Same-Sign Tetralepton Signature at Large Hadron Collider, and future $pp$ Collider

We analyze a novel signature of the type II seesaw model - same-sign tetra-lepton signal arising from the mixing of neutral Higgs bosons and their subsequent decays to singly and doubly charged Higgs bosons. For this, we consider wide ranges of the triplet vacuum expectation value (vev) and Yukawa couplings, that are consistent with the observed neutrino masses and mixing as well as the LHC search limits. We find that a doubly charged Higgs boson with mass around 250 GeV and triplet vev around $10^{-4}-10^{-2}$ GeV can give significantly large number of events through it decay to same-sign $W$ gauge bosons at High-Luminosity LHC with $3000 \text{fb}^{-1}$ of data. We also pursue the analysis for a future hadron collider with the c.m. energy of 100 TeV. Considering a heavy Higgs boson around 900 GeV and an intermediate region of the triplet vev, where both same-sign dilepton and gauge boson decays can occur, we identify a limited range of the parameters where the number of same-sign tetra-lepton events are as large as 1000.

appealing feature of this model. Hence a discovery of this exotic particle will be a smoking gun signature of this model.
Most of the works in the literature explored di-lepton or gauge boson decay modes of the doubly charged Higgs, leading to multi-lepton final states. Due to the possible cascade decays of the charge neutral Higgs into a singly charged Higgs, and the cascade decay of a singly charged Higgs into a doubly charged Higgs, the model can also lead to a very unique signature, same-sign tetra-lepton final states. This has been first proposed in [53], and explored for the lower triplet vev, where di-lepton decay is pre-dominant. In this work, we consider a wide range of triplet vev, in particularly, focussing on gauge boson decay modes, and explore the signature for 14 TeV LHC. For higher range of triplet vev, as the LHC constraint on the mass of doubly charged Higgs is relatively relaxed, we therefore perform the analysis for lighter Higgs state, as low as M H ±± ∼ 247 GeV. In addition, we also consider a very high energy pp collider, that can operate with c.m.energy √ s = 100 TeV, and explore this unique signature for a heavy doubly charged Higgs. We show that for heavier doubly charged Higgs, there is a very narrow region of triplet vev, which can accommodate significantly large O(10 3 ) same-sign tetra-lepton signatures.
Our paper is organized as follows: we briefly review the basics of the type-II seesaw model in Sec. 2. In Sec. 3, we discuss branching ratios of doubly and singly charged Higgs, and the relation between H ±± and H ± decays. In Sec. 4, and in Sec. 5, we present the simulation of same-sign tetra-lepton signal at √ s = 14 TeV LHC, and √ s = 100 TeV. Finally, we present our conclusions in Sec. 6.

Model Description
One of the most simplest seesaw models is the type-II seesaw model [11][12][13][14], that, in addition to the SM particle contents, also contains one SU (2) L triplet Higgs field The neutral components of the SM doublet (Φ) and triplet Higgs fields are denoted as The kinetic term for the triplet has the following form The new triplet scalar field ∆, being a triplet under SU (2) L interacts with the SM gauge bosons. In addition to the kinetic term, ∆ has Yukawa interaction with the SM lepton doublet. The Yukawa interactions of ∆ with the lepton fields are where Y ∆ is a 3 × 3 matrix and c denotes charge conjugation. The scalar potential of the Higgs fields Φ and ∆ is where m Φ andM ∆ are real parameters with mass dimension 1, and λ, λ 1−4 are dimensionless quartic Higgs couplings. Note that, µ is the parameter with positive mass dimension. The triplet field ∆ carries lepton number +2 and hence the Yukawa term conserves lepton number. However, the lepton number is violated 2-units by a non-zero µ. Therefore, together a non-zero µ and a non-zero Y ν violate lepton number symmetry. The scalar potential that generates scalar mass matrix, includes tri-linear as well as quartic couplings among the scalar fields. The scalar mass matrix, after diagonalization, generates seven physical Higgs states. They are: the charged Higgs bosons H ±± , H ± , and the neutral Higgs bosons h 0 , H 0 and A 0 . The two charged scalar fields Φ ± of Φ and ∆ ± of ∆ mix to give singly-charged states H ± and the charged Goldstone χ ± bosons. Similarly, the mixing between the two CP-odd fields (χ 0 and η 0 ) gives rise to A 0 , and the neutral Goldstone boson ρ 0 . Finally, we obtain the SM Higgs boson (h) and a heavy Higgs boson (H) via the mixing of the two neutral CP-even states Φ 0 and δ 0 . For the detail description of the charged and neutral mass matrix, see [16].
