Displaced vertex signatures of doubly charged scalars in the type-II seesaw and its left-right extensions

The type-II seesaw mechanism with an isospin-triplet scalar $\Delta_L$ provides one of the most compelling explanations for the observed smallness of neutrino masses. The triplet contains a doubly-charged component $H_L^{\pm\pm}$, which dominantly decays to either same-sign dileptons or to a pair of $W$ bosons, depending on the size of the triplet vacuum expectation value. However, there exists a range of Yukawa couplings $f_L$ of the triplet to the charged leptons, wherein a relatively light $H_L^{\pm\pm}$ tends to be long-lived, giving rise to distinct displaced-vertex signatures at the high-energy colliders. We find that the displaced vertex signals from the leptonic decays $H_L^{\pm\pm} \to \ell_\alpha^\pm \ell_\beta^\pm$ could probe a broad parameter space with $10^{-10} \lesssim |f_L| \lesssim 10^{-6}$ and 45.6 GeV $<M_{H_L^{\pm\pm}} \lesssim 200$ GeV at the high-luminosity LHC. Similar sensitivity can also be achieved at a future 1 TeV $e^+e^-$ collider. The mass reach can be extended to about 500 GeV at a future 100 TeV proton-proton collider. Similar conclusions apply for the right-handed triplet $H_R^{\pm\pm}$ in the TeV-scale left-right symmetric models, which provide a natural embedding of the type-II seesaw. We show that the displaced vertex signals are largely complementary to the prompt same-sign dilepton pair searches at the LHC and the low-energy, high-intensity/precision measurements, such as neutrinoless double beta decay, charged lepton flavor violation, electron and muon anomalous magnetic moments, muonium oscillation and M{\o}ller scattering.


Introduction
To account for the tiny neutrino masses, as suggested by the neutrino oscillation experiments, the Standard Model (SM) has to be extended in the scalar, fermion and/or gauge sector; see Ref. [1] for a review. By simply extending the SM scalar sector by an SU (2) Ltriplet scalar, the type-II seesaw [2][3][4][5][6][7] is one of the most economical frameworks to generate the observed neutrino masses and mixing. In this paper we study the displaced vertex (DV) signatures from the doubly-charged scalars in the type-II seesaw and its left-right extensions, which could enrich the searches for new physics behind neutrino mass generation at the high-energy frontier, beyond the standard prompt decays currently being probed at the Large Hadron Collider (LHC).
In the pure type-II seesaw, the doubly-charged component H ±± L of the SU (2) L -triplet ∆ L couples to the SM charged leptons and the W boson [8][9][10]. The strength of these interactions can be potentially suppressed by either the small Yukawa couplings f L or the small vacuum expectation value (VEV) of the neutral component of ∆ L . Therefore, in a sizable parameter space, the doubly-charged scalar H ±± L tends to be long-lived at the hadron and lepton colliders, and the decay products of H ±± L form distinct displaced vertex (DV) signatures. In this paper we study only the simplest DV scenario, i.e. the displaced same-sign dilepton pairs H ±± L → ± α ± β (with α, β = e, µ, τ being the lepton flavor indices), though in principle we could also have the displaced multi-body final states through the (off-shell) W bosons: H ±± L → W ± ( * ) W ± ( * ) →f f f f (with the f 's being the SM fermions). For the sake of concreteness and illustration purpose, we estimate the DV prospects at the highluminosity LHC (HL-LHC) with the center-of-mass energy of 14 TeV and an integrated luminosity of 3000 fb −1 [11], as well as the proposed 100 TeV Future Circular Collider (FCC-hh) with a luminosity of 30 ab −1 [12] and the International Linear Collider (ILC) with the center-of-mass energy of 1 TeV and a luminosity of 1 ab −1 [13]. The sensitivities at other proposed facilities such as the Super Proton-Proton Collider (SPPC) [14], Circular Electron-Positron Collider (CEPC) [15], FCC-ee [16] and Compact LInear Collider (CLIC) [17] might be somewhat different, depending on the colliding energies and luminosities, but could be easily derived following our analysis.
