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A search is presented for decays of Z and Higgs bosons to a J$/\psi$ mesonand a photon, with the subsequent decay of the J$/\psi$ to $\mu^+\mu^-$. Theanalysis uses data from proton-proton collisions with an integrated luminosityof 35.9 fb$^{-1}$ at $\sqrt{s} =$ 13 TeV collected with the CMS detector at theLHC. The observed limit on the Z $\to$ J$/\psi \gamma$ decay branchingfraction, assuming that the J$/\psi$ meson is produced unpolarized, is 1.4$\times$ 10$^{-6}$ at 95% confidence level, which corresponds to a rate higherthan expected in the standard model by a factor of 15. For extreme-polarizationscenarios, the observed limit changes from -13.6 to +8.6% with respect to theunpolarized scenario. The observed upper limit on the branching fraction for H$\to$ J$/\psi \gamma$ where the J$/\psi$ meson is assumed to be transverselypolarized is 7.6 $\times$ 10$^{-4}$, a factor of 260 larger than the standardmodel prediction. The results for the Higgs boson are combined with previousdata from proton-proton collisions at $\sqrt{s} =$ 8 TeV to produce an observedupper limit on the branching fraction for H $\to$ J$/\psi \gamma$ that is afactor of 220 larger than the standard model value.

Search for rare decays of Z and Higgs bosons to J$/\psi$ and a photon in proton-proton collisions at $\sqrt{s} =$ 13 TeV