Study of the decay pattern of $f_0 (1370)$ as a $κ\bar{κ}$ molecular state

Abstract

Assuming that the $f_0(1370)$ is a $κ\barκ$ molecular state, the partial widths of its various decay channels are calculated, including the two-body decay $K \bar{K}$, $ππ$, $ηη$ and the four-body decay $ρρ/ σσ\to 4 π$ and $K \bar{K} ππ$. The coupling of $g_{f_0(1370) κ\barκ}\approx 13$ GeV estimated from the Weinberg criterion appears to be significantly underestimated. If this coupling is adjusted to $25 \sim 40$ GeV, the total width of $f_0(1370)$ can be fitted to the measured value $200\sim 500$ MeV. At the center-of-mass energy $\sqrt{s}=1.37$ GeV, the channels that mainly contribute to the total width are $K \bar{K}$, $ππ$ and $4 π$ ranked as $Γ(K \bar{K }) > Γ(4 π) \approx Γ(ππ) $ with $g_{f_0(1370) κ\barκ}= 35$ GeV. Around $1.37$ GeV, the decay widths of the two-body channels $K \bar{K}$, $ππ$ and $ηη$ remain stable with variation in $\sqrt{s}$, whereas the decay widths of the four-body channels $4 π$ and $K \bar{K }ππ$ increase continuously with $\sqrt{s}$. Most current data are model-dependent and conflicting, such as the $4 π$ dominant conclusion and the $K \bar{K}$ to $ππ$ ratios. The current data can not rule out the $κ\barκ$ assignment for $f_0(1370)$. Further reliable theoretical and experimental analyses of $f_0(1370)$ are required to reveal its nature.

Figures (4)

• Figure 1: The mechanism for $f_0(1370)$ decay as a $\kappa \bar{\kappa}$ molecule state. (a) $\pi^+ \pi^-$/$\eta \eta$ final states. (b) $K^+ K^-$ final states. (c) $4 \pi$ final states. $S/V$ indicates the $t$-exchange meson could be $\kappa$ or $K^*$. (d) $K \bar{K } \pi \pi$ final states.
• Figure 2: The total width of $f_0(1370)$ dependent on the parameter $g_{F \kappa \bar{\kappa}}$ with $\sqrt{s}=1.37$ GeV and the form factor parameter $\alpha$ varying from $1\sim 3$. The red dashed vertical line indicates the value of $g_{F\kappa \bar{\kappa}}$ calculated via the Weinberg criterion using a complex threshold. The light purple band indicates the range of the Breit-Wigner width of $f_0(1370)$ measured in experiments.
• Figure 3: The dependence of the partial widths of different channels on the initial energy $\sqrt{s}$ with fixed initial coupling $g_{F\kappa\bar{\kappa}}=35$ GeV and cut-off parameter $\alpha=2$. Because the scales in widths for some channels are much smaller than others, we present them in two separate figures with distinct scale for clarity. (a) $K\bar{K}$, $\pi \pi$, $4 \pi$ and $K \bar{K} \pi \pi$ channels. Different specific $4 \pi$ states are also presented. (b) $K \bar{K} \pi \pi$ and $\eta \eta$ channels. Different specific $K \bar{K} \pi \pi$ states are also presented. Note that $\Gamma(K^+ K^- \pi^+ \pi^-)= \Gamma(K^0 \bar{K}^0 \pi^+ \pi^-)$, $\Gamma(K^+ \bar{K}^0 \pi^0 \pi^-)=\Gamma(K^0 K^- \pi^+ \pi^0)$ and $\Gamma(K^+K^- \pi^0 \pi^0 )=\Gamma(K^0 \bar{K}^0 \pi^0 \pi^0)$. For the $\eta \eta$ channel, $\Gamma(F\to \eta \eta )$ is presented.
• Figure 4: The dependence of the partial widths of $4 \pi$ states on the initial energy $\sqrt{s}$. Contributions from different intermediate channels $\sigma \sigma$ and $\rho \rho$ are also plotted. (a) For the $2 \pi^+ 2\pi^-$ states. (b) For the $\pi^+ \pi^- 2 \pi^0$ states.