The left-right symmetric model (LRSM) [18][19][20], based on the gauge group SU (3) C × SU (2) L × SU (1) R × U (1) B−L , provides a natural ultraviolet (UV)-completion of the type-II seesaw mechanism. In this case, as a "partner" of H ±± L under parity, there exists a right-handed (RH) doubly-charged scalar H ±± R originating from the SU (2) R -triplet scalar ∆ R , which couples predominantly to the RH charged leptons via the Yukawa couplings f R and the heavy W R boson via the RH gauge interaction [21,22]. For sufficiently heavy W R boson and small couplings f R , the lifetime of H ±± R could also be large enough to give rise to DV signatures in H ±± R → ± α ± β at future colliders. The DV signatures from the off-shell W R bosons: H ±± R → W ± * R W ± * R →f f f f (here the f 's stand for the SM fermions as well as the RH neutrinos (RHNs) in the LRSM) will not be covered in this paper, as a proper analysis of displaced jets and RH neutrinos from W R decay is more involved than the simplest case of displaced same-sign dilepton pairs.
As we will show below, for both the H ±± L in the pure type-II seesaw and H ±± R in the LRSM, the DV signatures from the decay of doubly-charged scalars are sensitive to relatively small Yukawa couplings, typically f L, R 10 −7 . These are largely complementary to the searches of prompt same-sign dilepton pair signals [8,[22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38] performed at LEP [39][40][41], Tevatron [42][43][44][45] and LHC [46,47], wherein the Yukawa couplings are assumed to be large such that the doubly-charged scalars decay promptly after production. If the Yukawa couplings f L,R happen to be very small, then we will not expect any prompt leptons at hadron and lepton colliders, and all these dilepton constraints are no longer applicable. Furthermore, with the lepton-flavor violating (LFV) couplings (f L, R ) αβ (α = β), both H ±± L and H ±± R could induce rare LFV processes like µ → eee and µ → eγ [48][49][50][51][52][53][54][55][56][57][58][59][60][61][62][63][64], electron and muon anomalous magnetic moment [21,61,65,66], and muonium-antimuonium oscillation [50,[67][68][69], which are all highly suppressed in the SM [70]. The current low-energy high-intensity experiments searching for these rare processes set severe constraints on the LFV couplings [37,71,72], which could go down to 10 −3 for a 100 GeV-scale doublycharged scalar. The DV signatures probing much lower LFV couplings are complementary to these low-energy high-intensity experiments as well. The rest of the paper is organized as follows: Section 2 is devoted to the left-handed (LH) doubly-charged scalar H ±± L in the pure type-II seesaw, where after sketching the basic properties of H ±± L we collect all the LFV constraints, re-interpret the high-energy same-sign dilepton constraints, and estimate the DV prospects at HL-LHC, FCC-hh and ILC. In Section 3 we focus on the RH doubly-charged scalar H ±± R in parity-conserving LRSM, where the symmetry relation f L = f R sets stringent constraints on the couplings f R . Setting the RH scale v R = 5 √ 2 TeV, the DV prospects are found to be somewhat different from those of H ±± L in the type-II seesaw. The parity-violating case of LRSM follows in Section 4. Without parity in the Yukawa sector, i.e. f L = f R , all the elements of f R can be considered as free parameters. Taking as an explicit example the benchmark scenario of (f R ) ee = 0 with all other elements vanishing, the DV prospects turn out to be quite similar to those in the parity conserving case. We conclude in Section 5. The details of four-body decays of H ±± (R) →f f f f can be found in Appendix A, and the LFV decay formulas are collected in Appendix B.

Left-handed doubly-charged scalar in type-II seesaw
In the type-II seesaw model [2][3][4][5][6][7], there exists a new complex scalar multiplet which transforms as a triplet under the SM SU (2) L gauge group. It can be written in terms of its components as (2.1) The most general scalar potential for the SM doublet φ = φ + , φ 0 T and the triplet ∆ L reads [73][74][75] V(φ, with all the couplings being real (λ 6 having the mass dimension) and σ 2 being the second Pauli matrix. A non-zero VEV for the doublet field φ 0 = v EW / √ 2 (with v EW 246 GeV being the electroweak scale) induces a tadpole term for the scalar triplet field ∆ L via the λ 6 term in Eq. (2.2), thereby generating a non-zero VEV for its neutral component, 2, and breaking lepton number by two units, which is responsible for neutrino mass generation at tree-level.
