Role of the triangle singularity in $\mathrm{\Lambda }\left(1405\right)$ production in the ${\pi }^{-}p\to K^0\pi \mathrm{\Sigma }$ and $pp\to p{K}^{+}\pi \mathrm{\Sigma }$ processes

We have investigated the cross section for the ${\pi }^{-}p\to K^0\pi \mathrm{\Sigma }$ and $pp\to p{K}^{+}\pi \mathrm{\Sigma }$ reactions, paying attention to a mechanism that develops a triangle singularity. The triangle diagram is realized by the decay of a $N^{*}$ to $K^{*}\mathrm{\Sigma }$ and the $K^{*}$ decay into $\pi K$, and the $\pi \mathrm{\Sigma }$ finally merges into $\mathrm{\Lambda }\left(1405\right)$. The mechanism is expected to produce a peak around $2140\mathrm{MeV}$ in the $K\mathrm{\Lambda }\left(1405\right)$ invariant mass. We found that a clear peak appears around $2100\mathrm{MeV}$ in the $K\mathrm{\Lambda }\left(1405\right)$ invariant mass, which is about $40\mathrm{MeV}$ lower than the expectation, and that is due to the resonance peak of a $N^{*}$ resonance which plays a crucial role in the $K^{*}\mathrm{\Sigma }$ production. The mechanism studied produces the peak of the $\mathrm{\Lambda }\left(1405\right)$ around or below 1400 MeV, as is seen in the $pp\to p{K}^{+}\pi \mathrm{\Sigma }$ HADES experiment.

Figure 1

Triangle diagram for $\mathrm{\Lambda }\left(1405\right)$ production from a $N^{*}$ resonance.

Figure 2

Diagrams for the reaction ${\pi }^{-}p\to K^0\pi \mathrm{\Sigma }$ that contains the triangle mechanism, where $\pi \mathrm{\Sigma }$ can be ${\pi }^{-}{\mathrm{\Sigma }}^{+}$, ${\pi }^0{\mathrm{\Sigma }}^0$, or ${\pi }^{+}{\mathrm{\Sigma }}^{-}$.

Figure 3

Diagram of $ab\to 1R\to 123$.

Figure 4

Diagrams for reaction $pp\to p{K}^{+}\pi \mathrm{\Sigma }$ that contains the triangle mechanism, where $\pi \mathrm{\Sigma }$ can be ${\pi }^{-}{\mathrm{\Sigma }}^{+}$, ${\pi }^0{\mathrm{\Sigma }}^0$, or ${\pi }^{+}{\mathrm{\Sigma }}^{-}$.

Figure 5

Re($t_T$), Im($t_T$), and $|t_T|$ of Eq. (26).

Figure 6

The $d\sigma /d{m}_{\mathrm{inv}}$ mass distribution as a function of $m_{\mathrm{inv}}\left({\pi }^0{\mathrm{\Sigma }}^0\right)$, $m_{\mathrm{inv}}\left({\pi }^{+}{\mathrm{\Sigma }}^{-}\right)$, and $m_{\mathrm{inv}}\left({\pi }^{-}{\mathrm{\Sigma }}^{+}\right)$ for the ${\pi }^{-}p\to K^0\pi \mathrm{\Sigma }$ scattering with several fixed values of $M_{\mathrm{inv}}$.

Figure 7

The $d^2{\sigma }_{p{K}^{+}\pi \mathrm{\Sigma }}/d{M}_{\mathrm{inv}}d{m}_{\mathrm{inv}}$ as a function of $m_{\mathrm{inv}}\left({\pi }^0{\mathrm{\Sigma }}^0\right)$, $m_{\mathrm{inv}}\left({\pi }^{+}{\mathrm{\Sigma }}^{-}\right)$, and $m_{\mathrm{inv}}\left({\pi }^{-}{\mathrm{\Sigma }}^{+}\right)$ for the $pp\to p{K}^{+}\pi \mathrm{\Sigma }$ scattering with several fixed values of $M_{\mathrm{inv}}$ and $\sqrt{s}=3179\mathrm{MeV}$.

Figure 8

Cross section of ${\pi }^{-}p\to K^0\pi \mathrm{\Sigma }$ process ${\sigma }_{K^0\pi \mathrm{\Sigma }}$ as a function of $M_{\mathrm{inv}}$ for ${\pi }^{-}p\to K^0\pi \mathrm{\Sigma }$ scattering. The red solid line corresponds to ${\pi }^0{\mathrm{\Sigma }}^0$, the black dash line corresponds to ${\pi }^{+}{\mathrm{\Sigma }}^{-}$, and the blue dash-dotted line corresponds to ${\pi }^{-}{\mathrm{\Sigma }}^{+}$.

Figure 9

$d{\sigma }_{p{K}^{+}\pi \mathrm{\Sigma }}/d{M}_{\mathrm{inv}}$ as a function of $M_{\mathrm{inv}}$ for $pp\to p{K}^{+}\pi \mathrm{\Sigma }$ scattering with fixed value of $\sqrt{s}=3179\mathrm{MeV}$. The red solid line corresponds to ${\pi }^0{\mathrm{\Sigma }}^0$, the black dash line corresponds to ${\pi }^{+}{\mathrm{\Sigma }}^{-}$, and the blue dash-dotted line corresponds to ${\pi }^{-}{\mathrm{\Sigma }}^{+}$.

Figure 10

Tree-level diagram corresponding to Fig. 4 with the fixed $\pi \mathrm{\Sigma }$ being ${\pi }^{-}{\mathrm{\Sigma }}^{+}$.

Figure 11

Triangle mechanism with $\varphi n{\overline{K}}^0$ intermediate state.

Figure 12

Source of $K^{*}\mathrm{\Sigma }\to K^{*}\mathrm{\Sigma }$ interaction through vector exchange.

Figure 13

Representation of Fig. 12 in terms of the resonance generated by the mechanism of Fig. 12.

Figure 14

Effective $R\to K^{*}\mathrm{\Sigma }$ vertex.

Figure 15

Effective mechanism for $R\to \pi N$.

Figure 16

Mechanism for $\pi N\to K^{*}\mathrm{\Sigma }$ combining the vertices of Figs. 13 and 15.