Revisiting type-II see-saw: present limits and future prospects at LHC

The type-II see-saw mechanism based on the annexation of the Standard Model by weak gauge triplet scalar field proffers a natural explanation for the very minuteness of neutrino masses. Noting that the phenomenology for the non-degenerate triplet Higgs spectrum is substantially contrasting than that for the degenerate one, we perform a comprehensive study for an extensive model parameter space parametrised by the triplet scalar vacuum expectation value (VEV), the mass-splitting between the triplet-like doubly and singly charged scalars and the mass of the doubly charged scalar. Considering all Drell-Yan production mechanisms for the triplet-like scalars and taking into account the all-encompassing complexity of their decays, we derive the most stringent 95% CL lower limits on the mass of the doubly charged scalar for a vast model parameter space by implementing already existing direct collider searches by CMS and ATLAS. These estimated limits are stronger by approximately 50–230 GeV than those reported by CMS and ATLAS. Strikingly, we also find a specific region of the parameter space that is beyond the reach of the existing LHC search strategies. Then, we forecast future limits by extending an ATLAS search at high-luminosity, and we propose a search strategy that yields improved limits for a part of the parameter space.


Introduction
The Standard Model (SM) falls short of offering explanations of tiny neutrino masses and mixings. Although plausible, it seems philosophically displeasing that the tiny neutrino masses are effectuated via the usual Brout-Englert-Higgs mechanism as it entails extremely small Yukawa couplings causing hierarchy among them. Conversely, the widely-studied seesaw mechanisms seem to proffer a natural explanation for the very minuteness of neutrino masses. The type-II see-saw model based on the annexation of the SM by weak gauge triplet of scalar field [1][2][3][4][5][6] is one such variant. Yukawa interaction of the scalar triplet with the SM lepton doublet leads to neutrino masses after its neutral component procure a non-zero vacuum expectation value (VEV). The Yukawa coupling driving the leptonic decays of the non-standard scalars in the model pans out to be determined by the neutrino oscillation parameters up to the triplet VEV (v t ). Though ad hoc, this prognostic characteristic of the present scenario makes the same an appealing one beyond the SM (BSM). Not only this model holds out a riveting rationale for the neutrino masses, but it also put forward an elaborated electroweak symmetry breaking (EWSB) mechanism and rich phenomenology at the Large Hadron Collider (LHC). This model contains several triplet-like physical Higgs states, namely doubly charged scalars (H ±± ), singly charged scalars (H ± ) and CP-even and CP-odd neutral scalars (H 0 and A 0 ). Phenomenological outcome of this model has been studied all-encompassingly in the literature. The main dynamical features of the Higgs potential have been discussed in detail in refs. [7][8][9][10][11]. The Yukawa interaction of the scalar triplet with the SM lepton doublet leads to charged lepton flavour violating decays. This has been addressed in detail in refs. [12][13][14][15][16][17]. Refs. [18][19][20][21][22] have investigated -1 -
A wealth of BSM models such as the present model [1][2][3][4][5][6], left-right symmetric models [55][56][57] Higgs triplet models [58,59], little Higgs model [60][61][62], Georgi-Machacek model [63,64], Zee-Babu model [65,66] and other extensions of SM [67][68][69][70][71][72] envisage presence of doubly charged scalar bosons and their illustrious signatures. This is why, a number of searches have been carried out at the LHC by CMS and ATLAS [73][74][75][76][77][78][79][80][81][82]. In view of the observations being consistent with the SM background expectations, these searches derived stringent limits with 95% confidence level (CL) on the doubly charged scalar mass. Collider phenomenology of this model, by and large, is governed by three parameters onlym H ±± , ∆m = m H ±± − m H ± and v t (see section 2). For degenerate scenario (∆m = 0), H ±± decays to same-sign dilepton for v t < 10 −4 GeV and to same-sign W -boson for v t > 10 −4 GeV. For H ±± decaying 100% into same-sign dilepton, a search in three and four lepton final states with an integrated luminosity of 12.9 fb −1 of pp collisions at √ s = 13 TeV LHC by the CMS collaboration [78] has excluded them with mass below 716-761 GeV considering four benchmark points targeting four possible neutrino mass hypotheses. In addition, considering 100% decay of H ±± into lepton (e, µ, τ ) pair, the same search has set a limit of 535-820 GeV. Another search in multilepton final states with an integrated luminosity of 36.1 fb −1 of pp collisions at √ s = 13 TeV LHC by the ATLAS collaboration [79] has set a limit of 770-870 GeV and 450 GeV for H ±± decaying, respectively, 100% and 10% into same-sign light lepton (e, µ) pair. A recent search in multilepton final states, optimised for H ±± decaying exclusively into same-sign W -boson pair, with an integrated luminosity of 139 fb −1 of pp collisions at √ s = 13 TeV LHC by the ATLAS collaboration [82] has excluded them with masses up to 350 GeV and 230 GeV, respectively, for the pair and associated production modes assuming v t = 0.1 GeV and the mixing between the CP-even scalars to be 10 −4 .
Evidently, the above-cited limits are not befitting to the entire parameter space, rather valid only for a constrained parameter space of the model. For instance, the CMS search in ref. [78] is only valid for ∆m = 0 and v t < 10 −4 GeV, whereas the ATLAS search in ref. [82] is only valid for ∆m = 0 and v t > 10 −4 GeV. Though in a realistic type-II see-saw scenario, the branching fractions of the triplet-like scalars into different lepton flavours are dictated by the neutrino oscillation parameters, most of the aforecited limits are derived in the context of simplified scenarios without reckoning the footprints of the low-energy neutrino parameters. Furthermore, these limits are often conservative as these searches do not incorporate all the Drell-Yan production channels for the triplet-like scalars. However, all the Drell-Yan processes are of sizeable cross-sections, and thus, all of them entail to be incorporated into the analyses. Moreover, the triplet components in this model are conceivably non-degenerate in mass. For moderate v t and passably large ∆m, cascade decays quickly dominate over the leptonic and diboson decay modes, see section 3. Not only does the mass-splitting overwhelm the decays of the triplet-like scalars, but it also affects their production cross-sections at the LHC. Thus, the pheneomenology for the non-degenerate scenario substantially is contrasting than that for the degenerate one [39-42, 45, 46]. Bearing the aforesaid discussion in mind, we perform a systematic and comprehensive collider study of this model. Incorporating all the Drell-Yan production modes for the triplet-like scalars and taking into account the all-encompassing complexity of their decays, we derive the most stringent 95% CL lower limit on m H ±± for a wide range of v t and ∆m by implementing already existing direct collider searches by CMS and ATLAS. Then, we forecast future limits on m H ±± by extending the ATLAS search at high-luminsity, and we propose a search strategy that yields improved limits on m H ±± for a part of the parameter space of v t and ∆m.
The rest of this work is structured as follows. In section 2, we briefly describe the theoretical structure of the type-II see-saw model. Production of the triplet-like scalars and their decays are discussed in section 3. In section 4, we discuss the LHC phenomenology of this model and obtain stringent limits on m H ±± for a wide region of model parameter space.