The minimization conditions of the potential are These give the following conditions for m 2 Φ , M 2 : The diagonalization conditions for the neutral and charged scalar fields are, where the mixing angles . (2.8) All these mixings being proportionl to the ratio of v ∆ v Φ is very small. The physical masses of the doubly and singly charged Higgs bosons H ±± and H ± can be written as (2.9) The CP-even and CP-odd neutral Higgs bosons h, and H have the physical masses m 2 h 0 = T 2 11 cos 2 α + T 2 22 sin 2 α − T 2 12 sin 2α, (2.10) m 2 H 0 = T 2 11 sin 2 α + T 2 22 cos 2 α + T 2 12 sin 2α. (2.11) In the above T 11 , T 22 and T 12 have the following expressions: The CP-odd Higgs field A 0 has the following mass The difference between H ±± and H ± masses is dictated by the coupling λ 4 of the scalar potential. For a positive λ 4 , the H ±± is lighter than H ± . The mass difference ∆M 2 is (2.14) Throughout our analysis, we consider the mass hierarchy M H ±± < M H ± . Among the neutral Higgs fields, we identify h 0 as the SM Higgs with mass M h 0 = 125 GeV. The mass of h 0 is primarily decided by λ, where the mass of H 0 is primarily decided by M ∆ . The neutral Higgs mixing angle α is very small, and hence, cos α 1. On the other hand, the charged Higgs and CP odd Higgs mixing angles tan β ± and tan β 0 being proportional v ∆ /v Φ , is very small, tan β ∼ 10 −3 . Note that, the mass square difference between H ± and Therefore, the mass difference between M H ±± , M H ± and the mass difference between M H 0 , M H ± are almost similar, and dictated by the same set of parameters λ 4 , and electroweak vev v Φ . The mass square difference betwteen H 0 and A 0 is extremely small, as this is proportional to the triplet vev, We denote the mass difference between H 0 and A 0 by M H 0 − M A 0 ∼ δM ∼ v ∆ , and the mass difference between H ± and H 0 by M H ± − M H 0 ∼ ∆M . As we will discuss in the next subsequent sections, the later parameter is important for few of the decay modes that depend on charged Higgs and neutral Higgs mass splitting, and is one of the key parameter for our discussion.
Due to the non-trivial representations of ∆, the Higgs triplet has interactions with a number of SM fermions and gauge bosons. This opens up a number of possible decay modes that can be explored at the LHC, and at other future colliders. In the next section, we summarise the different direct experimental constraints on the charged Higgs states.

Decay Modes and Experimental Constraints
We assume the neutral Higgs H 0 and A 0 are more massive than the charged Higgs. Among the charged Higgs, H ± is heavier than H ±± . The doubly-charged Higgs boson H ±± of this model can decay into the leptonic or bosonic states and gives unique signatures at high energy colliders. The partial decay widths and branching ratios of the H ±± depend on the triplet vev v ∆ . For smaller triplet vev, the H ±± predominantly decays into the same-sign leptonic states H ±± → l ± l ± , whereas for larger v ∆ , the gauge boson mode H ±± → W ± W ± becomes dominant [20,22,23]. The relevant decay widths are calculated to be, Here, M ν denotes the neutrino mass matrix, i, j are the generation indices, Γ l i l j and Γ W ± W ± are the partial decay widths for the H ±± → l ± i l ± j , and H ±± → W ± W ± channels, respectively. The parameter r W represents the ratio of H ±± and the W gauge boson masses, Other than the doubly charged Higgs, the model also contains a singly charged Higgs. The singly charged Higgs H ± can decay to lν, W Z, W h, tb final states. Additionally, for non-degenerate charged Higgs masses, and triplet vev v ∆ in between 10 −6 GeV and 10 −2 GeV, the cascade decay H ± → H ±± W * can also become dominant. The partial width for the charged Higgs decaying into H ±± W − * have these following form: In the above β ± is the charged Higgs mixing angle. For the expression of the function G and other partial decay widths of H ± into two fermion, gauge bosons, see [26]. We show the branching ratio of H ±± and H ± in Fig A number of searches have been proposed at the LHC to discover H ±± using multilepton signatures. The searches in [22][23][24]45] focussed on the pair and associated production with the H ±± decaying into leptonic, gauge boson states. Below we discuss the existing constraints on H ±± from LEP and LHC searches. • Constraints from pair and associated production: Stringent constraint on M H ±± have been placed by the 13 TeV LHC searches. These searches analysed H ±± → l ± l ± channel. The CMS collaboration looked for different leptonic flavors including ee, eµ, eτ, µµ, µτ and τ τ . In