},W_{\mathrm{\Sigma }})=$ ($+30$ MeV, $-26$ MeV) and $f_s=$ 1.23 as a guide. All the spectra for ($V_{\mathrm{\Sigma }},W_{\mathrm{\Sigma }}$) are renormalized at the point of the arrow corresponding to the ${\mathrm{\Sigma }}^0$ threshold ($M_x=$ 5882.4 MeV/$c^2$) to be compared among the shapes of them. The spectra are folded with a detector resolution of 3 MeV FWHM.

Figure 11

Calculated spectra of the ${}^6\mathrm{Li}$(${\pi }^{-},K^{+}$) reaction at $p_{{\pi }^{-}}=1.2$ GeV/$c$ and $K^{+}$ forward-direction angles of (a) ${\theta }_{\mathrm{lab}}=$ $3^{\circ }$, (b) $5^{\circ }$, (c) $7^{\circ }$, (d) $9^{\circ }$, (e) ${11}^{\circ }$, and (f) ${13}^{\circ }$, as a function of the missing mass $M_{\mathrm{x}}$. The strengths of $(V_{\mathrm{\Sigma }},W_{\mathrm{\Sigma }})=$ ($+30$ MeV, $-26$ MeV) are used in the WS potential with $r_0=$ 0.835 fm and $a=$ 0.706 fm. Solid, dot-dashed, and dashed curves denote total, $s$-hole, and $p$-hole contributions, respectively, where the calculated spectra are normalized by a common factor $f_s=$ 1.19. The spectra are folded with a detector resolution of 3 MeV FWHM. The data are taken from Ref. [2].