Flavor-changing decay $h\to \tau\mu$ at super hadron colliders

We study the flavor-changing decay $h\to \tau \mu$ with $\tau=\tau^-+\tau^+$ and $\mu=\mu^-+\mu^+$ of a Higgs boson at future hadron colliders, namely: a) High Luminosity Large Hadron Collider, b) High Energy Large Hadron Collider and c) Future hadron-hadron Circular Collider. The theoretical framework adopted is the Two-Higgs-Doublet Model type III. The free model parameters involved in the calculation are constrained through Higgs boson data, Lepton Flavor Violating processes and the muon anomalous magnetic dipole moment; later they are used to analyze the branching ratio of the decay $h\to\tau\mu$ and to evaluate the $gg\to h$ production cross section. We find that at the Large Hadron Collider is not possible to claim for evidence of the decay $h\to\tau\mu$ achieving a signal significance about of $1.46\sigma$ by considering its final integrated luminosity, 300 fb$^{-1}$. More promising results arise at the High Luminosity Large Hadron Collider in which a prediction of 4.6$\sigma$ when an integrated luminosity of 3 ab$^{-1}$ and $\tan\beta=8$ are achieved. Meanwhile, at the High Energy Large Hadron Collider (Future hadron-hadron Circular Collider) a potential discovery could be claimed with a signal significance around $5.04\sigma$ ($5.43\sigma$) for an integrated luminosity of 3 ab$^{-1}$ and $\tan\beta=8$ (5 ab$^{-1}$ and $\tan\beta=4$).