Displaced Higgs production in Type-III Seesaw at the LHC/FCC, MATHUSLA and Muon collider

In this article, we explore the possibility of displaced Higgs production from the decays of the heavy fermions in the Type-III seesaw extension of the Standard Model at the LHC/FCC and the muon collider. The displaced heavy fermions and the Higgs boson can be traced back by measuring the displaced charged tracks of the charged leptons along with the $b$-jets. A very small Yukawa coupling can lead to two successive displaced decays which makes the phenomenology even more interesting. The prospects of the transverse and longitudinal displaced decay lengths are extensively studied in the context of the boost at the LHC/FCC. Due to the parton distribution function, the longitudinal boosts leads to larger displacement compared to the transverse one, which can reach MATHUSLA and beyond. The longitudinal measurements are indeed possible by the visible part of the finalstate, which captures the complete information about the longitudinal momenta. The comparative studies are made at the LHC/FCC with the centre of mass energies of 14, 27 and 100 TeV, respectively. A futuristic study of the muon collider where the collision happen in the centre of mass frame is analysed for centre of mass energies of 3.5, 14 and 30 TeV. Contrary to LHC/FCC, here the transverse momentum diverges, however, the maximum reach in both the direction are identical due to the constant total momentum in each collision. The reach of the Yukawa couplings and fermion masses are appraised for both the colliders. FCC at 100 TeV can probe a mass of 4.25 TeV and a lowest Yukawa coupling of $\mathcal{O}(5 \times 10^{-11})$.