Elementarity of composite systems

The “compositeness” or “elementarity” is investigated for $s$-wave composite states dynamically generated by energy-dependent and independent interactions. The bare mass of the corresponding fictitious elementary particle in an equivalent Yukawa model is shown to be infinite, indicating that the wave function renormalization constant $Z$ is equal to zero. The idea can be equally applied to both resonant and bound states. In a special case of zero-energy bound states, the condition $Z=0$ does not necessarily mean that the elementary particle has the infinite bare mass. We also emphasize arbitrariness in the “elementarity” leading to multiple interpretations of a physical state, which can be either a pure composite state with $Z=0$ or an elementary particle with $Z\ne 0$. The arbitrariness is unavoidable because the renormalization constant $Z$ is not a physical observable.

Figure 1

Sum of the infinite set of diagrams that contributes to the meson-meson scattering amplitude (1).

Figure 2

Scattering amplitude in the Yukawa model.

Figure 3

(a) Interaction kernel $v\left(s\right)$ in Eq. (45) and (b) its inverse $v{\left(s\right)}^{-1}$, with $x=0$ (the WT term) denoted by the red solid line and $x=1$ (WT + pole) by the blue dashed line, as functions of $\sqrt{s}$. The parameters are set to be $m=140$ MeV, $M_0=500$ MeV, $f=92.4$ MeV. The arrows in the figure indicate the divergent behaviors of the kernels.

Figure 4

(a) Interaction kernel $v\left(s\right)$ in Eq. (45) and (b) its inverse $v{\left(s\right)}^{-1}$ with $x=0.2$ denoted by the green solid line and with $x=1$ by the blue dashed line as functions of $\sqrt{s}$. The parameters are set to be $m=140$ MeV, $M_0=500$ MeV, $f=92.4$ MeV.

Figure 5

Tree amplitudes in (a) the Yukawa model in Eq. (52), (b) the “nonlinear” model in Eq. (53), and (c) the “linear” model in Eq. (54). The solid and open circles are for the three-point vertices and for the four-point interactions, respectively.

Figure 6

Self-energy for the fictitious particle in the linear and nonlinear models. The solid and open circles indicate the couplings in the same manner as in Fig. 5.

Figure 7

(a) Flow of the pole position of the sigma [$\sigma \left(500\right)$] resonance in the scattering amplitude obtaining by changing the bare mass of the elementary particle (sigma meson) from 0.5 to 3 GeV. The solid square indicates the pole position in the limit $m_{\mathrm{fic}}\to \infty$ [22]. (b) The real part of the wave function renormalization constants as functions of the bare mass of the elementary particle. The blue dash-dotted line is for the Yukawa model $\left(Z_Y^{1/2}\right)$ with the interaction kernel (61), the green solid line for the nonlinear model $\left(Z_{\mathrm{NL}}^{1/2}\right)$ with (62), and the red dashed line for the linear model $\left(Z_L^{1/2}\right)$ with (63).

Figure 8

Wave function renormalization constants as functions of the binding energy. The blue dash-dotted line is for the Yukawa model $\left(Z_Y^{1/2}\right)$ with the interaction kernel (61), the green solid line for the nonlinear model $\left(Z_{\mathrm{NL}}^{1/2}\right)$ with (62), and the red dashed line for the linear model $\left(Z_L^{1/2}\right)$ with (63). The parameters are set to be $f=f_{\pi }=92.4$ and $m_{\mathrm{fic}}=550$ MeV. The corresponding pion mass is indicated in the upper horizontal axis.