As the VEV v L is in charge of the tiny neutrino masses, it is expected to be much smaller than the electroweak scale, or even close to the eV scale, depending on the corresponding Yukawa couplings. On the other hand, the electroweak precision data, and in particular, the ρ-parameter constraint requires that v L 2 GeV [76]. In the limit of v L v EW , after spontaneous symmetry breaking, the neutral component from the SM doublet has a mass m 2 h λv 2 EW and identified as the SM Higgs, whereas the neutral, singly-charged and doubly-charged components of the triplet ∆ L give rise to the additional physical scalars with "Re" and "Im" denoting respectively the real and imaginary parts. Their masses are respectively The mass splitting of the triplet scalars is dictated by the quartic coupling λ 5 in Eq. (2.2) and tends to be small (compared to the triplet scalar masses), in particular when the electroweak precision data is taken into consideration [77,78]. The triplet ∆ L couples to the SM lepton doublet ψ L = (ν, ) T L via the Yukawa interactions where α, β = e, µ, τ denote the lepton flavors and C is the charge conjugation matrix. Then the tiny neutrino mass matrix is obtained with the induced VEV v L : The Yukawa coupling matrix f L is fixed by the active neutrino data, i.e. the observed neutrino mass squared differences and mixing angles, up to the unknown lightest neutrino mass m 0 , the neutrino mass hierarchy and the Dirac and Majorana CP violating phases. In Eq. (2.8) m ν = diag{m 1 , m 2 , m 3 } with m 1,2,3 the physical neutrino masses, and U is the standard PMNS matrix, which can be parameterized as [70] where c ij ≡ cos θ ij , s ij ≡ sin θ ij with θ ij the mixing angles, δ CP the Dirac CP phase and α 1,2 the two Majorana phases. For the convenience of the calculations below, a recent global fit results [79,80] on the mass squared differences and mixing angles are collected in Table 1, including the central values and 1σ uncertainties for both the normal hierarchy (NH) and inverted hierarchy (IH) spectra of neutrino masses. We vary the CP phases within the whole range of [0, 2π] (unless otherwise specified). Note that the recent T2K [81] and NOνA [82] results indicate a mild preference for non-zero δ CP , but this has not been established at 5σ level yet.

Decay Length
In the type-II seesaw, the doubly-charged scalar H ±± L has the following decay modes: β which depends on the Yukawa couplings in Eq. (2.7). The partial width is given by with M H ±± L the mass of H ±± L , δ αβ the Kronecker δ-function, and (m ν ) αβ the neutrino mass matrix elements given by Eq. (2.8).
The partial width is given by in which case the partial width calculation is a bit involved [10] and is detailed in Appendix A.
In principle, H ±± L has other diboson decay modes as follows: > M H ± which implies that λ 5 > 0 in Eqs. (2.4)-(2.6), the mass splitting larger than 60 GeV is disfavored by current electroweak precision data [78], thus the on-shell decay into H ± W ± is not kinematically allowed. For is smaller than that for the dilepton and W boson pair channels given by Eqs. (2.10) and (2.11) respectively [9,10].
• H ±± L → H ± * H ± * which is subject to the trilinear scalar couplings in the potential (2.2). In light of the electroweak precision data [78], both the singly-charged scalars H ± in the final state are expected to be off-shell.