The type-II see-saw model
The scalar sector of the minimal type-II see-saw model employs a SU(2) L triplet scalar field with hypercharge 1, ∆ in addition to the SM Higgs doublet, Φ: The most general renormalizable gauge invariant scalar potential invloving Φ and ∆ is given by [7] where m 2 Φ , m 2 ∆ and µ are the mass parameters, λ and λ i (i = 1, . . . , 4) are the independent dimensionless couplings. The neutral components of Φ and ∆ can be parametrised as GeV. The degrees of freedom carrying identical electric charges mix after the EWSB,. The neutral states Φ 0 and ∆ 0 mix into two CP-even states h 0 and H 0 , and two CP-odd states G 0 and A 0 , whereas the singly charged states Φ ± and ∆ ± mix into mass states G ± and A ± . The doubly charged gauge state ∆ ±± is aligned with its mass state H ±± . Therefore, the mixings result into several massive physical states (h 0 , H 0 , A 0 , H ± and H ±± ) and Nambu-Goldstone bosons (G 0 and G ± ) eaten by the longitudinal -3 -

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modes of Z and W ± . The mixing angles in the CP-even, CP-odd and singly-charged Higgs sectors (denoted by α, β 0 and β ± , respectively) are given by [7] For v t v d , the CP-even Higgs mixing angle and masses of the physical states reduces to and their mass-squared differences 1 are given by For usefulness, we define the mass-splitting between H ±± and H ± as ∆m = m H ±± − m H ± . Thereby, the masses of all the physical Higgs states can be traded in terms of just two parameters-m H ±± and ∆m. The value (sign) of λ 4 , thus ∆m predicts three characteristic mass spectra: We refer to these mass spectra as degenerate, positive and negative scenario, respectively.
The Yukawa interaction of the scalar triplet with the SM lepton doublet L = (ν L , L ) T is given by where Y ν is a 3 × 3 symmetric complex matrix, i and j are the generation indices (i, j = 1, 2, 3), and C is the charge-conjugation matrix. This interaction leads to majorana masses for the neutrinos after the EWSB: m ν can be diagonalised using the Pontecorvo-Maki-Nakagawa-Sakata matrix U which is parametrised by three mixing angles, one dirac phase and two Majorana phases: . For simplicity, we set the phases to zero as they are either poorly measured or hitherto not measured. Measurements of large scale structure in the universe by the Planck satellite has put a bound i m i < 0.12 eV when combined with baryon acoustic oscillation data [84]. The best fit values for the neutrino oscillation parameters used in this work are taken from ref. [85].

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Phenomenologically relevant parameters. While the Yukawa couplings are determined by the neutrino oscillation parameters 2 up to v t (see eq. 2.2), all the scalar potential parameters can be framed in terms of the physical Higgs masses, v t and α [7]. The mixing angle is further determined in terms of the others (see eq. 2.1). Moreover, the masses can be traded in terms of just two parameters-m H ±± and ∆m. Therefore, the phenomenology of this model, by and large, is governed by three parameters only-m H ±± , ∆m and v t . Before concluding this section, we briefly discuss the relevant constraints on these parameters: (i) The value of the ρ parameter from the electroweak precision data, ρ = 1.00038(20) [86], which is 1.9σ above the SM expectation at tree level leads to an upper bound of (ii) The electroweak precision data observables, namely S, T and U parameters tightly constrain the mass-splittings requiring |∆m| 40 GeV [9,11,49,87].

