Probing doubly and singly charged Higgs bosons at the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>p</mml:mi><mml:mi>p</mml:mi></mml:mrow></mml:math> collider HE-LHC

We analyze the signal sensitivity of multilepton final states at colliders that can arise from doubly and singly charged Higgs decay in a type-II seesaw framework. We assume the triplet vacuum expectation value to be very small and degenerate masses for both the charged Higgs states. The leptonic branching ratios of doubly and singly charged Higgs states have a large dependency on the neutrino oscillation parameters and the lightest neutrino mass scale, as well as the neutrino mass hierarchy. We explore this, as well as the relation between the leptonic branching ratios of the singly and doubly charged Higgs states, in detail. We evaluate the effect of these uncertainties on the production cross section of the multilepton signal. Finally, we present a detailed analysis of multilepton final states for a future hadron collider, HE-LHC, that can operate with center-of-mass energy $\sqrt{s}=27\text{ }\text{ }\mathrm{TeV}$.