Hadron mass scaling near the $s$-wave threshold

The influence of a two-hadron threshold is studied for the hadron mass scaling with respect to some quantum chromodynamics parameters. A quantum mechanical model is introduced to describe the system with a one-body bare state coupled with a single elastic two-body scattering. The general behavior of the energy of the bound and resonance state near the two-body threshold for a local potential is derived from the expansion of the Jost function around the threshold. It is shown that the same scaling holds for the nonlocal potential induced by the coupling to a bare state. In $p$ or higher partial waves, the scaling law of the stable bound state continues across the threshold describing the real part of the resonance energy. In contrast, the leading contribution of the scaling is forbidden by the nonperturbative dynamics near the $s$-wave threshold. As a consequence, the bound state energy is not continuously connected to the real part of the resonance energy. This universal behavior originates in the vanishing of the field renormalization constant of the zero-energy resonance in the $s$ wave. A proof is given for the vanishing of the field renormalization constant, together with a detailed discussion.

Figure 1

Schematic illustration of the near-threshold eigenenergy $E_h$ for $l=0$ (a) and $ł\ne 0$ (b) as a function of the variation of the potential $\delta \lambda$ or the variation of the bare mass $\delta M$. The solid lines represent the bound state energy, the dotted lines stand for the real part of the resonance energy ($l\ne 0$) and the energy of the virtual state ($l=0$), and the dashed line represents the imaginary part of the resonance energy ($l\ne 0$).

Figure 2

Schematic illustration of the field renormalization constant $Z$ as a function of the coupling strength $g_0$ with a fixed binding energy $B$. Solid (dotted) line represents the $B=0$ ($B>0$) case.

Figure 3

Near-threshold eigenenergies as functions of $\delta M$ for $l=0$ (a) and for $l=1$ (b). Solid, dotted, and dashed lines represent the energy in the first Riemann sheet, the real part of the energy in the second Riemann sheet, and the imaginary part of the energy in the second Riemann sheet, respectively.

Figure 4

Eigenenergy for $l=0$ as a function of $\delta M$ in the region $\left|\delta M\right|\le 0.2{\Lambda }^2/\mu$. Solid, dotted, and dashed lines represent the energy in the first Riemann sheet, the real part of the energy in the second Riemann sheet, and the imaginary part of the energy in the second Riemann sheet, respectively.