Production and decays of triplet scalars
The TeV scale triplet-like scalars are pair produced copiously at the LHC by quarkantiquark annihilation via the neutral current and charged current Drell-Yan mechanisms: 3 We implement the model in SARAH [94,95] to generate UFO modules, and use MadGraph [96,97] with the NNPDF23_lo_as_0130_qed parton distribution function [98,99] for numerical evaluation of the leading order (LO) production cross-sections of the triplet 2 Some of the neutrino oscillation parameters, namely the lightest neutrino mass and the CP phases, are either poorly measured or hitherto not measured. In this work, we set the phases to be zero for simplicity. However, note that these parameters could substantially change the leptonic decays and thereby the phenomenology of the triplet-like scalars [22,37]. 3 Also, the triplet-like scalars are produced via t/u-channel photon fusion [90,91] and vector boson-fusion processes [23,92,93], with two associated forward jets at the LHC. However, their production through such processes is sub-dominant for the mass range of our interest, and thus neglected. That said, the photon fusion process become important for large masses of the triplet scalars, and thus, entail to be incorporated into the analyses for multi-TeV scalar masses at high-energy LHC.
Refs. [25,100] have estimated the QCD corrections to the production of doubly charged scalars at hadron colliders which result in a next-to-leading (NLO) K-factor of 1.2-1.3. Considering that the QCD corrections to the production of singly charged scalars are similar to those of doubly charged ones, we apply an overall QCD K-factor of 1.25 to the LO cross-section.
We next discuss the decays of the triplet-like scalars. Their decays have been well studied in the literature [22,40,[101][102][103][104]. 4 The doubly charged scalars have three possible decay modes: (i) leptonic decay, i.e. ± ± , (ii) gauge boson decay, i.e. W ± W ± , and (iii) cascade decay, i.e. H ± W ± * . The latter decay mode kinematically opens up only for ∆m > 0. For m 2 H ±± m 2 W , the ratio of the braching fractions for these modes are obtained as H ±± decays into ± ± and W ± W ± for ∆m < O(1) GeV. These two decay modes are comparable for v t ∼ O(10 −4 ) GeV, and the former dominates over the latter for v t < 10 −4 GeV and vice versa. The cascade mode starts to contribute for ∆m O(1) GeV and become dominant for large ∆m.

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The CP-odd (CP-even) heavy neutral scalar has four decay modes: (i) leptonic decay, i.e. νν, (ii) hadronic decay, i.e. qq with q b, t, (iii) diboson decay, i.e. h 0 Z (W W ,ZZ,h 0 h 0 ), and (iv) cascade decay, i.e. H ± W ∓ * . The latter decay mode kinematically opens up only for ∆m < 0. For m 2 H 0 m 2 W , the ratio of the braching fractions for νν, qq, h 0 h 0 , W W , ZZ and H ± W ∓ * decay modes of H 0 can be estimated as Likewise, the ratio of the braching fractions for νν, qq, h 0 Z and H ± W ∓ * decay modes of A 0 are evaluated as decays into neutrinos and hadrons/dibosons, respectively, for v t < 10 −4 GeV and v t > 10 −4 GeV. Further, the diboson mode dominates over the hadronic one for m 2 The cascade mode starts to contribute for −∆m O(1) GeV, and shortly dominates for larger −∆m.

Collider searches
Profuse Drell-Yan production of the triplet-like scalars and their subsequent prompt decays 5 to SM particles lead to a variety of final state signatures at the LHC. Possible final states include smoking gun signatures like two pairs of same-sign lepton or two pairs of same-sign W -boson. Phenomenological consequence of the present model at the LHC has been studied extensively in the literature [9-48, 48-54, 90, 92, 105-109]. Rightfully, central attention of most of those studies pivots around the doubly charged scalars because of their distinct decay signatures. For the very same reason, both the CMS and ATLAS collaborations have carried out a number of collider searches at the LHC [73][74][75][76][77][78][79][80][81][82]. Hitherto no significant excess over the SM background expectations has been observed in any of these direct seraches. These seraches thereupon have set stringent limits with 95% CL on the masses of the doubly charged scalars. As argued in section 1, these limits are not befitting to the entire model parameter space. Also, these limits are often conservative as these searches do not incorporate all Drell-Yan production channels for the triplet-like scalars. Furthermore, most of these limits are derived in the context of simplified scenarios without reckoning the footprints of the low-energy neutrino parameters.
The quartic scalar interaction λ 4 Φ † ∆∆ † Φ entitles the triplet components to split in mass. For moderate v t and passably large ∆m, cascade decays quickly dominate over the leptonic and diboson modes. Not only does the mass-splitting overwhelm the decays of the triplet-like scalars, but it also affects their production cross-sections at the LHC. Thus, the

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pheneomenology for the non-degenerate scenario substantially is contrasting than that for the degenerate one [39-42, 45, 46]. We next briefly discuss the possible final state signatures, and outline already existing direct collider searches by CMS and ATLAS which are potentially sensitive in constraining different parts of the model parameter space.
Degenerate scenario. All the Drell-Yan production mechanisms for the triplet-like scalars except H + H − are of sizeable cross-sections. For v t < O(10 −4 ) GeV, H ±± , H ± and H 0 /A 0 decay to ± ± , ± ν and νν, respectively. Production of H ±± H ∓ and H ++ H −− lead to, respectively, three and four light leptons (e, µ) in the final state. Though H ± H 0 /A 0 and H 0 A 0 have sizeable cross-sections, they fall through to complement the multilepton final state because of their invisible decays. The already existing multilepton searches by CMS and ATLAS in refs. [78,79,110,111] are expected to constrain this part of the parameter space. For All the production channels give rise multiboson leading into multilepton final states. Therefore, one anticipates this part of the parameter space to be probed by the existing multiboson leading into multilepton searches by ATLAS in refs. [81,82].

