Discriminating HTM and MLRSM models in colliders studies of a doubly charged Higgs boson pair production and its subsequent leptonic decays.

We present a case study for the doubly charged Higgs bosons $H^{\pm\pm}$ pair production in $e^+e^-$ and $pp$ colliders with their subsequent decays to four charged leptons. We consider the Higgs Triplet Model (\texttt{HTM}) not restricted by the custodial symmetry and the Minimal Left-Right Symmetric Model (\texttt{MLRSM}). The models include scalar triplets with different complexity of scalar potentials and, due to experimental restrictions, completely different scales of non-standard triplet vacuum expectation values. In both models, a doubly charged Higgs boson $H^{\pm\pm}$ can acquire a mass of hundreds of gigaelectronvolts, which can be probed at HL-LHC, future $e^+e^-$, and hadron colliders. We find that after taking into account the constraints on the parameters of the models coming from neutrino oscillations, LHC, $e^+e^-$ and low-energy lepton flavour violating data, for the same mass of $H^{\pm\pm}$, the $H^{\pm\pm}$ pair production in lepton and hadron colliders is comparable in both models. However, decay branching ratios can be different, leading to distinguishable four lepton final signatures. Setting model parameters to maximise the $4e$ and $4\mu$ signals separately, after suitable kinematic cuts, the maximal possible signals for the process $pp \to H^{++} H^{--} \to 4l$ are connected with detection of four muon events in \texttt{MLRSM}. Typically, it exceeds the \texttt{HTM} signal by one order of magnitude for $M_{H^{++}}=1$ TeV with significance of the $pp \to 4\mu$ signal $S \simeq 8$ for HL-LHC, and $S \simeq 200$ for FCC-hh. Additionally, we discuss dilepton invariant mass distributions and leptons separations which can be useful in the identification of non-standard signals.