Rare Decays of the Z-Boson

We study radiative decay modes of the Z-boson into heavy quark bound states. We find that the widths for these decays are extremely small. We conclude that these decays will not be detectable for the time being unless there is a significant increase in the number of Z-bosons produced at the electronpositron colliders. Rare decays of the Z-boson have recently been studied at the ee colliders, CERN’s LEP and the SLAC Linear Collider, where branching ratios down to 10 corresponding to a few Kev can be measured [1]. The partial decay widths of the Zboson to a photon and bound states of heavy quarks (J/ψ) have been previously calculated by using Bethe-Salpeter wavefunctions for the quarkonium state and modeldependent potentials and found to be about .2 Kev [2] which might be observebale soon. However, using a model-independent method, we show that these decay widths are actually much smaller than previously reported. Calculation of the amplitudes A(Z→ J/ψγ) and A(Z→ Υγ) We are considering the decays Z→ J/ψγ and Z→ Υγ. The quark diagrams for the first decay are shown below with similar diagrams for the second case (replace c and c̄ by b and b̄). Fig.1. The quark diagrams for Z→ J/ψγ decay Here we are assuming that the quarks inside the meson are at rest with respect to each other. In other words, we are making the approximation that P (c) = 1 2 P (J/ψ)+ O( μ mc ), where μ = .5 Gev is the characteristic hadronic scale here. This is not a very good approximation for the c quark, but is much better for the b quark [3]. Then we will have mJ/ψ = 4m 2 , (P +K) = M/2 and Q = 2P +K where m and P are the quark mass and momentum, M and Q are the Z-boson mass and momentum and K