Role of a triangle singularity in the $\pi \mathrm{\Delta }$ decay of $N\left(1700\right)(3/2^{-})$

We show the important role played by the $\pi \mathrm{\Delta }\left(1232\right)$ channel in the build up of the $N\left(1700\right)(3/2^{-})$ resonance due to the nontrivial enhancement produced by a singularity of a triangular loop. The $N\left(1700\right)$ is one of the dynamically generated resonances produced by the coupled-channel vector-baryon interaction. The $\pi \mathrm{\Delta }$ channel was neglected in previous works but we show that it has to be incorporated into the coupled-channel formalism due to an enhancement produced by a singularity in the triangular loop with $\rho$, nucleon, and $\pi$ as internal loop lines and $\pi$ and $\mathrm{\Delta }$ as external ones. The enhancement is of nonresonant origin but it contributes to the dynamical generation of the $N\left(1700\right)$ resonance due to the nonlinear dynamics involved in the coupled-channel mechanisms. We obtain an important increase of the total width of the $N\left(1700\right)$ resonance when the $\pi \mathrm{\Delta }$ channel is included and provide predictions for the partial widths of the $N\left(1700\right)$ decays into $VB$ and $\pi \mathrm{\Delta }$.

Figure 1

Unitarization of the vector-baryon interaction.

Figure 2

Diagonal vector-baryon amplitudes without the $\pi \mathrm{\Delta }$ channel.

Figure 3

Triangular loop for the $\pi \mathrm{\Delta }$ channel.

Figure 4

Functions $t_T^{\left(1\right)}$ and $t_T^{\left(2\right)}$ of the triangle diagram.

Figure 5

Transition potentials for $\rho N$.

Figure 6

Full amplitudes with and without the $\pi \mathrm{\Delta }$ channel.