Structure of the $\mathrm{\Lambda }\left(1405\right)$ and the $K^{-}d\to \pi \mathrm{\Sigma }n$ reaction

The $\mathrm{\Lambda }\left(1405\right)$ resonance production reaction is investigated within the framework of the coupled-channels Alt-Grassberger-Sandhas (AGS) equations. We perform full three-body calculations for the $\overline{K}NN\text{−}\pi YN$ amplitudes on the physical real energy axis and investigate how the signature of the $\mathrm{\Lambda }\left(1405\right)$ appears in the cross sections of the $K^{-}d\to \pi \mathrm{\Sigma }n$ reactions, also in view of the planned E31 experiment at J-PARC. Two types of meson-baryon interaction models are considered: an energy-dependent interaction based on chiral $SU\left(3\right)$ effective field theory, and an energy-independent version that has been used repeatedly in phenomenological approaches. These two models have different off-shell properties that imply correspondingly different behavior in the three-body system. We investigate how these features show up in differential cross sections of $K^{-}d\to \pi \mathrm{\Sigma }n$ reactions. Characteristic patterns distinguishing between the two models are found in the invariant mass spectrum of the final $\pi \mathrm{\Sigma }$ state. The $K^{-}d\to \pi \mathrm{\Sigma }n$ reaction, with different (${\pi }^{\pm }{\mathrm{\Sigma }}^{\mp }$ and ${\pi }^0{\mathrm{\Sigma }}^0$) charge combinations in the final state, is thus demonstrated to be a useful tool for investigating the subthreshold behavior of the $\overline{K}N$ interaction.

Figure 1

(a) One particle exchange interaction $Z_{\alpha \beta ,L}^{\left(I\right)\left(I^{\prime }\right)}(p_i,p_j,W)$. (b) Isobar propagator ${\tau }_{\alpha \beta }^{\left(I\right)}(W-E_i\left({\mathit{p}}_i\right),{\mathit{p}}_i)$.

Figure 2

Results of the fit with the E-dep. model (solid lines) and E-indep. model (dashed lines). Total cross sections of (a) $K^{-}p\to K^{-}p$, (b) $K^{-}p\to {\pi }^{+}{\mathrm{\Sigma }}^{-}$, (c) $K^{-}p\to {\pi }^{-}{\mathrm{\Sigma }}^{+}$, (d) $K^{-}p\to {\pi }^0{\mathrm{\Sigma }}^0$, and (e) $K^{-}p\to {\pi }^0\mathrm{\Lambda }$. Data are from Refs. [23, 24, 25, 26, 27].

Figure 3

(a) Real and (b) imaginary parts of the $\overline{K}N$ amplitude in the isospin $I=0$ channel as functions of the total $\overline{K}N$ center-of-mass energy. Solid curve: E-dep. model based on the chiral SU(3) potential (17); dashed curve: E-indep. model using the potential (18). The shaded area show for comparison the $I=0\overline{K}N$ amplitude from NLO chiral SU(3) dynamics including uncertainties as described in Ref. [36].

Figure 4

Results of the fit with the E-dep. model. Total cross sections of (a) $K^{-}p\to K^{-}p$, (b) $K^{-}p\to {\pi }^{+}{\mathrm{\Sigma }}^{-}$, (c) $K^{-}p\to {\pi }^{-}{\mathrm{\Sigma }}^{+}$, (d) $K^{-}p\to {\pi }^0{\mathrm{\Sigma }}^0$, and (e) $K^{-}p\to {\pi }^0\mathrm{\Lambda }$. Data are from Refs. [23, 24, 25, 26, 27]. The shaded areas reflect variations of the cutoff ${\mathrm{\Lambda }}_{\alpha }^{\left(I\right)}$ as listed in Table 4. In (c) the dashed curve indicates the best consistent fit to the $K^{-}p\to {\pi }^{-}{\mathrm{\Sigma }}^{+}$ cross section. Its implication for the other cross sections is shown by the dashed curves in subfigures (a), (b), (d), and (e).

Figure 5

Phase shifts of the $NN$ scattering in the ${}^3{S}_1$ channel. The solid line shows the phase shift with our model, and the triangles show the phase shifts with the model of Ref. [58].

Figure 6

Differential cross sections $d\sigma /d{M}_{\pi \mathrm{\Sigma }}$ for $K^{-}d\to \pi \mathrm{\Sigma }n$. The initial kaon momentum is set to $p_{\mathrm{lab}}=1$ GeV. (a) E-dep. model; (b) E-indep. model. Solid curves: ${\pi }^{+}{\mathrm{\Sigma }}^{-}n$; dashed curves: ${\pi }^{-}{\mathrm{\Sigma }}^{+}n$; dotted curves: ${\pi }^0{\mathrm{\Sigma }}^{0}n$ in the final state, respectively.