addition, the CMS searches also include the associated  production pp → H ±± H ∓ and the subsequent decays, H ± → l ± ν. This combined channel of pair-production and associated production gives the stringent constraint M H ±± > 820 GeV [39] at 95% C.L for e, µ flavor. The realistic bound depends on the neutrino mass matrix [20]. Similar constraint from ATLAS searches have been placed on the mass of doubly charged Higgs, that takes into account only pair-production. The bound is M H ±± > 870 GeV at 95% C.L [38]. Note that these limits are valid only for a small triplet vev v ∆ < 10 −4 GeV. Additionally, ATLAS looked into the pair-production of doubly charged Higgs, with subsequent decays into gauge bosons, resulting in multi-lepton final states. The search in [54], have constrained the mass of doubly charged Higgs M H ±± in between 200-220 GeV at 95% C.L. This is valid for the triplet vev v ∆ > 10 −4 GeV, where the gauge boson decay is most dominant.
• Constraint from VBF: For larger values of the triplet vev v ∆ > 10 −4 GeV, the leptonic branching ratio becomes smaller. Instead the decay mode H ±± → W ± W ± is dominant. Therefore the searches in vector boson fusion (VBF) become more important. A search for pp → jjH ±± → jjW ± W ± at the 8 TeV LHC in the VBF channel sets a constraint on the triplet vev v ∆ ∼ 25 GeV for M H ±± ∼ 300 GeV [40]. This constraint has been updated [41] using 13 TeV data at the LHC. Such a large triplet vev is anyway excluded by the ρ parameter bound [19] in the minimal type-II seesaw model.
The above mentioned constraints imply that a large range of triplet vev v ∆ > 10 −4 GeV exists, where low mass of M H ±± > 220 GeV is still allowed. For lower triplet vev v ∆ < 10 −4 GeV, the mass constraint is more conservative M H ±± > 870 GeV. In our  analysis of tetra-lepton signatures, we therefore choose both the lighter and heavier mass points.
4 Large triplet vev and same-sign tetra-lepton signature for √ s = 14 TeV We explore the tetra-lepton signature arising from a lighter charged Higgs and neutral Higgs decay. We consider associated production of H ± along-with H 0 /A 0 . For triplet vev in between 10 −5 GeV < v ∆ < 10 −3 GeV, and assuming mass hierarchy between singly and doubly charged Higgs M H ± > M H ±± , the cascade decay of H ± into H ±± W * is predominant. In the same triplet vev region, H 0 /A 0 → H ± W * decay is also significantly large. We furthermore consider the gauge boson decay modes of H ±± → W ± W ± , that has large branching ratio for v ∆ > 10 −4 GeV and subsequent leptonic decay of the produced on-shell W ± . For the signal, therefore, the complete process is [53], The Feynman diagrams for these above two processes have been shown in Fig. 2. Note that this phenomenon of wrong sign leptons production occurs as ∆ 0 can oscillate to ∆ 0 † and vice versa. As a result, H 0 and A 0 , sharing the same final states, can mix together like in the B 0 − B 0 system. Finally we can write the cross-section for these signals as: In the above F 1,2 are When the two decay widths Γ H 0 and Γ A 0 are nearly equal, i.e., Γ H 0 Γ A 0 . The generatisation of these two processes to the case of where G 1 and G 2 have the following forms: Note that, to compute the tetra-lepton signature, one needs to take into account the leptonic branching ratios from W . In our analysis, we consider both the W → lν, with l = e, µ, as well as W → τ ν, with the leptonic decays of τ included. To compute the cross-section, we implement the model in FeynRules(v2.3) [55]. The UFO output is then fed into MadGraph5 aMC@NLO(v2.6) [56] that generates the parton-level events. We use the default pdf NNPDF23LO1 [57] for computation. We perform parton showering and hadronization with Pythia8 [58] and analyse the HepMC [59] event files. The above cross-sections pp → H ± H 0 and pp → H 0 A 0 depend on the masses of the neutral and charged Higgs. We therefore show the variation of associated production cross-section of pp → H ± H 0 /A 0 and pp → H 0 A 0 with the mass of H 0 in Fig. 3. For c.m.energy √ s = 14 TeV, the cross-section for pp → H 0 A 0 varies in between 1 − 70 fb, for neutral Higgs mass between 200 − 500 GeV. For pp → H + H 0 /A 0 , the cross-section is very similar, only lower than by a factor of O(1.5). For pp → H − H 0 /A 0 , cross-section is smaller due to the parton distribution function. In addition, we also show the production cross-section for a future pp collider, with c.m.energy √ s = 100 TeV. As is evident from the right panel of Fig. 3, the production cross-section is quite large for higher c.m.energy, and multi-TeV Higgs mass can be probed.
