Searching for the doubly-charged Higgs bosons in the Georgi-Machacek model at the ep colliders

The Georgi-Machacek model is one of many beyond Standard Model scenarios with an extended scalar sector which can group under the custodial $\rm SU(2)_C$ symmetry. There are 5-plet, 3-plet and singlet Higgs bosons under the classification of such symmetry in addition to the Standard Model Higgs boson. Here we study the prospects for detecting the doubly-charged Higgs boson ($\rm H_5^{\pm\pm}$) through the vector boson fusion production at the electron-proton colliders. Typically, we concentrate on our analysis through $\mu$-lepton pair production via pair of same-sign W bosons decay. The discovery significance are calculated as the functions of the triplet vacuum expectation value and necessary luminosity.