Figure 7

Contributions from each two-body reaction process to the $K^{-}d\to \pi \mathrm{\Sigma }n$ differential cross section $d\sigma /d{M}_{\pi \mathrm{\Sigma }}$ using the E-dep. model. (a) ${\pi }^{+}{\mathrm{\Sigma }}^{-}n$; (b) ${\pi }^{-}{\mathrm{\Sigma }}^{+}n$; (c) ${\pi }^0{\mathrm{\Sigma }}^{0}n$ in the final state. The solid curve represents the summation of all reaction processes of Eq. (10); dashed curve: $X_{Y_K,d}$ component; dotted curve: $X_{Y_{\pi },d}$ component; dotted-dotted curve: $X_{N^{*},d}$ component; dashed-two-dotted curve: $X_{d_y,d}$ component, respectively. The initial kaon momentum is set to $p_{\mathrm{lab}}=1$ GeV.

Figure 8

Contributions of partial wave components to the differential cross section $d\sigma /d{M}_{\pi \mathrm{\Sigma }}$ for the E-dep. model. (a) ${\pi }^{+}{\mathrm{\Sigma }}^{-}n$; (b) ${\pi }^{-}{\mathrm{\Sigma }}^{+}n$; (c) ${\pi }^0{\mathrm{\Sigma }}^{0}n$ in the final state. The solid curve represents the summation of total orbital angular momentum $L=0$ to 10. The dashed curve represents the $L=0$ component. The dotted curve represents the $L=1$ component. The dashed-dotted curve represents the $L=2$ component. The dashed-two-dotted curve represents the $L=3$ component. The contribution from $L\ge 4$ components which we omit is small. The initial kaon momentum is set to $p_{\mathrm{lab}}=1$ GeV.

Figure 9

Angular dependence of the double differential cross sections $d^2\sigma /d{M}_{\pi \mathrm{\Sigma }}dcos{\theta }_{p_N}$ for the E-dep. model. (a) ${\theta }_{p_N}=0^{\circ }$; (b) ${\theta }_{p_N}={90}^{\circ }$; (c) ${\theta }_{p_N}={180}^{\circ }$, respectively. Here ${\theta }_{p_N}$ is scattering angle of the neutron in the center-of-mass frame. The illustration of curves and the initial kaon momentum are the same as those in Fig. 6.

Figure 10

Angular dependence of the double differential cross sections $d^2\sigma /d{M}_{\pi \mathrm{\Sigma }}dcos{\theta }_{p_N}$ for the E-indep. model. (a) ${\theta }_{p_N}=0^{\circ }$; (b) ${\theta }_{p_N}={90}^{\circ }$; (c) ${\theta }_{p_N}={180}^{\circ }$. The illustration of curves and the initial kaon momentum are the same as those in Fig. 6.

Figure 11

Differential cross sections $d\sigma /d{M}_{\pi \mathrm{\Sigma }}$ for $K^{-}d\to \pi \mathrm{\Sigma }n$. (a) The E-dep. model; (b) the E-indep. model. The illustration of curves and the initial kaon momentum are the same as those in Fig. 6.

Figure 12

Double differential cross sections $d^2\sigma /d{M}_{\pi \mathrm{\Sigma }}dcos{\theta }_{p_N}$ for $K^{-}d\to \pi \mathrm{\Sigma }n$ with the neutron emitted in forward direction, ${\theta }_{p_N}=0^{\circ }$. (a) The E-dep. model; (b) the E-indep. model. The illustration of curves are the same as in Fig. 6. The incident $K^{-}$ momentum is $p_{\mathrm{lab}}=1$ GeV.

Figure 13

Uncertainty bands reflecting cutoff parameter dependence of the differential cross section $d\sigma /d{M}_{\pi \mathrm{\Sigma }}$ for the E-dep. model. (a) ${\pi }^{+}{\mathrm{\Sigma }}^{-}n$; (b) ${\pi }^{-}{\mathrm{\Sigma }}^{+}n$; (c) ${\pi }^0{\mathrm{\Sigma }}^{0}n$ in the final state. The initial kaon momentum is set to $p_{\mathrm{lab}}=1$ GeV. The dashed curves refer to the choice of “optimized” fit to the $K^{-}p\to {\pi }^{-}{\mathrm{\Sigma }}^{+}$ cross section in Fig. 4.

Figure 14

Uncertainty bands reflecting cutoff parameter dependence of the double differential cross section $d^2\sigma /d{M}_{\pi \mathrm{\Sigma }}dcos{\theta }_{p_N}$ for the E-dep. model, with neutrons emitted in forward direction. (a) ${\pi }^{+}{\mathrm{\Sigma }}^{-}n$; (b) ${\pi }^{-}{\mathrm{\Sigma }}^{+}n$; (c) ${\pi }^0{\mathrm{\Sigma }}^{0}n$ in the final state. The initial kaon momentum is set to $p_{\mathrm{lab}}=1$ GeV. The dashed curves refer to the choice of “optimized” fit to the $K^{-}p\to {\pi }^{-}{\mathrm{\Sigma }}^{+}$ cross section in Fig. 4.