For simplicity, we neglect the H ± W ± and H ± H ± diboson channels, e.g. by choosing appropriate quartic couplings such that M H ±± L < M H ± and the trilinear coupling H ±± H ∓ H ∓ is small. Then the total width is given by The dilepton width in Eq. (2.10) is suppressed by the tiny neutrino masses when the VEV v L is comparatively large, while the W pair width in Eq. (2.11) is suppressed by the VEV v L , which leads to a maximal total width at a VEV value of v L ∼ 1 MeV, depending on the doubly-charged scalar mass [8], as shown in Fig. 1. For sufficiently light H ±± L , roughly of order 100 GeV, the proper lifetime cτ 0 (H ±± L ) is of order millimeter to meter and could thus generate DV signal at colliders. For the illustration purpose, the proper decay length cτ 0 of 1 mm, 1 cm, 10 cm and 1 m are shown in Fig. 1, as functions of the Yukawa coupling |f L | and the doubly-charged scalar mass M H ±± L . The left and right panels are respectively for the NH and IH cases with the lightest neutrino mass m 1 = 0 (NH) and m 3 = 0 (IH). As all the elements of f L are strongly correlated by the neutrino mass and mixing data, as shown in Eq. (2.8), for concreteness we take the largest element |f L | max on the left y-axes. Also we take only the central values of the neutrino mixing angles and mass-squared differences in Table 1 and choose the Dirac CP phase δ CP = 3π/2, as suggested from the best-fit central value of the recent T2K [81] and NOνA [82] data. The corresponding values of v L from Eq. (2.8) are also shown on the right y-axes of the plots, with the relation v L |f L | max = 0.027 eV , for NH with m 1 = 0 , 0.048 eV , for IH with m 3 = 0 . (2.14) The contours of branching ratios (BRs) BR(H ±± L → ± α ± β ) = 1−BR(H ±± L → W ±( * ) W ±( * ) ) = 1%, 10%, 50%, 90%, 99% are also depicted in Fig. 1 as respectively the long-dashed red, short-dashed red, thick solid black, short-dashed blue and long-dashed blue lines. It is clear from Fig. 1   → W ±( * ) W ±( * ) ) = 1%, 10%, 50%, 90%, 99%. The left and right panels are respectively for the neutrino spectra of NH and IH with zero lightest neutrino mass. The corresponding values of the VEV v L are also shown in the plots. signatures), the LH doubly-charged scalar H ±± L in the type-II seesaw is required to have a mass from m Z /2 (see Section 2.2.3 for the mass limit M H ±± L m Z /2) to roughly 150 GeV, with Yukawa couplings |f L | max ∼ 10 −10 to 10 −7 (or effectively v L = 10 5 to 10 8 eV). For larger values of lightest neutrino masses m 1,3 > 0, the neutrino mass elements in Eq. (2.10) tend to be larger; however, as the (proper) lifetime of H ±± L is only sensitive to the total width, the lifetime contours, and as a result the DV sensitivities, do not change too much in Fig. 1.
Given the physical mass spectrum in the triplet sector, cf. Eqs. (2.4)-(2.6), we could also envision DV signatures due to the singly-charged Higgs bosons H ± or the neutral CP-even (odd) Higgs bosons H (A). As for the singly-charged one, the dominant decay modes are H ± → ± ν (depending on the Yukawa coupling f L ) and H ± → W ± Z, W ± h (depending on the VEV v L ). However, the Drell-Yan production cross section for H ± pair at colliders is smaller than that of H ±± pair, due to the smaller electric charge, and for our choice M H ±± L < M H ± , other production modes are also smaller for H ± [8]. Therefore, the DV signatures of H ± are expected to be sub-dominant compared to that of H ±± L . As for the neutral CP-even scalar H, the dominant decays are into νν (depending on f L ) and hh, ZZ, tt, bb (depending on v L ). Similarly, for the CP-odd scalar A, the dominant decays are into νν (depending on f L ) and hZ, tt, bb (depending on v L ). Thus, in the parameter space of interest where H ±± L gives rise to DV dilepton decays, the neutral scalars would most likely lead to DV missing energy signal, which is not very promising. We postpone a detailed investigation of the possible DV prospects of singly-charged and neutral scalars in the type-II seesaw model at future colliders to a follow-up work.

Low and high-energy constraints
In this section, we discuss various experimental constraints on the Yukawa couplings (f L ) αβ from both low-and high-energy observables.
