Probing eV-TeV vacuum scales by doubly charged Higgs bosons with LFV low energy and collider signals

We present a case study for the doubly charged Higgs bosons H++ pair production in e+e- and pp colliders with their subsequent decays to four charged leptons. We consider the Higgs Triplet Model (HTM) with a type-II seesaw mechanism which is not restricted by the custodial symmetry and the Minimal Left-Right Symmetric Model (MLRSM). Taking into account theoretical and present experimental bounds, those models differ drastically in values of allowed non-standard triplet vacuum expectation values (VEV). In HTM the triplet VEV can be at the electronvolt level, in MLRSM the relevant non-standard VEV scale $v_R$ can be from the teraelectronvolt level up. Nonetheless, in both models, a doubly charged Higgs boson H++ can acquire a mass of hundreds of gigaelectronvolts, which can be probed at HL-LHC, future e+e-, and hadron colliders. We present calculations showing which aspects of collider measurements will allow us to differentiate the two models. We find that after taking into account the constraints on the parameters of the two models coming from neutrino oscillations, LHC, e+e- and low-energy lepton flavour violating data, for the same mass of H++, the H++H-- pair production is comparable in both models, both in lepton and hadron colliders. However, decay branching ratios can be quite different, leading to substantially different 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 MLRSM. Typically, it exceeds the 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.