Negative scenario. For ∆m
O(1), this scenario resembles the degenerate one. For passably large ∆m and moderate v t , the cascade decays H 0 /A 0 → H ± W ∓ * and H ± → H ±± W ∓ * dominate over the other decays, thereby enhancing the effective production crosssection for H ±± . Then, depending on v t , H ±± decays into ± ± and/or W ± W ± . Therefore, this scenario can be probed using the multilepton searches in refs. [78,79,110,111] and/or multiboson leading into multilepton searches in refs. [81,82].
Positive scenario. Again, this scenario resembles the degenerate one for small ∆m. For passably large enough ∆m and moderate v t , the cascade decays H ±± → H ± W ± * and H ± → H 0 /A 0 W ± * dominate over the other decays. This enhances the effective production cross-section for H 0 and A 0 . For v t > 10 −4 GeV, H 0 and A 0 decay to ZZ/W W/h 0 h 0 and h 0 Z, respectively. This gives rise to multiboson final state signatures. Therefore, one expects this part of the parameter space to be probed by the existing ATLAS searches in refs. [81,82]. 6 For v t < 10 −4 GeV, both H 0 and A 0 decay invisibly into neutrinos. The relevant production mechanisms H ±± H ∓ and H ++ H −− yield soft leptons or jets resulting from the off-shell W -bosons and neutrinos. Being very soft, these final state leptons/jets are very difficult to reconstruct at the LHC. Therefore, in this scenario, the most optimistic final states would be an energetic jet resulting from initial state radiation plus large missing

Multilepton final states search by CMS [110]
The CMS collaboration has published a multilepton final states search [110] with an integrated luminosity of 137.1 fb −1 of pp collisions at √ s = 13 TeV. This search targeted the triplet fermions in the type-III see-saw model [117]. However, because of similar multilepton final state signatures, this search is conjectured to be sensitive in probing the type-II see-saw model. Hitherto, there is no multilepton search targeting the type-II see-saw model using the full Run-2 dataset by CMS or ATLAS. Thereupon, we set forth to implement this search meticulously.
We simulate the signal events using MadGraph [96,97] with the NNPDF23_lo_as_0130_qed parton distribution function [98,99]. The subsequent decays, initial state radiation (ISR), final state radiation (FSR), showering, fragmentation and hadronisation are simulated with PYTHIA [118]. Hadronized events are passed into Delphes [119] for object reconstruction and selection, defining signal regions and event selection. In doing so, we rigorously follow the search strategy in ref. [110]. Lastly, we use a hypothesis tester which uses a library of C++ classes RooFit [120] in the ROOT environment to estimate CL.
The selected events are categorised into several mutually exclusive signal regions (SRs), namely 3LOSSF0, 3LOSSF1, 4LOSSF0, 4LOSSF1 and 4LOSSF2, based on the multiplicity of light leptons, the multiplicity and mass of opposite-sign same-flavour (OSSF) lepton pairs, N OSSF and M OSSF . The names of the SRs are self-explanatory, see ref. [110] for details. The events in the 3LOSSF1 SR are further classified as 3L below-Z, 3L on-Z and 3L above-Z when M OSSF is below, within and above the Z-boson mass window (M Z ± 15), respectively. All the SRs are further divided into several independent signal bins using a primary kinematic discriminant, thereby resulting in 40 signal bins in total. For 3L on-Z SR, this search uses transverse mass (M T ), 8 as the primary discriminant, whereas for all other SRs, scalar sum of the transverse momenta of all charged leptons (L T ) plus the missing transverse momentum (p miss T ) is used. These variables, exploiting the relatively high momenta of the decay products, are found to be useful in discriminating the signal from the background. For the detailed description of our implementation of this search [110], see refs. [121,122]. The implementation of this search has been validated in 7 It turns out that both the monojet search by ATLAS and soft leptons search by CMS fail to constrain this scenario. Monojet search usually requires a larger signal cross-section to suppress the vast SM background and is thus usually applicable to the strongly produced particles decaying into soft leptons/jets plus missing particles. Furthermore, the soft lepton final states are suppressed by W -leptonic branching fractions. Inconsiderably small signal cross-section compared to the SM background in the present scenario makes the same very challemging to probe. 8 The transverse mass is defined as MT = 2p miss  previously published work [121] (see figure 10 in ref. [121]). This successful implementation enables us to use the distributions of expected SM backgrounds and observed events in ref. [110] to constrain the type-II see-saw scalars in various v t -∆m regions. Figure 1 shows the L T + p miss T distributions of the expected SM background events (histograms with black line), 9 the observed events (big black dots) and the expected signal events corresponding to 137.1 fb −1 integrated luminosity data at the 13 TeV LHC for three SRs-3L above-Z (left), 3OSSF0 (middle) and 4OSSF0 (right). For brevity, we do not show similar distributions for the other SRs. The magenta dotted, dark yellow solid and blue dashed histrograms show the expected signal events for three benchmark masses -600, 800 and 1000 GeV for v t = 10 −8 GeV 10 and ∆m = 0 assuming NH neutrino mass spectrum with m 1 = 0.03 eV. 11

