Compositeness of dynamically generated states in a chiral unitary approach

The structure of dynamically generated states in the chiral unitary approach is studied from a viewpoint of their compositeness. We analyze the properties of bound states, virtual states, and resonances in a single-channel chiral unitary approach, paying attention to the energy dependence of the chiral interaction. We define the compositeness of a bound state using the field renormalization constant which is given by the overlap of the bare state and the physical state in the nonrelativistic quantum mechanics, or by the residue of the bound state propagator in the relativistic field theory. The field renormalization constant enables one to define a normalized quantitative measure of compositeness of the bound state. Applying this scheme to the chiral unitary approach, we find that the bound state generated by the energy-independent interaction is always a purely composite particle, while the energy-dependent chiral interaction introduces the elementary component, depending on the value of the cutoff parameter. This feature agrees with the analysis of the effective interaction by changing the cutoff parameter. A purely composite bound state can be realized by the chiral interaction only when the bound state lies at the threshold or when the strength of the two-body attractive interaction is infinitely large. The natural renormalization scheme, introduced by the property of the loop function and the matching with the chiral low-energy theorem, is shown to generate a bound state which is dominated by the composite structure when the binding energy is small.

Figure 1

The compositeness of the bound state as a function of the binding energy $B=M+m-M_B$ with $a=a_{\text{natural}}$, $m=496$ MeV, and $M=939$ MeV. The solid line denotes expression (42), while the dotted line represents the approximate form (46) valid for the small binding energy.

Figure 2

Same with Fig. 1 with $m=138$ MeV and $M=939$ MeV [$\pi N$ system, panel (a)], $m=496$ MeV and $M=2286$ MeV [$\overline{K}{\Lambda }_c$ system, panel (b)], and $m=138$ MeV and $M=2286$ MeV [$\pi {\Lambda }_c$ system, panel (c)].

Figure 3

The factor $A\left(M_B\right)$ defined in Eq. (47) which shows the accuracy of the approximation. The dashed line represents the result with $m=496$ MeV and $M=939$ MeV ($\overline{K}N$ system), the solid line with $m=138$ MeV and $M=939$ MeV ($\pi N$ system), the dotted line with $m=496$ MeV and $M=2286$ MeV ($\overline{K}{\Lambda }_c$ system), and the dash-dotted line with $m=138$ MeV and $M=2286$ MeV ($\pi {\Lambda }_c$ system).

Figure 4

The compositeness of the bound state as a function of the deviation of the subtraction constant from the natural value $\Delta a$ (a) and the effective pole mass $M_{\text{eff}}$ (b) with $B=10$ MeV, $m=496$ MeV, and $M=939$ MeV. The vertical dotted line represents the composite subtraction constant $a_{\text{comp}}$, while the vertical dashed line in the panel (a) corresponds to the natural value of the subtraction constant $a_{\text{natural}}$. The horizontal solid line in the panel (b) indicates the compositeness at $a=a_{\text{natural}}$ ($M_{\text{eff}}\to \infty$).

Figure 5

Pole trajectories with $M=939$ MeV, $m=496$ MeV, and the natural subtraction constant. Pole 1 is the most relevant to the amplitude above the threshold. The poles on the real axis are slightly shifted to positive (negative) direction of the imaginary axis when they are on the first (second) Riemann sheet.