In light of the neutrino mass relation in Eq. (2.8), all these limits on the couplings |(f L ) αβ | or |f † L f L | can be traded for constraints on the VEV v L in the type-II seesaw, up to the unknown lightest neutrino mass m 0 , the Dirac CP phase δ CP and the neutrino mass hierarchy [71]. To be concrete, we utilize only the central values of the neutrino data in Table 1, with the Dirac CP violating phase δ CP = 3π/2. To set limits on v L , we consider both the NH and IH spectra, and adopt two benchmark values of m 0 = 0 and 0.05 eV in each case. All the corresponding constraints on the product v L M H ±± L are collected in the last two columns of Table 2, in unit of (eV) (100 GeV). The limits for these four benchmark scenarios (NH and IH, m 0 = 0 and 0.05 eV) are graphically depicted in Fig  to electrons also contributes to the Møller scattering e − e − → e − e − and could be probed by the upcoming MOLLER experiment [85,86], but in the pure type-II seesaw case, the MOLLER sensitivity is precluded by the LFV constraints [72]. Table 2. Current experimental limits on the BRs of α → β γ δ , α → β γ [70,87], anomalous electron [88] and muon [89] magnetic moments, muonium oscillation [90], and LEP e + e − → + − data [84], along with the corresponding constraints on the Yukawa couplings |f | or |f † f |, in unit of (M ±± /100 GeV) 2 (third column). These limits apply to both H ±± L in the type-II seesaw and H ±± R in the LRSM (so the subscript "L" has been removed). The data in the last two columns are the resultant constraints on (v L M H ±± L ) in the type-II seesaw, in unit of (eV)(100 GeV), for both NH and IH with the lightest neutrino mass m 0 = 0 (0.05 eV).

Process Experimental
Constraint × M±± 6.9 (6.9) 6.9 (6.9) Thus in the fourth and fifth columns of Table 2, the limits involving the coupling f ee , like µ → eee, for the case of NH (m 1 = 0) is weaker than that for IH (m 3 = 0). When the three active neutrinos becomes heavier, for instance in the scenarios of NH with m 1 = 0.05 eV and IH with m 3 = 0.05 eV in Table 2, the matrix elements (m ν ) αβ tend to be larger (though some of them would get smaller due to mild cancellation in  and the largest Yukawa coupling |f L | max in the type-II seesaw, for the NH (left) and IH (right) with the lightest neutrino mass m 0 = 0 (upper) or 0.05 eV (lower). All the shaded regions are excluded by either the lowenergy LFV constraints on BR( α → β γ δ ), BR( α → β γ) [70], anomalous muon g − 2, muonium oscillation, or the LEP e + e − → + − data [84]. More constraints can be found in Table 2. The horizontal black line represents the perturbative limit of |f L | max < √ 4π.
the summation i m i U αi U βi ), and most of the limits in the parentheses of Table 2 are somewhat stronger than the cases with a massless neutrino.

Neutrinoless double beta decay
Due to its direct interaction with the SM W boson and the electrons, the LH doublycharged scalar H ±± L in type-II seesaw contributes to 0νββ, in addition to the canonical light neutrino contribution. The parton-level Feynman diagrams of 0νββ from the light neutrinos ν i and H ±± L are presented in Fig. 3. The corresponding half lifetime of 0νββ can be factorized as [57] T 0ν with G the phase space factor, M ν the nuclear matrix element (NME) for the light neutrino contribution, and the dimensionless term η ν = (m ν ) ee /m e is the amplitude of the canonical Compared to the canonical η ν term, the extra H ±± L contribution is highly suppressed by the doubly-charged scalar mass. Therefore we can not set any limits on H ±± L in the pure type-II seesaw.

High-energy collider constraints
The doubly-charged scalar H ±± L couples directly to the SM Z boson, with the coupling proportional to the factor (1 − 2 sin 2 θ w ), where sin θ w the weak mixing angle. For M H ±± L < m Z /2, it contributes to the total width of Z via the decay and is thus stringently constrained by the high precision Z-pole data [70]. This puts a lower bound on M H ±± L > m Z /2 45.6 GeV, irrespective of how the doubly-charged scalar decays or whether it is long-lived or not [10].