Multiboson leading to multilepton final states search by ATLAS [82]
The ATLAS collaboration has recently published a search for doubly and singly charged Higgs bosons decaying into vector bosons in multilepton final states with an integrated luminosity of 139 fb −1 of pp collisions at √ s = 13 TeV [82]. As mentioned earlier, this AT-LAS search considered either pair or associated production modes for the doubly charged scalars, but not both at once. Also, this search does not incorporate the Drell-Yan production channels for the singly charged and neutral triplet-like scalars. Thus, the limits set by this ATLAS search are conservative. Furthermore, these limits are not befitting to the entire parameter space, rather valid only for ∆m = 0 and v t > O(10 −4 ) GeV. Therefore, we set forth to recast this search by incorporating all the Drell-Yan production modes for the triplet-like scalars to constrain them for a vast v t -∆m region. For the implementation, we minutely follow the search strategy in ref. [82].
After object reconstruction and selection (see ref. [82]), the events are categorised into three mutually exclusive analysis channels, namely same-sign dilepton (2 sc ), three 9 The gray bands represent the total (systematic + statistical) uncertainty on the expected SM background. 10 The CMS multilepton search in ref. [110] is designed to probe final states with hard-pT leptons, and hence sensitive to small vt region where the scalars directly decay to leptons, and results into hard signal leptons in the final state. 11 For NH, 0.03 eV is the maximum possible value for the lightest neutrino mass consistent with the bound from cosmology.   For defining SRs, several other variables such as the invariant mass of all selected leptons, the invariant mass of all the jets, the distance between two same-sign leptons in the η-φ plane, the azimuthal distance between the dilepton system and p miss T , the smallest distance between any lepton and its closest jet in the η-φ plane, etc. are used (see ref. [82] for details). 12 These variables, exploiting the boosted decay topology of the triplet-like Higgs bosons as well as the high energy of their decay products, are useful to discriminate between the signal and the background. Events in the 2 sc SRs are further divided into ee, µµ and eµ final states, whereas those in the 3 SRs are separated into two categories (3 1 and 3 0) based on whether or not an OSSF lepton pair exists in the event. This enhances the sensitivity of this search by distinguishing the lepton-flavour composition between signal and background.
We, then, move forward to validate our implementation of this search by reproducing the ATLAS 95% CL bound on the total pair production cross-section times branching fraction for two scenarios corresponding to the pair production H ±± H ±± and the associated production H ±± H ∓ in ref. [82]. The left (right) plot in figure 2 shows the ATLAS observed and expected 95% CL upper limits on the H ±± H ±± (H ±± H ∓ ) production cross-section 12 Though different sets of kinematic variables and selection cuts are used to define the SRs (SR1, SR2, SR3 and SR4 ), they are not mutually exclusive. They are designed by optimising the sensitivity for the H ±± pair production mode, respectively, for the m H ±± = 200, 300, 400 and 500 GeV mass hypotheses. ref. [82] considers SR1 (SR2 ){SR3 }[SR4 ] for 200-300(300-400){400-500}[> 500] GeV mass hypothesis. Here, we differ from ref. [82]; for a given mass hypothesis, we conider all the SRs disjointly, and eventually, choose the most sensitive one. However, for the validation of our implementation, we adhere to the ref. [82]'s approach.  times branching fraction. The green and yellow bands represent the expected exclusion curves within one and two standard deviations, respectively. The NLO QCD corrected [25] theoretical prediction is shown by the solid red curve. The reproduced 95% CL upper limit is represented by the blue dashed curve. The reproduced result is found to be in agreement with the ATLAS one, thereby validating our implementation of this search. This entitles us to use the distributions of expected SM backgrounds and observed events in ref. [82]. Figure 3 shows the expected SM background events(histograms with black line), 13 the observed events (big black dots) and the expected signal events corresponding to 139 fb −1 integrated luminosity data at the 13 TeV LHC for four different SRs -SR1, SR2, SR3 and SR4. For each SR, the yields are shown for all the relevant channels, namely ee, eµ, µµ, 3 0, 3 1 and 4 . The magenta dotted, dark yellow solid and blue dashed histograms show the expected signal events for three benchmark masses -200, 350 and 500 GeV for v t = 1 GeV and ∆m = 0 assuming NH neutrino mass spectrum with m 1 = 0.03 eV.

95% CL lower limit on m H ±±
In view of the observations being consistent with the SM background expectations, we derive limits on m H ±± using the LHC searches. In what follows, we present stringent limits with 95% CL on m H ±± for a wide range of v t and ∆m using the above-described CMS multilepton and ATLAS multiboson leading into multilepton searches. In deriving the present limits, we incorporate all the Drell-Yan production modes for the triplet-like scalars. Table 1 displays five benchmark points in different region of the model parameter space along with their exclusion significances from both the CMS and ATLAS searches. Also displayed is the exclusion significances from the CMS and ATLAS combined search. These benchmark points corroborate that the exclusion limits vary significantly across different regions of the v t -∆m parameter space. The ATLAS and CMS searches fail to probe the triplet-like scalars with mass as low as 200 GeV for the v t -∆m region characterised by BP5, whereas for that characterised by BP4, the limit on m H ±± is significantly larger than the previous ones.
The left plot in figure 4 shows 95% CL lower limits on m H ±± as a function of v t for ∆m = 0 assuming NH neutrino mass spectrum with m 1 = 0.03 eV. The khaki shaded region (on the left) is excluded from the ρ parameter measurement from the electroweak precision data, whereas the coral shaded region (on the right) is excluded from the lepton flavour violating decay constraints. The dark goldenrod and pink shaded regions are excluded, respectively, from the CMS multilepton and the ATLAS multiboson leading to multilepton searches. For small v t , the triplet-like scalars with masses below 950 GeV are excluded from the CMS search. This exclusion limit is stronger than those from the previous LHC searches [78,79] by approximately 200-230 GeV. For large v t , the above-described ATLAS search excludes the triplet-like scalars up to 400 GeV masses which is stronger by approximately 50 GeV than the ATLAS limit in ref. [82]. Given that the CMS and ATLAS searches are mutually exclusive, it is reasonable to combine them. That said, as these two searches are primarily designed to target different regions in the parameter space, viz.
small v t and large v t , we expect only marginal improvement on the limits while combining them. The purple shaded region shows excluded parameter space when these two searches are combined.  On the contrary, for H 0 /A 0 decaying into h 0 h 0 , ZZ/h 0 Z, the signal cross-section is small compared to the overwhelming background from either QCD jets or Drell-Yan processes. This makes such a scenario challenging to probe. Note that for v t ∼ O(10 −2 )-O(10 −3 ) GeV, the ATLAS search manages to put some bounds in the ∆m = 30 case, but it fails in the ∆m = 10 case. This is because for larger ∆m, some of the leptons from the off-shell W ± 's pass the object reconstruction and selection criteria to contribute to the signal yields, whereas the leptons are too soft to do so for smaller ∆m. As one approaches towards small v t , the leptonic decays retrieve their dominance over the cascade one, and give rise to multilepton final states; this occurs at v t ∼ O(10 −6 ) and O(10 −7 ) GeV, respectively, for ∆m = 10 and 30 GeV. This has been reflected in the middle plot. On the contrary, in negative scenario, H ± and H 0 /A 0 decay into off-shell W ± 's and H ±± , thereby enhancing the effective production cross-section for H ±± . Therefore, in such a scenario, the limit gets enhanced compared to the degenerate case, see the right plot in figure 4. For ∆m = −10(−30) GeV, the exclusion limit extends up to 1115(1076) GeV compared to 955 GeV for ∆m = 0. Note that for a given m H ±± , H ± and H 0 /A 0 are lighter in the ∆m = −10 GeV case compared to those in the ∆m = −30 GeV case. Thus, the signal cross-section is larger for ∆m = −10 GeV than for ∆m = −30 GeV. This explains the stronger limits for ∆m = −10 GeV than ∆m = −30 GeV.

