Triangle mechanisms in the build up and decay of the $N^{*}\left(1875\right)$

We studied the $N^{*}\left(1875\right)(3/2^{-})$ resonance with a multichannel unitary scheme, considering the $\mathrm{\Delta }\pi$ and ${\mathrm{\Sigma }}^{*}K$, with their interaction extracted from chiral Lagrangians, and then added two more channels, the $N^{*}\left(1535\right)\pi$ and $N\sigma$, which proceed via triangle diagrams involving the ${\mathrm{\Sigma }}^{*}K$ and $\mathrm{\Delta }\pi$ respectively in the intermediate states. The triangle diagram in the $N^{*}\left(1535\right)\pi$ case develops a singularity at the same energy as the resonance mass. We determined the couplings of the resonance to the different channels and the partial decay widths. We found a very large decay width to ${\mathrm{\Sigma }}^{*}K$, and also observed that, due to interference with other terms, the $N\sigma$ channel has an important role in the $\pi \pi$ mass distributions at low invariant masses, leading to an apparently large $N\sigma$ decay width. We discuss justifying the convenience of an experimental reanalysis of this resonance, in light of the findings of the paper, using multichannel unitary schemes.

Figure 1

The triangle diagram for the $i(\mathrm{\Delta }\pi ,{\mathrm{\Sigma }}^{*}K)\to N^{*}\left(1535\right)\pi$ transition. The momenta of the lines are in parentheses.

Figure 2

Different diagrams leading to $\mathrm{\Delta }\pi \to N^{*}\pi$ ($I=\frac{1}{2}$, $I_3=\frac{1}{2}$).

Figure 3

Different diagrams leading to ${\mathrm{\Sigma }}^{*}K\to N^{*}\pi$ ($I=\frac{1}{2}$, $I_3=\frac{1}{2}$).

Figure 4

Transition diagram from $\mathrm{\Delta }\pi \left({\mathrm{\Sigma }}^{*}K\right)\to \mathrm{\Delta }\pi \left({\mathrm{\Sigma }}^{*}K\right)$ via the intermediate $N^{*}\pi$.

Figure 5

Triangle diagram for the transition of $i(\mathrm{\Delta }\pi ,{\mathrm{\Sigma }}^{*}K)$ to $N\sigma$.

Figure 6

Diagrams for the $|\mathrm{\Delta }\pi ,I=\frac{1}{2},I_3=\frac{1}{2}\rangle$ transition to $p\sigma$.

Figure 7

Diagrams for the $|{\mathrm{\Sigma }}^{*}K,I=\frac{1}{2},I_3=\frac{1}{2}\rangle$ transition to $p\sigma$.

Figure 8

$\mathrm{Re}\left(t_T\right)$, $\mathrm{Im}\left(t_T\right)$ for Figs. 2 and 3 as a function of $\sqrt{s}$ for different values of $M_{N^{*}}$.

Figure 9

Real and imaginary parts of $t_T$ of the triangle diagram, Eq. (4), with ${\mathrm{\Sigma }}^{*}K$ intermediate state.

Figure 10

${\tilde{V}}_{i,\pi N^{*}}/V_{i,{\mathrm{\Sigma }}^{*}K}^{(I=1/2)}$ of Eq. (17).

Figure 11

Real and imaginary parts of $t_T^{\prime }$ corresponding to the triangle diagram of Fig. 5.

Figure 12

${\tilde{V}}_{i,\sigma N}/V_{i,\mathrm{\Delta }\pi }^{(I=1/2)}$ as a function of the energy corresponding to Eq. (B9).

Figure 13

$|T_{ii}{|}^2$ for the $\mathrm{\Delta }\pi$ and ${\mathrm{\Sigma }}^{*}K$ channels alone as a function of the energy. The wider curve corresponds to considering the width of $\mathrm{\Delta }$ and ${\mathrm{\Sigma }}^{*}$.

Figure 14

$|T_{22}{|}^2$ with two channels ($\mathrm{\Delta }\pi ,{\mathrm{\Sigma }}^{*}K$), three channels ($\mathrm{\Delta }\pi ,{\mathrm{\Sigma }}^{*}K,N^{*}\pi$), and four channels ($\mathrm{\Delta }\pi ,{\mathrm{\Sigma }}^{*}K,N^{*}\pi ,N\sigma$).

Figure 15

$|T_{12}{|}^2$ as a function of the energy, for the $\mathrm{\Delta }\pi \to {\mathrm{\Sigma }}^{*}K$ transition.

Figure 16

$|T_{32}{|}^2$ as a function of energy for the ${\mathrm{\Sigma }}^{*}K\to N^{*}\pi$ transition.

Figure 17

$|T_{42}{|}^2$ as a function of energy for the ${\mathrm{\Sigma }}^{*}K\to N\sigma$ transition.

Figure 18

Diagrams for the decay of the $N^{*}\left(1875\right)$ resonance into $\pi \mathrm{\Delta }$ [(a) and (b)], $\pi N^{*}$ (c), and $\sigma p$ (d).

Figure 19

The $\frac{d\mathrm{\Gamma }}{d{M}_{12}}$, $\frac{d\mathrm{\Gamma }}{d{M}_{13}}$ and $\frac{d\mathrm{\Gamma }}{d{M}_{23}}$ mass distributions for the $N^{*+}\left(1875\right)\to {\pi }^{+}{\pi }^{-}p$ decay with 1,2,3 denoting ${\pi }^{+}$, ${\pi }^{-}$, and $p$, with the red dashed lines indicating the case considering only $\pi \mathrm{\Delta }$ production [i.e., $A$ and $B$ terms in Eq. (30)], the blue dash-dotted lines the case considering $\pi \mathrm{\Delta }$ and $\pi N^{*}$ production [i.e., $A$, $B$, and $C$ terms in Eq. (30)], the black solid lines the case considering $\pi \mathrm{\Delta }$, $\pi N^{*}$, and $\sigma N$ production [i.e., all the $A$, $B$, $C$, and $D$ terms in Eq. (30)], and the green dotted lines the case of changing the sign of $g_{\sigma ,\pi \pi }$ in Eq. (B7). The purple dash-dot-dotted line in the $\frac{d\mathrm{\Gamma }}{d{M}_{12}}$ distribution corresponds to the case considering only $\sigma N$ production [i.e., the $D$ term in Eq. (30)] fitted approximately to the $M_{12}$ distribution at low masses.