On the pole trajectory of the subthreshold negative parity nucleon with varying pion masses

TL;DR

The paper investigates how the subthreshold $N^*(920)$ pole evolves as the pion mass $m_\pi$ is varied, using a renormalizable linear $\sigma$ model with nucleons and an $N/D$ unitarization framework. It employs a [1,1] Padé approach to the $\pi\pi$ sector to trace the $\sigma$ pole and a separate $N/D$ treatment for the $\pi N$ sector to locate the $N^*(920)$ pole, with $m_\pi$-dependent inputs anchored by chiral perturbation theory constraints. The results show that the $\sigma$ pole trajectory is in line with Roy-Steiner analyses, while the $N^*(920)$ trajectory is novel: at tree level the pole moves toward and crosses the $u$-channel cut into an adjacent Riemann sheet as $m_\pi$ increases, but at one-loop it remains complex up to $m_\pi=0.36$ GeV, highlighting the rich analytic structure of $\pi N$ amplitudes. These findings provide benchmarks for lattice QCD and point to future studies including finite temperature and chemical potential effects and potential parity-doublet implications.

Abstract

We study the pole trajectory of the recently established subthreshold negative parity nucleon pole, namely the $N^*(920)$, with varying pion masses, in the scheme of linear $σ$ model with nucleons using the $N/D$ unitarization method. We find that as the pion mass increases, the pole moves toward the real axis. For larger pion masses, at tree level, the pole falls to a specific point on $u$-channel cut and crosses to the adjacent Riemann sheet defined by the logarithmic $u$ channel cut. At one-loop level, the pole does not meet the $u$-cut up to $m_π=0.36$GeV. We also re-examined the $σ$ pole trajectory and find it in good agreement with Roy equation analysis result.

Figures (6)

• Figure 1: The tree-level Feynman diagrams contributing to $\pi\pi$ scatterings.
• Figure 2: The trajectory of $\sigma$ resonance with $m_\pi$ variation. The vertical dashed line denotes the physical threshold. Right: the contribution of $\sigma$ self-energy correction to $\pi \pi$ amplitude.
• Figure 3: The tree-level Feynman diagrams contributing to $\pi N$ scatterings.
• Figure 4: The trajectories of $N^*(920)$ as $m_\pi$ ranges from the physical value to $0.27\mathrm{GeV}$ for both cases of fixed $m_\sigma$ and fixed $\lambda$ at tree level. The points with $\operatorname{Im}[W]>0$ are the zeros (RSI) of $S$ matrix for different pion masses and those with $\operatorname{Im}[W]<0$ are the zeros of the $S_+$ function (see the text). The vertical dashed lines correspond to roots obtained by solving equation \ref{['eq:Imt-zero']} for $m_\pi = 0.236\mathrm{GeV}$ and $0.253\mathrm{GeV}$, respectively.
• Figure 5: The $S$ matrix values for $\pi\pi$ scatterings between the left hand cut and the threshold using the results of the previous section. Initially, no $S$-matrix zeros exist in this region (upper-left subfigure). As $m_\pi$ increases, a virtual-state zero (VSIII) emerges from the left-hand cut (upper-right). The sigma resonance turns into virtual-state zeros when the $S$ matrix's local minimum contacts the real axis, generating a second-order zero precisely at this point. Subsequently, the two virtual-state zeros (VSI and VSII) split apart along the real axis (lower-left). VSI later becomes a bound-state pole of the $S$-matrix, leaving two virtual-state zeros (VSII and VSIII). These two zeros then coalesce on the real axis and move into the complex plane as resonance zeros (lower-right).