Proposed multilepton final states search for small v t
For small v t and ∆m = 0, the triplet-like scalars up to 950 GeV masses are excluded from the CMS multilepton search with 139 fb −1 of data, see figure 4. Given the small signal crosssection for m H ±± > 1 TeV and comparatively large background in the afore-discussed CMS multilepton search, a similar search at high-luminosity is deemed non-optimal in probing the triplet-like scalars much heavier than 1 TeV. Also, the said search, which results in the most stringent limits in the small v t region, is not designed to probe the triplet-like scalars. In this section, we design a multilepton search that is optimised to probe the triplet-like scalars much heavier than 1 TeV in the small v t region. In what follows, we give a brief description about reconstruction and selection of various objects (jets, leptons, etc.), event selection and classification of selected events into mutually exclusive signal regions (SRs) for our proposed multilepton final states search.
Object reconstruction and selection. Different physics objects, viz. jets, electrons, muons and missing transverse energy, are reconstructed in Delphes [119]. Jets are reconstructed using the anti-kT algorithm [123] with a distance parameter ∆R = 0.4 as implemented in the FastJet package [124]. Reconstructed jets are required to have transverse momentum p T > 30 GeV within the central pseudorapidity range |η| < 2.5. Electron (muon) candidates with p T > 10 GeV and |η| < 2.5(2.4) are considered for further analysis. For the electron candidates within barrel (endcap), we demand a maximum 5-10% (5-15%) p T -dependent relative isolation with ∆R = 0.4, 14 whereas we demand a maximum 15% relative isolation with ∆R = 0.4 for muons. In addition, the following set of lepton displacement requirements on the transverse and longitudinal impact parameters, d z and d xy , with respect to the primary vertex are enforced. For electron candidate within barrel (endcap), we demand d z < 1 mm and d xy < 0.5 mm (d z < 2 mm and d xy < 1 mm), whereas muon candidates require d z < 1 mm and d xy < 0.5 mm. Lepton isolation, which trims hadronic activity inside the isolation cone, along with impact parameter requirements suppress the reducible backgrounds such as Z+jets and tt+jets, where a jet is misidentified as lepton or additional leptons originate from heavy quark decays. Finally, the missing transverse momentum p miss T is estimated as the magnitude of the negative vector sum of the transverse momenta of all reconstructed particle-flow objects in an event.
Overlaps between reconstructed objects resulting in ambiguities among them lead to object double counting. To sidestep that, all selected jets within a cone of ∆R < 0.4 of a selected lepton are thrown away. In addition, all selected electrons within a cone of ∆R < 0.05 of a selected muon are discarded as these are likely due to bremsstrahlung interactions of the muon with the inner detector material. Some of the jets, especially those on the tail of the detector response, and single pions could mimic lepton signatures and could be misidentified as leptons. Though the composition of the fake-lepton background differs substantially among the analysis channels, without going into the intricacy of modelling the fake-lepton contributions, we straightforwardly take the probability of 0.1-0.3% [125] JHEP03(2022)195 for a jet to be misidentified as a lepton. Furthermore, bremsstrahlung interactions of the electrons with the inner detector material could lead to charge misidentification. The radiated photon converts to e − e + pair near the primary electron trajectory leading to charge misidentification ambiguity. Also, the photon could traverse the inner detector without creating any track. In such a case, the electron usually has a short lever arm on its curvature. This could lead to incorrect determination of the electron charge. We adopt the charge misidentification probability from ref. [126]: is found to be 0.02-0.1 and f (η) is found to be 0.03-1 such that P (p T , η) ranges from 0.02% to 10%. Note that the high-p T electrons are more likely to be affected by charge misidentification as they have almost straight tracks, thereby making the curvature measurement very challenging. Also, the electrons with larger η have a larger misidentification probability as they traverse through a higher amount of inner detector material.