Note that the production cross-sections for pp → H ± H 0 /A 0 depends on both the parameters λ 4 and the triplet vev v ∆ . For a fixed value of µ, the triplet vev primarily governs the masses of the Higgs H ± , H 0 /A 0 , while the parameter λ 4 determines their mass difference. In the left panel of Fig. 4, we show the variation of production cross section  4 for the process pp → H ± H 0 /A 0 with mass of A 0 being fixed as M A 0 = 253 GeV. For the second process pp → H 0 /A 0 , behaviour of the product of branching ratio will be same.
for the process pp → H + H 0 in the v ∆ − λ 4 plane for a benchmark value of neutral Higgs, M H 0 ∼ 253 GeV. For the process pp → H − H 0 , the plot is very similar, only the production cross section is relatively smaller by a factor of two. The channel pp → H 0 A 0 has the largest cross-section, larger than pp → H + H 0 by almost a factor of O(1.4 − 1.7). Since λ 4 has a very nominal effect on the mass splitting of H 0 , A 0 , the cross-section of this channel is almost fixed in the entire plane of λ 4 − v ∆ , and thus does not vary.
The doubly, singly charged, and neutral Higgs bosons will decay through a number of subsequent decay modes, leading to the same-sign tetra-lepton final states. The two key parameters are again triplet vev v ∆ and the coupling λ 4 . Since a number of branching ratios are involved in the same-sign tetra-lepton process, we show the product of these branching ratios. In the right panel of Fig. 4, we show the variation of the product of branching ratios 4 for the process pp → H ± H 0 /A 0 in the v ∆ − λ 4 plane. From the top panel of Fig. 1, it is evident that the doubly charged Higgs H ±± decays predominantly to same sign W ± W ± state. For smaller range of the triplet vev it entirely decays to l ± l ± final state. This is reflected in Fig. 4, where there is a sharp change in branching ratio around 10 −4 GeV. The product goes to zero in the left side of this line (as shown by the black region). In the right side of this line, the product can be large, as indicated by the colour bar. We stress that, the product of the branching ratios has a significantly large value for a wide range of the triplet vev, 10 −4 GeV < v ∆ < 10 −2 GeV. Therefore, in this region, there will be handful of events for same-sign tetra-lepton final states, that can be tested at LHC. In the next section, we will see how this large range of triplet vev shrinks to a very narrow range for higher masses of the charged and neutral Higgs. This occurs due to significant change in branching ratios of the channel H ± → H ±± W − * for the same value of λ 4 .