Given the gauge interactions to the SM photon and Z bosons, the doubly-charged scalar can be pair produced from the electron-positron and quark annihilation processes:  β), which is almost SM background free. Direct same-sign dilepton pair searches of this kind have been performed at LEP [39][40][41], Tevatron [42][43][44][45] [46,47]. All these limits are presented in Fig. 4, as functions of M H ±± L and the BRs into six distinct flavor combinations α β = ee, µµ, eµ, eτ, µτ, τ τ . 2 With more data taken at LHC 13 TeV and future 14 TeV and high-luminosity stages, the doubly-charged scalars could be probed up to about 1 TeV [22,33]. Future 100 TeV 2 Including the photon fusion process γγ → H ++ H −− , these limits could be slightly strengthened [34].
Limited by the center-of-mass energy, the LEP data could only probe H ±± L up to the masses of ∼100 GeV in the pair production mode. Furthermore, in the data analysis of Refs. [39,40] the Yukawa couplings are assumed to be larger than 10 −7 , otherwise the reconstruction efficiency of the charged leptons would be affected by the non-prompt decays of H ±± L . The doubly-charged scalar has also been searched in the single production mode, via the process e + e − → e ∓ e ∓ H ±± [83]. Analogous searches have also been performed at the lepton-hadron collider HERA [98] in the process e + p → − pH ++ . The single production is dictated by the Yukawa interaction (f L ) ee but not the gauge couplings, thus these experimental data can be used to directly constrain the Yukawa couplings, but not the BRs like BR(H ±± → e ± µ ± ) [37]. Therefore, the limits on the Yukawa couplings derived in Ref. [83] are not shown in the plots of BR constraints in Fig. 4.
Benefiting from the higher energy and larger luminosity, the H ±± L mass limits from the LHC data are much more stringent, up to ∼ 800 GeV in the ee, eµ and µµ channels and ∼ 500 GeV if the τ lepton is involved. The BRs are probed up to ∼ 10 −2 for lighter H ±± L in the e and µ channels, whereas in the channels involving the τ flavor, the limits are much weaker, at most up to the level of 0.2. In hadron collisions, H ±± L could also be produced in association with the singly-charged scalar H ± , i.e. pp → W ± * → H ∓ H ±± L . This depends however on the mass splitting M H ± − M H ±± L [77] and the decay of H ± [8], which involves the couplings in the scalar potential (2.2). For simplicity, the associated production channel is not considered in this paper, though the corresponding production cross section σ(pp → H ∓ H ±± L ) tends to be larger than that of the Drell-Yan pair production cross section σ(pp → H ++ L H −− L ). As shown in Eqs. (2.10) and (2.11) and in Fig. 1, the triplet VEV v L plays a crucial rule in determining the BRs of H ±± L into same-sign leptons and W boson pairs. In the limit of small v L , i.e. v L 0.1 MeV [10], the W pair channel is suppressed and H ±± L decays predominantly into the same-sign dileptons. Since the Yukawa coupling f L is related to the neutrino mass matrix m ν via Eq. (2.8), the leptonic BRs can be readily obtained from Eq. (2.10) in terms of the neutrino masses:  Table 1 are rather precise, when the uncertainties are taken into account, some of the decay BRs in Eq. (2.23) like BR(H ±± L → µ ± µ ± ) might vary significantly. Therefore, we consider the central values of the neutrino oscillation parameters as shown in Table 1, as well as their 3σ uncertainties, and take the whole range of [0, 2π] for the Dirac CP phase δ CP . In Fig. 5 the dashed curves correspond to the central values of neutrino data, while the shaded bands are due to the 3σ uncertainties. The gray shaded region in these plots is excluded by the cosmological limit on the sum of light neutrino masses i m i < 0.23 eV [99].   Table 1, and the colorful shaded bands are due to the 3σ uncertainties. The gray shaded region is excluded by the cosmological constraint on the sum of light neutrino masses We see from Fig. 5 that for both NH and IH the dilepton limits are the most stringent in the ee (upper panels) and µµ (middle panels) channels, whereas those involving τ lepton are much less constraining, mainly limited by the τ lepton reconstruction efficiency at colliders. Similarly, the eµ channel is suppressed by the solar mixing angle (sin 2 θ 12 ) when compared to the ee and µµ decay modes. For the NH case, when the lightest neutrino mass gets small, say m 1 0.01 eV [cf. Eq. (2.23)], the branching fraction BR(H ±± L → e ± e ± ) is so small that it goes out of the range of the LEP and LHC data (see the upper left panel in Fig. 4). Thus there is no dilepton limit for m 1 0.01 eV in the ee channel for the NH case, as shown in the upper left panel of Fig. 5. Same thing happens for the eµ limits in the IH case (lower right panel). By the same token, there is no limit in the µτ channel for m 0 0.01 in both the NH and IH cases (cf. the two lower panels in Fig. 5), as the dilepton limits in this channel is comparatively weaker than those without the tau lepton. The constraints from the Tevatron data in Fig. 4 are much weaker and are not shown in Fig. 5.
