Constraints on the Higgs-charm coupling by CMS

The discovery of the Higgs~(H) boson ten years ago and successful measurements of its couplings to third generation fermions by ATLAS and CMS experiments mark great milestones for the field of High Energy Physics. The much weaker coupling of the H boson to second generation quarks predicted by the standard model (SM) makes the measurement of the H-charm coupling much more challenging. With the full Run-2 data collected at the LHC a lot of progress has been made by the CMS Collaboration to constrain this coupling. In this report I present the latest results of the direct and indirect measurements of the H-charm coupling by CMS. Direct measurements are performed by searching for the $H \to c\bar{c}$ decay. It is done in gluon fusion and VH production modes with stronger limits coming from VH mode, $\mu_{VH(H\to c\bar{c})}<14.4$ at 95\% CL. In the ggH channel the limit is set at $\mu_{ggH(H\to c\bar{c})}<45$. Indirect constraints on the charm coupling come from the measurement of the H boson $p_T$ distribution, which leads to a limit of $\vert\kappa_c\vert<35$. A~rare H boson decay, $H \to J/\psi \gamma$, is also searched for at CMS, though giving a limited sensitivity of $\sim220$ x SM. Prospects for future measurements at HL-LHC are also discussed. From the projections it seems feasible to constrain the H-charm coupling better than 2 times the SM value.