Improved analysis of double $J/\psi$ production in $Z$-boson decay

In this paper, we present an improved calculation for the decay rate of the rare $Z$-boson decay into $J/\psi + J/\psi$. This decay is dominated by the photon fragmentation mechanism, i.e., the transition $Z\to J/\psi + \gamma^{*}$ followed by the fragmentation $\gamma^{*}\to J/\psi$. In our calculation, the amplitude of $\gamma^{*}\to J/\psi$ is extracted from the measured value of $\Gamma(J/\psi \to e^+ e^-)$, and the amplitude of $Z\to J/\psi + \gamma^{*}$ is calculate through the light-cone approach. The higher-order QCD and relativistic corrections in the amplitude of $\gamma^{*}\to J/\psi$ and the large logarithms of $m_{_Z}^2/m_c^2$ that appear in the amplitude of $Z\to J/\psi + \gamma^{*}$ are resummed in our calculation. Besides, the non-fragmentation amplitude is calculated based on the NRQCD factorization, and the next-to-leading order QCD and relativistic corrections are included. The obtained branching fraction for this $Z$ decay channel is $8.66 ^{+1.48} _{-0.69}\times 10^{-11}$.