Composite and elementary nature of a resonance in the $\sigma$ model

We analyze the mixing nature of the low-lying scalar resonance consisting of the $\pi \pi$ composite and the elementary particle within the $\sigma$ model. A method to disentangle the mixing is formulated in the scattering theory with the concept of the two-level problem. We investigate the composite and elementary components of the $\sigma$ meson by changing a mixing parameter. We also study the dependence of the results on model parameters such as the cut-off value and the mass of the elementary $\sigma$ meson.

Figure 1

Feynman diagrams contributing for the function $A\left(s\right)$ in Eq. (7).

Figure 2

Sum of the infinite set of diagrams that contributes to the meson-meson scattering amplitude (10). If a pole is developed, then it is interpreted as the composite state in Eq. (22) which is depicted as in the right-hand side.

Figure 3

Full propagators of (a) composite and (b) elementary $\sigma$ mesons defined in Eq. (25). The solid line indicates the $\pi$ propagator, while the curly and double lines are those of the composite and elementary $\sigma$ mesons.

Figure 4

Full scattering amplitude in terms of the full propagators $\widehat{D}$. The first two diagrams denote the amplitudes through the full propagators of composite and elementary $\sigma$ mesons ($D^{11}$ and $D^{22}$), respectively, and the last two diagrams those of off-diagonal propagators ($D^{21}$ and $D^{12}$).

Figure 5

Trajectories of the poles of $\widehat{D}$ by changing the parameter $x$. The open circle ($\circ$) indicates the mass of the elementary $\sigma$ meson, $m_0$, and the solid square ($\blacksquare$) the pole position of the composite $\sigma$ generated by the four-pion interaction only. The solid circle (•) indicates the pole position of the physical state at $x=1$.

Figure 6

Trajectories of the poles of $\widehat{D}$ by changing the mixing parameter $X$. The open circle ($\circ$) indicates the mass of the elementary $\sigma$ meson, $m_0$, the solid square ($\blacksquare$) the pole position of the composite $\sigma$ generated by the four-pion interaction only. The solid triangle ($▴$) indicates the pole position of the composite $\sigma$ meson developed by the four-pion interaction and the $\sigma$ exchange in $t$ and $u$ channels. The solid circle (•) indicates the pole position of the physical state at $X=1$. The thick arrow shows the shift of the composite pole position by adding the $\sigma$-exchange term as explained in text.

Figure 7

Residues $z^{11}$ and $z^{22}$ of the full propagator $D^{11}$ and $D^{22}$ at the pole-a shown in Fig. 6 as functions of the mixing parameter $X$. (a) The real parts of the residues, (b) those of the imaginary parts, (c) the modulus of the residues, (d) the real parts of the square-root of the residues, and (e) those of the imaginary parts are shown, respectively.

Figure 8

Mass of pole-b as functions of mixing parameter $X$, defined by the real part of the pole position in complex-energy plane, plotted (a) in linear plot and (b) in double logarithmic plot.

Figure 9

Trajectories of the poles of $\widehat{D}$ by changing the mixing parameter $X$ for different mass of the elementary $\sigma$ (A) $m_0=550$ MeV (solid lines), (B) $m_0=700$ MeV (dashed lines), and (C) $m_0=1$ GeV (dotted lines). The open circles indicate the mass of the elementary $\sigma$ meson, $m_0$, and the solid circles indicate the pole position of the physical state at $X=1$ for each $m_0$ case.

Figure 10

(a) Trajectory of the poles of $\widehat{D}$ at $X=1$ by changing the mass of the elementary $\sigma$, $m_0$. The solid square indicates the pole position of the naive-composite $\sigma$ meson, which is also obtained in $m_0\to \infty$ limit. The (blue) hatched area denotes the energy range of the experimental value of the $\sigma$ mass $m_{\sigma }=400$–550 MeV and width ${\Gamma }_{\sigma }=400$–700 MeV in the PDG [1]. (b) The real parts of the square-root of the residues at the resulting pole at $X=1$ as functions of $m_0$. The area between two vertical dashed-lines corresponds to the case in which the corresponding pole are within the error bars of the PDG value [1]. (c) The possible range of Re$\sqrt{z^{11}}$ and Re$\sqrt{z^{22}}$ when the resulting pole position is within error bars of PDG value [1].

Figure 11

(a) Trajectories of the poles of $\widehat{D}$ at $X=1$ by changing the mass of the elementary $\sigma$, $m_0$, for different cut-off values $\Lambda$. The solid squares indicate the pole positions of the naive-composite $\sigma$ meson, which are also obtained in $m_0\to \infty$ limit. The (blue) hatched area denotes the energy range of experimental value of the $\sigma$ mass $m_{\sigma }=400$–550 MeV and width ${\Gamma }_{\sigma }=400$–700 MeV in PDG [1]. (b) The real parts of the square-root of the residues at the poles at $X=1$ as functions of $m_0$ for different $\Lambda$. The thick lines corresponds to the case in which corresponding poles are within error bars of the PDG value [1]. (c) The possible range of Re$\sqrt{z^{11}}$ and Re$\sqrt{z^{22}}$ for different $\Lambda$ when the resulting pole position is within error bars of the PDG value [1].