When the W boson channel becomes important i.e. Γ(H ±± β ) on the VEV v L , or equivalently the dependence on the magnitudes of Yukawa couplings (f L ) αβ , has to be taken into consideration. In this case, the leptonic branching fractions could be obtained from Eq.  Table 1, and the "widths" of the curves are due to the 3σ uncertainties. All the regions above the curves are excluded by the same-sign dilepton data from LEP and LHC just aforementioned, which set upper bounds on the Yukawa coupling |f L | (or lower bounds on the VEV v L ). To be concise, the limits in Fig. 6 and 7 are all expressed as functions of the largest Yukawa coupling element |f L | max , which is the µµ (ee) element in the case of NH (IH).
As shown in Fig. 1, for a light H ±± L 150 GeV, when the Yukawa coupling is small, say 10 −7 , the decay length of H ±± L is sizable at the high-energy colliders, and the prompt dilepton limits can not be used to set limits on the mass of H ±± L and the Yukawa couplings f L , because the prompt lepton efficiencies are significantly affected [39,40,100]. This is because of two reasons: (i) the algorithm for reconstructing particle tracks in the electromagnetic calorimeter or the muon spectrometer has a loose requirement of extrapolation to the interaction point, and (ii) the opening angle between the two leptons decreases as the boost increases. To this end, we show in Figs. 6 and 7 the regions of proper decay length cτ 0 > 1 mm and 0.1 mm in the darker and lighter gray color, which can be considered respectively as the aggressive and conservative estimates of the regions, within which the prompt dilepton limits are not applicable. In the analysis of the LHC data [46,47,[92][93][94][95], the doubly-charged scalars are assumed to decay promptly, with a lifetime cτ < 10 µm, corresponding to a coupling f ∼ 10 −6 for a doubly-charged scalar with mass of 200 GeV. For smaller couplings, a sizable fraction of H ±± L tends to be non-prompt, and the LHC sensitivities would be significantly weakened and even not applicable. For simplicity we just exclude the LHC limits inside the shaded gray regions in Figs. 6 and 7 to make sure that H ±± L decay promptly at the LHC. It is clear that in all the four benchmark scenarios considered above, the higher-energy data tend to be more sensitive to large couplings and larger H ±± L mass. This could be easily understood by looking closer at the two partial widths in Eqs.      Table 1, and the colorful bands are due to the 3σ uncertainties. The corresponding lower limits on the VEV v L are also shown in these plots. The darker and lighter gray regions correspond to the proper decay lengths cτ 0 > 1 mm and 0.1 mm respectively; within these regions the prompt dilepton limits are not applicable.
