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Studies on quark-mass dependence of the $N^*(920)$ pole from $πN$ $χ$PT amplitudes

Xu Wang, Kai-Ge Kang, Qu-Zhi Li, Zhiguang Xiao, Han-Qing Zheng

Abstract

The quark-mass dependence of the $N^*(920)$ pole is analyzed using $K$-matrix method, with the $πN$ scattering amplitude calculated up to $O(p^3)$ order in chiral perturbation theory. As the quark mass increases, the $N^*(920)$ pole gradually approaches the real axis in the complex $w$-plane (where $w=\sqrt{s}$). Eventually, in the $O(p^2)$ case, it crosses the $u$-cut on the real axis and enters the adjacent Riemann sheet when the pion mass reaches $526~{\rm MeV}$. At order $O(p^3)$, the rate at which it approaches the real axis slows down; however, we argue that it will ultimately cross the $u$-cut and enter the adjacent Riemann sheet as well. Additionally, the trajectory of the \(N^*(920)\) pole is in qualitative agreement with the results from the linear $σ$ model calculation.

Studies on quark-mass dependence of the $N^*(920)$ pole from $πN$ $χ$PT amplitudes

Abstract

The quark-mass dependence of the pole is analyzed using -matrix method, with the scattering amplitude calculated up to order in chiral perturbation theory. As the quark mass increases, the pole gradually approaches the real axis in the complex -plane (where ). Eventually, in the case, it crosses the -cut on the real axis and enters the adjacent Riemann sheet when the pion mass reaches . At order , the rate at which it approaches the real axis slows down; however, we argue that it will ultimately cross the -cut and enter the adjacent Riemann sheet as well. Additionally, the trajectory of the \(N^*(920)\) pole is in qualitative agreement with the results from the linear model calculation.

Paper Structure

This paper contains 4 sections, 22 equations, 5 figures.

Table of Contents

  1. Introduction
  2. A brief introduction to B$\chi$PT and PWAs of $\pi N$ scatterings
  3. Analysis of the $N^*(920)$ Pole Trajectory and Its Quark Mass Dependence
  4. Summary

Figures (5)

  • Figure 1: Cuts in $\pi N$ PWAs, represented by the bold lines. $s_L=(m_N-m_\pi)^2,c_L=(m_N^2-m_\pi^2)^2/m_N^2,c_R=m_N^2+2m_\pi^2,s_R=(m_N+m_\pi)^2$
  • Figure 2: Dependencies of the nucleon mass $m_N$, axial-vector coupling $g_A$, and pion decay constant $F_{\pi}$ on the pion mass $m_{\pi}$.
  • Figure 3: Variation of the $N^*(920)$ pole position with the pion mass in the $\mathcal{K}^{(2)}$ and $\mathcal{K}^{(3)}$ amplitudes. The units for the pole positions are in GeV. The results obtained in this work are shown in red ($O(p^2)$ tree-level) and green ($O(p^3)$ one-loop), corresponding to pion masses in the range $m_\pi=0.1396$–$0.60\ \text{GeV}$. The results from Ref. Li:2025fvg are displayed in orange (Tree–Li) and blue (Loop–Li), covering the range $m_\pi=0.138$–$0.360\ \text{GeV}$.
  • Figure 4: $m_\pi$ dependence of $N^*(920)$ pole from the full $O(p^3)$ amplitude including loop corrections, using parameters from Eq. (\ref{['second']}).
  • Figure 5: The dependence of the $N^*(920)$ pole position on pion mass, as determined from the $\mathcal{K}^{(2)}$ and $\mathcal{K}^{(3)}$ amplitudes (with the dependencies of $m_N$, $g_A$, and $F_\pi$ on $m_\pi$ taken from lattice–data–based fits). The unit is ${\rm GeV}$. The $\mathcal{K}^{(2)}$ results are indicated by red triangles, while the $\mathcal{K}^{(3)}$ results are shown as green circles. The pion mass $m_\pi$ varies from 0.1396 to 0.44 GeV.