Dipionic transitions of $Y(4500)$ to $J/ψ$

TL;DR

This paper addresses how the dipionic transition $Y(4500)\to J/\psi$ occurs by modeling the $Y(4500)$ as a $5S$-$4^3D_1$ mixed state with mixing angle $\theta \approx 30^\circ$ and computing the decay via charmed-meson loops (box and triangle topologies). Using an effective Lagrangian framework, it includes intermediate mesons $f_0(500)$, $f_0(980)$, and $f_2(1270)$ in triangle loops and constructs amplitudes from box, FB, and $\mathcal{R}$-loop contributions, incorporating a monopole form factor with cutoff $\Lambda=M+\alpha\Lambda_{QCD}$. The study finds that the invariant-mass spectra from box and $\mathcal{R}$ loops are largely insensitive to $\alpha$, while FB loops dominate the partial width, yielding $\Gamma_{FB} \approx 64.6$--$960.2$ keV (with similar ranges for box and $\mathcal{R}$ loops), and that interference among loop types can significantly shape the $\pi^+\pi^-$ and $\pi^+ J/\psi$ spectra. A representative total width of $\Gamma(Y(4500)\to \pi^+\pi^- J/\psi) \approx 250$--$700$ keV is obtained for $\alpha=3.0$ and phases set to zero, consistent with BESIII-based estimates when combined with $Y(4500)\to K^+K^- J/\psi$ data. The work provides testable predictions for future BESIII or Belle II measurements and offers a means to discriminate the underlying loop mechanism and validate the $5S$-$4^3D_1$ mixing scenario and coupled-channel effects in higher charmonia.

Abstract

We, using an effective Lagrangian approach, investigated the dipionic transition of the $Y(4500)$ newly well established in the process $e^+e^-\to K^+K^- J/ψ$. In this study, the $Y(4500)$ was considered as a mixture of the $ψ(5S)$ and $ψ(4\,{^3D_1})$, and its dipionic decay to the $J/ψ$ was assumed to occur via charmed meson loops including box and two kinds of triangle loops. The calculated invariant mass spectra due to the different loop mechanisms exhibit distinct differences. These spectrum patterns are found to be nearly independent of the cutoff parameter, especially for the box and $\mathcal{R}$ loops. By means of comparing our calculated results with the future experimental measurements of the $π^+π^-$ and $π^\pm J/ψ$ invariant mass distributions, we could judge which loop mechanism is of more importance in this dipionic transition. The interference among the different kinds of loops is also exhibited. Despite the unknown phase angles between different kinds of loop diagrams, the partial decay width for the process $Y(4500)\toπ^+π^- J/ψ$ is estimated to be 250--700 keV, aligning with the estimation obtained by combining BESIII Collaboration measurements and theoretical predictions for $Y(4500)\to K^+K^- J/ψ$. We hope that the present calculations would be tested by the future BESIII or Belle II experiments.

Figures (7)

• Figure 1: Feynman diagrams for the $Y(4500)\to\pi^+\pi^- J/\psi$ decay process via the intermediate meson loops. As indicated, the diagrams (a)--(j) describe the box loops, while the diagrams (k)--(r) are the triangle loops. The $Y$ represents the $Y(4500)$, and the $\mathcal{R}$ stands for the $f_0(500),\,f_0(980)$, and $f_2(1270)$. The relevant kinematics [$p,\,p_{1(2,3)},\,q,\,q_{0(1,2,3)}$] are explicitly indicated in the graphs. The charge conjugated loops are not shown here but included in the calculations.
• Figure 2: The Dalitz plots and distributions of the invariant mass of the $\pi^+\pi^-$ ($m_{12}$) and $\pi^+ J/\psi$ ($m_{23}$) due to the box diagrams in Figs. \ref{['fig:feyndiags']}(a)--1(j) at $\alpha=1.5,\,2.5$, and $3.5$. The solid lines are the corresponding spectra projected onto the $m_{12}$ (red) and $m_{23}$ axis, namely, $(\mathrm{d} \Gamma / \mathrm{d} m_{12})$ and $(\mathrm{d} \Gamma / \mathrm{d} m_{23})$. For graphical reasons, the projected spectra for $\alpha=1.5$ are increased by $10^5$, while for $\alpha=2.5$ and $3.5$ they are increased by $10^4$. The color bars are not the same.
• Figure 3: As in Fig. \ref{['fig:boxdgdm']}, but these data are obtained using the triangle loops shown in Figs. \ref{['fig:feyndiags']}(k)--1(n). For graphical reasons, $(\mathrm{d}\Gamma/\mathrm{d}m_{12})\times 10^5$ and $(\mathrm{d}\Gamma/\mathrm{d}m_{23})\times 10^4$ for $\alpha=1.5$, while $(\mathrm{d}\Gamma/\mathrm{d}m_{12})\times 10^4$ and $(\mathrm{d}\Gamma/\mathrm{d}m_{23})\times 10^4$ for $\alpha=2.5$ and $3.5$.
• Figure 4: Distributions of the invariant mass of the $\pi^+\pi^-$ ($m_{12}$) and $\pi^+ J/\psi$ ($m_{23}$) for different combinations of the $\phi_1$ and $\phi_2$ (in units of degree) at $\alpha=3.5$, obtained using the triangle $\mathcal{R}$ loops (o)--(r) in Fig. \ref{['fig:feyndiags']}. The solid lines are the corresponding spectra projected onto the $m_{12}$ (red) and $m_{23}$ axis, namely $(\mathrm{d} \Gamma / \mathrm{d} m_{12})\times 10^{3}$ and $(\mathrm{d} \Gamma / \mathrm{d} m_{23})\times 10^{4}$. The color bars for different combinations are not the same.
• Figure 5: Partial decay width as a function of the model parameter $\alpha$. The closed circles, triangles, and squares represent results contributed the FB, box, and $\mathcal{R}$ loops, respectively. The calculations for $\mathcal{R}$ loops were conducted at phases $\phi_1=\phi_2=0^\circ$. The two bands depict the estimations using the BESIII experiments BESIII:2018iop and the theoretical predictions for the $Y(4500)\to K^+K^- J/\psi$Wang:2022jxj (see the text).