In Fig. 5, we show the variation of number of events for the same-sign tetra-lepton signature, where we consider integrated luminosity L = 3000 fb −1 . This has been obtained by folding the production cross-section with the overall branching ratio, and integrated luminosity. We also implement few basic cuts at the pythia level. These are p T (e ± /µ ± ) > 10 GeV, |η(e ± /µ ± )| < 2.5. We obtain a cut-efficiency c ef f = 0.62 for M H 0 = 253 GeV, that we include in our calculation of total number of events. We consider the processes pp → H + H 0 /A 0 (top left), pp → H − H 0 /A 0 (top right) and pp → H 0 A 0 (bottom).To calculate the number of events we followed the prescription given at the beginning of Section 4. As we can see from the bottom left plot of Fig. 1, that for the low mass range of the particle spectrum, the channel H ± → H ±± W − * has 100% branching ratio for a wide range of triplet vev. Hence in all these three plots, we get a reasonable number of events for triplet vev v ∆ ∼ 10 −4 − 10 −1 GeV. As exhibited in Fig. 3, the cross section for the different final states have the following hierarchies σ(pp . The same hierarchy also translates in the number of events. All the three plots have a similar kind of morphology in the v ∆ − λ 4 plane and the nature of the variation of the number of events can be understood in the following way. Since we are considering H ±± → W ± W ± channel which start contributing when triplet vev is v ∆ > 10 −4 GeV, so the number of events N evt > 5 starts around this region of triplet vev. As shown in Fig. 4, the cross section increases with larger λ 4 , while the branching ratio for the channel H ± → H ±± W − * decreases (bottom right plot of Fig. 1) for larger triplet vev, leading to the specific variation of the number of events shown in 5.
5 Inclusive same-sign tetra-lepton signature for √ s = 100 TeV We consider heavier Higgs, and analyse its discovery prospect at a future pp collider that can operate with c.m.energy √ s = 100 TeV. Due to the suppression from a number of branching ratios, observation of same-sign tetra-lepton final states will be beyond the scope of 13 TeV LHC. However, this can easily be observed in a future collider with higher c.m.energy. As a benchmark sample, we consider neutral Higgs mass M H 0 /A 0 = 900 GeV, and variation of doubly charged Higgs of mass at most by 5 GeV from M H 0 /A 0 . The chosen value of the doubly charged Higgs mass is consistent with the constraints from 13 TeV LHC searches for the entire range of triplet vev v ∆ ∼ 10 −9 − 1 GeV. Near the triplet vev v ∆ ∼ 10 −4 , both the di-lepton and gauge boson modes will substantially contribute. We therefore cover a large range of triplet vev v ∆ , and consider the doubly charged Higgs decaying into both the same-sign di-lepton, and gauge boson modes. Hence, in addition to the gauge bosons, discussed in Sec. 4, the total cross-section also contains the following contribution from di-lepton decay mode, In the above, l = e, µ, τ , and we finally consider the leptonic branching ratios of τ , while calculating the number of events. The functions G 1,2 have been described in Section. 4. We show the variation of cross-section in Fig. 6. The cross-section for the mass M A 0 = 900 GeV varies around 5 fb. We next show the variation of the product of branching ratios in Fig. 7 for heavier charged and neutral Higgs. For triplet vev smaller than v ∆ < 10 −4 GeV, the  figure). Lower panel: the sum of these two products of branching ratios for the process pp → H ± H 0 /A 0 with mass of A 0 being fixed as M A 0 = 900 GeV. For the process pp → H 0 A 0 , the product of branching ratio is very similar.
doubly charged Higgs H ±± → l ± l ± is dominant, while around 10 −4 GeV, both the gauge boson mode and di-lepton are dominant. For a heavier singly charged Higgs, the branching ratio for H ± → H ±± W − * decay channel is large for a large value of λ 4 . Note that, for λ 4 ∼ 0.1, the branching ratio becomes more than 1% in a very small range of the triplet vev (see Fig. 1). This in turns has large effect on the total branching ratio, and is clearly visible in all the three plots of Fig. 7. The region in v ∆ , in which the overall branching ratio is larger than 0.5% is now considerably smaller. The left plot in the top panel represents the overall branching ratio with only H ±± → W ± W ± decay included. The plot in the right panel shows the total branching ratio for H ±± → l ± l ± . The product of the branching ratio is smaller for the case of H ±± → W ± W ± due to additional suppression from Br(W ± → ± ν) 4 . In the lower panel, we show the sum of these two branching ratios. The higher values of the product of the branching ratios is governed by H ±± → l ± l ± decay mode (relevant for v ∆ < ∼ 10 −4 GeV). More explicitly we show the H ±± → W ± W ± dominated branching ratio by the blueish region, and H ±± → l ± l ± dominated branching ratio by yellowish region. The total cross-section has been computed by folding the branching ratios with the crosssection shown in Fig. 6. In addition, we also include few preliminary cuts, p T (e ± /µ ± ) > 10 GeV, |η(e ± /µ ± )| < 2.5. For M H 0 = 900 GeV and neutrino oscillation parameters to their best fit values [1], we obtain the cut-efficiencies c ef f = 0.64 in H ±± → l ± l ± mode and c ef f = 0.62 in H ±± → W ± W ± mode, that have been included in this analysis.
