Genuine quark state versus dynamically generated structure for the Roper resonance

In view of the recent results of lattice QCD simulation in the $P11$ partial wave that has found no clear signal for the three-quark Roper state we investigate a different mechanism for the formation of the Roper resonance in a coupled channel approach including the $\pi N$, $\pi \mathrm{\Delta }$, and $\sigma N$ channels. We fix the pion-baryon vertices in the underlying quark model while the $s$-wave sigma-baryon interaction is introduced phenomenologically with the coupling strength, the mass, and the width of the $\sigma$ meson as free parameters. The Laurent-Pietarinen expansion is used to extract the information about the $S$-matrix pole. The Lippmann-Schwinger equation for the $K$ matrix with a separable kernel is solved to all orders. For sufficiently strong $\sigma NN$ coupling the kernel becomes singular and a quasibound state emerges at around 1.4 GeV, dominated by the $\sigma N$ component and reflecting itself in a pole of the $S$ matrix. The alternative mechanism involving a ${\left(1s\right)}^{2}2s$ quark resonant state is added to the model and the interplay of the dynamically generated state and the three-quark resonant state is studied. It turns out that for the mass of the three-quark resonant state above 1.6 GeV the mass of the resonance is determined solely by the dynamically generated state, nonetheless, the inclusion of the three-quark resonant state is imperative to reproduce the experimental width and the modulus of the resonance pole.

Figure 1

Graphical representation of Eqs. (10, 11, 12) (from top to bottom).

Figure 2

Graphical representation of $M_{\alpha i,\gamma j}^{\beta }$: $M_{\alpha }$ and $B_{\alpha }$ denote respectively the meson ($\pi$ or $\sigma$) and baryon ($N$ or $\mathrm{\Delta }$) in channel $\alpha$.

Figure 3

The behavior of the lowest eigenvalue $w_{\min }$ as a function of $W$ for ${\mu }_{\sigma }={\mathrm{\Gamma }}_{\sigma }=600$ MeV for coupling constant $g$ from 1.55 to 2.3.

Figure 4

The weights of the $\pi N$, $\pi \mathrm{\Delta }$, and $\sigma N$ components of the eigenvector belonging to the lowest eigenvalue $w_{\min }$ for $g=2.3$ (thick lines), $g=2.0$ (medium lines), and $g=1.55$ (thin lines), and ${\mu }_{\sigma }={\mathrm{\Gamma }}_{\sigma }=600$ MeV.

Figure 5

The imaginary part of the $T$ matrix for $g$ from 1.55 to 2.3 and ${\mu }_{\sigma }={\mathrm{\Gamma }}_{\sigma }=600$ MeV.

Figure 6

The lowest eigenvalue of $\mathsf{A}$ as a function of $W$ for $g=2.0$, ${\mu }_{\sigma }={\mathrm{\Gamma }}_{\sigma }=600$ MeV (blue) and ${\mu }_{\sigma }={\mathrm{\Gamma }}_{\sigma }=500$ MeV (red); the thick solid lines correspond to the full calculation, medium lines to the system without $\mathrm{\Delta }$, and thin lines to the $\sigma N$ system alone.

Figure 7

The imaginary part of the $T_{\pi N\pi N}$ amplitude; the nonpole contribution is shown separately as well as the contribution from the lowest state of the $\mathsf{A}$ matrix (dashed lines) for $g=2.00$ and $g=1.95$.

Figure 8

${\lambda }_R\left(W\right)$ for masses of the $3q$ resonant state $m_R$ from 1480 to 2200 MeV and $g=1.55$, ${\mu }_{\sigma }={\mathrm{\Gamma }}_{\sigma }=600$ MeV.

Figure 9

Evolution of the Roper pole as the interaction strength is gradually switched on for two bare masses of the three-quark configuration, 1750 and 2000 MeV.

Figure 10

$\mathrm{Im}T\left(W\right)$ for ${\mu }_{\sigma }=600$ MeV when the $3q$ resonant state with $m_R=2000$ MeV is turned off (left) and on (right).