Revisiting the pion Regge trajectories

We propose a model-independent ansatz $M={\beta_x}\left(x+c_0\right)^{\nu}+c_1$ ($x=l,\,n_r$) and then use it to fit the orbital and radial pion Regge trajectories without the preset values. It is shown that nonzero $c_1$ is reasonable and acceptable. Nonzero $c_1$ gives an explanation for the nonlinearity of the pion Regge trajectories in the usually employed $(M^2,\,x)$ plane. As $m_R$ or $c_1$ is chosen appropriately, both the orbital and radial pion Regge trajectories are linear in the $((M-m_R)^2,\,x)$ plane whether the $\pi^0$ is included or not on the Regge trajectories. The fitted pion Regge trajectories suggest $0.45\le\nu\le0.5$, which indicates the confining potential $r^a$ with $9/11{\le}a\le1$. Moreover, it is illustrated in the appendix B that $m_R$ can be nonzero for the light nonstrange mesons. We present discussions in the appendix A on the structure of the Regge trajectories plotted in the $(M,\,x)$ plane and on the structure of the Regge trajectories in the $((M-m_R)^2,\,x)$ plane based on the potential models and the string models.