In Fig. 8, we present the number of events for heavier doubly charged Higgs, charged and neutral Higgs (∼ 900 GeV). In all the three plots which correspond to pp → H + H 0 /A 0 , pp → H − H 0 /A 0 and pp → H 0 A 0 processes, its possible to achieve a significantly large number of events in a very narrow region indicated by the yellow patch. This is in contrary to the low mass range, discussed in the previous section, where we get reasonable number of events for a wider range of triplet vev. Fig. 9 represents the variation of the total number of events for tetra leptons with either +ve or -ve sign of the leptons. The left panel shows the variation of the sum of the number of events (l + l + l + l + + l − l − l − l − ) for the low mass of the particles and its shape is exactly similar as discussed before (see Fig. 5). The figure in the right panel shows the events for higher mass and also has the similar shape as displayed in Fig. 8.

Conclusion
The type-II seesaw model is one of the most simplest models of neutrino mass generation, where the model is extended by an additional triplet scalar field. Due to an extended Higgs sector, and mixing between SM doublet scalar field and triplet scalar, the model contains few additional Higgs fields, including doubly charged and singly charged Higgs fields, as well as, CP even and odd neutral Higgs fields. In this work, we consider a type-II seesaw model for neutrino mass generation, and analyse an unique same-sign tetra-lepton signature at pp colliders. This arises from the associated production of Higgs fields H ± H 0 , H 0 A 0 , and the subsequent decay of neutral Higgs field into a singly charged Higgs state, and the decay of a singly charged field into a doubly charged Higgs state. More precisely, for nondegenerate Higgs masses, and for an intermediate triplet vev v ∆ in between 10 −5 GeV < v ∆ < 10 −2 GeV, the neutral and charged Higgs H 0 , A 0 , H ± decay predominantly to H 0 /A 0 → H ± W * , and H ± → H ±± W * . The subsequent decays of H ±± either to samesign di-leptons, or to same-sign gauge bosons lead to the same-sign tetra-lepton final states. We analyse this signature for a pp collider, taking into account two different c.m.energies, √ s = 14 TeV, and √ s = 100 TeV. In our analysis, we choose those benchmark mass points, that are consistent with the present limits from 13 TeV LHC. In particular, for the lower c.m.energy, we explore the tetra-lepton signatures from a lighter Higgs state, and for higher c.m.energy, we consider a heavier Higgs states.
As an illustrative example, we first consider a large triplet vev v ∆ > 10 −5 GeV, and a benchmark mass with M H 0 ,A 0 = 253 GeV. We vary the mass difference between doubly charged Higgs and charge neutral Higgs by at most 5 GeV. In this region of triplet vev, the gauge boson decay mode of H ±± is pre-dominant. The associated production cross-section pp → H + H 0 varies in between σ ∼ 17 − 20 fb. The product of different branching ratios become maximal in a region v ∆ ∼ 10 −4 GeV − 10 −2 GeV. To analyse the signal events, we implement few basic cuts, for which we get a cut efficiency c ef f = 0.6. With integrated luminosity of L = 3000 fb −1 , we find that a doubly charged Higgs of mass around 257 GeV can lead to 600 number of events at the future upgrade of LHC.
Additionally, we also consider heavier neutral, and doubly charged Higgs, for which tetra-lepton signature can be observed in a pp collider with higher c.m.energy. For illustration, we consider the mass M H 0 ,A 0 = 900 GeV, and vary the masses of doubly and singly charged Higgs at most by M H ± − M H ±± ∼ 5 GeV. We explore the signal sensitivity for this benchmark point at 100 TeV pp collider. We consider both the di-lepton and gauge boson decay modes of the doubly charged Higgs. For heavier mass, the branching ratio of H ± → H ±± W * is large for a very large λ 4 . We find that the production cross-section pp → H + H 0 varies nominally σ ∼ 5 fb. We find that in a narrow region region in λ 4 − v ∆ plane, the same-sign tetra-lepton events can be very large N evnt ∼ O(10 3 ).