Event selection and signal region definition.
Events with three or more light leptons are considered for this search. Events containing a lepton pair with ∆R < 0.4 or a sameflavour lepton pair with invariant mass below 12 GeV are vetoed. This suppresses background contributions from final-state radiations as well as low-mass resonances -Drell-Yan processes and neutral mesons. Furthermore, events containing a same-flavour lepton pair with an invariant mass within the nominal Z-boson mass window, i.e. M Z ± 15 GeV are discarded. 15 This suppresses background contributions from the Z → * → γ(→ ) process as well as the W Z production. Events with exactly three light leptons (3L) in one category and four or more light leptons (4L) in another category are considered for further analysis.
Noting that the triplet-like scalars, which are to be probed, are heavier than 1 TeV, we persuade to exploit the relatively high momenta of their decay products. Before continuing, let us briefly reckon the processes contributing to the 3L and 4L signal events. For 3L events, the dominant contribution arises either from the H ±± H ∓ → ± ± ∓ ν process or from the H ++ H −− → ± ± ∓ τ ∓ process with τ ± decaying hadronically. Therefore, the invariant mass distribution of the same-sign lepton pair is expected to peak at m H ±± . One would expect high-p T leptons, large p miss T and no high-p T jet (except for those coming from ISR and FSR) in the final states for the former. For the latter, one would expect final states with high-p T leptons, small p miss T and at least one high-p T jet. Then, the dominant contribution to the 4L signal events comes from the H ++ H −− → + + − − process. The invariant mass distributions of both the same-sign lepton pairs are expected to peak at m H ±± . Once again, one would expect high-p T leptons and small p miss   SM processes like Z/γ * +jets, tt+jets, etc., where a jet is misidentified as lepton or additional leptons originate from heavy quark decays. The irreducible ones are from diboson and triboson production and processes like ttW , ttZ and Higgs boson production, etc. Note that final state events with n leptons also contribute to those with n − 1 leptons when one of the leptons falls outside the detector coverage (in the high rapidity region) or is too soft to pass the object reconstruction and selection criteria or gets misidentified by the detector. All the background events are generated in association of up to two jets using MadGraph [96,97] at the leading order using the 5 flavour scheme followed by MLM matching in PYTHIA [118], and the corresponding cross-sections are taken at least upto NLO [127][128][129][130][131][132][133][134][135][136][137][138][139].
We plot different kinematic distributions for 3L events in figure 5 for a benchmark point  events, we require one same-charge lepton pair. The leading (subleading) lepton in the pair is required to have p T > 300(100) GeV. We discard events with m eff < 1500 GeV. To enhance the sensitivity of this search, the selected events are categorised into two mutually exclusive SRs, namely 3L0J 17 and 3L1J, based on whether or not at least one selected jet exists in the event. 3L1J events are further classified as 3L1J-1 and 3L1J-2 based on whether p miss T is larger or smaller than 150 GeV. The 3L0J events with p miss T < 150 GeV or m sc < 800 GeV are thrown away to get rid of the sizeable SM backgrounds. Furthermore, we reject 3L1J-1 events with p miss T /H T < 1.0. The cut on p miss T /H T turns out to be remarkably effectual in reducing the leftover backgrounds. Finally, to supplement the sensitivity of this search, the selected events in 3L1J-1 and 3L1J-2 SRs are divided into six bins each in the [600:1800] GeV range using m sc as the primary kinematic discriminant. 18 Different kinematic distributions for 4L events are plotted in figure 6 for BP1. The leftmost and middle plot in the top panel shows the transverse momentum distributions of the leading and subleading lepton in the positive-charge lepton pair. The leptons in the negative-charge lepton pair have similar p T -distributions, we avert to show them for brevity. The distributions of the scalar p T sum of the leptons is shown in the rightmost plot in the same panel. It is evident from these kinematic distributions that relatively stronger cuts on the same-sign leptons' p T is useful in suppressing the relevant backgrounds. Further, a cut on L T turns out to be efficacious in reducing the remaining background. For 4L events, we require two same-charge lepton pairs. The leading (subleading) lepton in both the pairs are required to have p T > 300(100) GeV. The events with L T < 1500 GeV are vetoed. Further, we require r = |m sc1 − m sc2 |/(m sc1 + m sc2 ) < 0.1, where m sc1 and m sc2 are the invariant masses of the same-charge lepton pairs. The last cut ensures correct pairing of the leptons.
Number of expected signal and background events in different signal regions after passing various selection cuts for BP1 for 1000 fb −1 of luminosity data at the 13 TeV LHC is are given in table 2. The 3L0J and 4L signal regions are free from any background, whereas some backgrounds remain after all the selection cuts in the other two signal regions. 17 Three leptons events with no reconstructed jet with pT > 30 GeV are considered in the 3L0J SR. 18 The overflow (underflow) events are contained in the last (first) bin in each signal region.   Figure 7 shows m sc distributions of signal and background events in 3L1J-1 signal region after passing various selection cuts for BP1 for 1000 fb −1 of luminosity data at the 13 TeV LHC. For brevity, we avert to show similar distribution for the 3L1J-2 signal region. This simple binning of the selected events enhances sensitivity of these two signal regions.

Future 95% CL lower limit on m H ±±
In this section, we present our forecasted 95% CL lower limits on m H ±± by using the ATLAS search [82] scaled at high-luminosity 19 as well as our proposed search described in section 4.4. We simply presume that not only the detector efficiencies and acceptances 19 The ATLAS search in ref. [82] is optimised for probing the large vt region where the type-II see-saw anchors decay into bosons. Thus, we use the same search strategy to forecast the future reach of the LHC in probing this part of the parameter space. but also the background uncertainties remain the same while scaling the ATLAS search at high luminosity. Given that both statistical and systematic contributions to the background uncertainties are expected to be reduced with increasing volume of LHC data, our forecasted limits are conservative. Also, to ensure robustness in statistical interpretations, we replace the less than one per-bin expected background yield at 3000 fb −1 , 20 with one background yield. For the proposed search, we assume an overall 20% total uncertainty on the estimated background.
The grey shaded region in figure 8 is excluded from the existing ATLAS and CMS combined search, see section 4.3. The regions below the green and cyan curves are expected to be excluded from the ATLAS search scaled at 500 and 3000 fb −1 of luminosity, respectively. Our proposed search is expected to probe the regions below the goldenrod and pink curves, respectively, at 500 and 3000 fb −1 of luminosity. For small (large) v t , the future reach extends up to 1220 and 1490 (520 and 640) GeV, respectively, for 500 and 3000 fb −1 of luminosity. We consider both the NH and IH neutrino mass spectrum while varying the lightest neutrino mass in accordance with the bound from cosmology, i m i < 0.12 eV. The effect of different possible neutrino mass hypotheses on the limits are reflected as bands for small v t regions. This is because, for small v t , the triplet-like scalars decay leptonically, and these decays are driven by the Yukawa couplings, which, in turn, are determined by the neutrino oscillation parameters up to v t . For large v t , the triplet-like scalars decay into diboson and hadrons, and these decays are independent of the Yukawa couplings and the neutrino oscillation parameters. The solid curves within the bands correspond to NH with m 1 = 0.03 eV. The plots in figure 9 show 95% CL future sensitivity of the LHC to probe as a function of v t assuming NH with m 1 = 0.03 eV for four 20 All the relevant backgrounds are generated in association of up to two jets using MadGraph [96,97] at the leading order using the 5 flavour scheme followed by MLM matching in PYTHIA [118] for 3000 fb −1 or more luminosity, and the corresponding cross-sections are taken at least upto NLO [127][128][129][130][131][132][133][134][135][136][137][138][139].

Summary and outlook
The type-II see-saw mechanism based on the annexation of the Standard Model by weak gauge triplet scalar field proffers a natural explanation for the very minuteness of neutrino masses. Because of the presence of the doubly charged scalar bosons and their illustrious signatures, a number of collider searches have been carried out at the LHC by CMS and ATLAS to look for the same. In view of the observations being consistent with the SM background expectations, these searches derived stringent limits with 95% CL on m H ±± . Most of these limits are derived in the context of simplified scenarios without reckoning the footprints of the low-energy neutrino parameters. Furthermore, the limits reported -21 -

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by ALTAS and CMS are often conservative as these searches do not incorporate all the Drell-Yan production channels for the triplet-like scalars. As discussed in section 3, other Drell-Yan processes such as pair and associated production of the triplet-like neutral scalars are also of sizeable cross-sections, thus, these too entail to be considered into the analyses. Therefore, the inclusion of these production processes, which are forsaken otherwise by both ATLAS and CMS, results in stronger bounds than those reported by them. Above all, in the non-degenerate scenario, the cascade decays are entitled to play a notable role in the phenomenology, thereby making the phenomenology for the non-degenerate scenario substantially contrasting than that for the degenerate one. Evidently, the ATLAS and CMS reported limits are not befitting to the entire parameter space, rather valid only for a constrained parameter space of the model. Bearing this discussion in mind, we perform a comprehensive study for a wide range of the model parameter space parametrised by v t , ∆m and m H ±± . Considering all the Drell-Yan production mechanisms for the triplet-like scalars and taking into account the all-encompassing complexity of their decays, we derive the most stringent 95% CL lower limit on m H ±± for a vast range of v t -∆m parameter space by implementing already existing direct collider searches by CMS and ATLAS. Further, we forecast future limits by extending the same ATLAS search at high-luminosity, and we propose a search strategy that yields improved limits for a part of the parameter space. To the best of our knowledge, such a study of up-to-the-minute collider limits for a vast range of parameter space is still lacking. This work is intended to fill this gap. In closing this section, we summarise the findings of this work.
(i) For ∆m = 0 and large (small) v t , doubly charged scalars with masses below 420(955) GeV are excluded from the existing ATLAS and CMS combined search. These exclusion limits are approximately 50-230 GeV stronger than those from the previous LHC searches [78,79,82]. This is attributed to the inclusion of all the Drell-Yan production processes in our analyses which is not the case for the LHC searches.
(ii) For large enough negative ∆m and moderate v t , the recasted limits extend up to 1115 GeV, which is 360-390 GeV stronger than those reported by ATLAS and CMS. In this region of parameter space, H ± and H 0 /A 0 decay to H ±± , thereby enhancing the effective production of the latter. This results in improved limits compared to the ∆m = 0 case.
(iii) For large enough positive ∆m, triplet-like Higgs as light as 200 GeV or even lighter is still allowed by the LHC data. In this region of parameter space, H ± and H ±± decay to H 0 /A 0 , thereby enhancing the effective production of H 0 /A 0 , which then decays invisibly into neutrinos or into h 0 h 0 , ZZ/h 0 Z depending on the value of v t . For H 0 /A 0 decaying into neutrinos, there are hardly visible objects in the final state, so much as the monojet search by ATLAS [112] and the soft leptons search by CMS [115,116] fall short in constraining this part of the parameter space. On the contrary, for H 0 /A 0 decaying into h 0 h 0 , ZZ/h 0 Z, the signal cross-section is small compared to the overwhelming background either from QCD jets or Drell-Yan processes. This makes -22 -

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such a scenario challenging to probe at the LHC. Note that e − e + colliders could have better prospects for probing such a nightmare scenario, which we left for future work.
(iv) For ∆m = 0 and large v t , the expected reach of the ATLAS search at 3000 fb −1 is 640 GeV.
(v) For ∆m = 0 and small v t , the expected reach from our proposed search extends up to 1490 GeV at 3000 fb −1 , while for ∆m = −10(−30) GeV and moderate v t , the reach is 1555(1550) GeV.