width is proportional to M H ±± L . Thus when H ±± L becomes heavier, the diboson channel is comparatively enhanced, and the dilepton channel needs a larger Yukawa coupling to compensate for the suppression. On the other hand, the production cross section times branching fractions σ(pp, e + e − → H ++ δ ) becomes smaller when the Yukawa couplings are smaller, with a sizable fraction of H ±± L decaying into same-sign W pairs. Thus the dilepton limits get to some extent weaker and    In the case of NH (m 1 = 0), the doubly-charged scalar decays predominantly into the muon and tau leptons, and the dilepton limits are mostly from the µµ and µτ channels, as shown in the upper left and middle left panels of Fig. 6. In contrast, for all the other three scenarios, i.e. the NH with m 1 = 0.05 eV (left panels in Fig. 7), the IH with m 3 = 0 (right panels in Fig. 6) and m 3 = 0.05 eV (right panels in Fig. 7), the most important constraints are from ee and µµ channels. As stated above, the eτ , µτ and τ τ channels are limited by the τ lepton reconstruction efficiency, while the eµ channel is comparatively suppressed by sin 2 θ 12 . The same-sign dilepton search results at LHC 8 TeV [94] were also interpreted as constraints on the (fiducial) cross section σ [10,101]. As a consequence of the small branching BR(W → ν), the diboson limits turn out to be much weaker than the "direct" dilepton limits in Figs. 5-7, and could only exclude a narrow mass range of m Z /2 M ±± L < 84 GeV. From Fig. 1, the dominance of the diboson decay mode (or suppression of the dilepton mode) implies that in the above mass range, |f L | < 10 −8 in both the NH and IH cases, which is similar to the LEP limits in Fig. 6 and 7. In this region H ±± L anyway tends to be long-lived (cf. Fig. 1), and therefore, the prompt diboson limits derived in Refs. [10,101] are not applicable.

Heavy stable charged particle search
For sufficiently small total width, the lifetime of H ±± L is sizable, even comparable to the detector sizes, as shown in Fig. 1. If H ±± L decays outside either the inner silicon tracker or the whole detector, it would be recorded at the detector as a heavy stable charged particle (HSCP) and leave a trail behind as it passes through the detector. The doubly-charged HSCP has been searched for by the CMS group [102]. Both the ionization energy loss in the tracker and the time-of-flight can be used to set limits on the HSCPs. Conservatively, we use only the "tracker-only" analysis in [102] to constrain H ±± L in the type-II seesaw, as it could hardly fly out of the whole detector if its mass is larger than the lowest value of 100 GeV for the HSCP mass used in the analysis of Ref. [102]. Requiring that the decay length 43 mm < bcτ 0 (H ±± L ) < 1100 mm [103] (b being the Lorentz boost factor), and rescaling the theoretical production cross section in Ref.
[102] for the center-of-mass energy of √ s = 13 TeV, we obtain the shaded orange and blue regions in Fig. 8 as the excluded regions respectively for the NH and IH cases, with the lightest neutrino mass m 0 = 0. This corresponds to the Yukawa coupling range 10 −8.5 |f L | 10 −7 , depending on the doubly-charged scalar mass within the narrow range 100 GeV < M H ±± L 140 GeV. For heavier active neutrinos, as long as they are within the cosmological constraints [99], the total width of H ±± L and the exclusion regions in Fig. 8 would not change too much. With better particle identification using the time-of-flight measurement at the upgraded LHC detectors, the HSCP search limits could in principle be improved by up to an order of magnitude [104].

Displaced vertex prospects
The decay of H ±± L in the dilepton and diboson channels are suppressed respectively by the small Yukawa couplings |f L | and the small VEV v L , and the widths are proportional respectively to v −2 L and v 2 L , as seen in Eq. (2.10) and (2.11). The total width of H ±± L reaches at a minimum when v L ∼ 1 MeV (depending also on the mass of H ±± L ) and the proper lifetime could go up to 1 m, as aforementioned and shown in Fig. 1. This would lead to DV signals from the decay of H ±± L → ± α ± β at the LHC and future higher energy hadron colliders like FCC-hh [12] and SPPC [14], as well as future lepton colliders such as CEPC [15], ILC [13], FCC-ee [16] and CLIC [17]. As a strikingly clean signature beyond the SM, this kind of fully reconstructible DV signal from the doubly-charged scalar is largely complementary to the prompt same-sign dilepton pair searches at the high energy colliders: the prompt decays apply to relatively large couplings, while the DVs are sensitive to smaller couplings. In addition, if H ±± L is produced from the gauge interactions, then the prompt decays can only be used to constrain the branching fractions, as shown in Fig. 4; for sufficiently small Yukawa couplings |f L |, the decay products from the DVs can, in principle, be used to measure the lifetime cτ 0 (H ±± L ), and even fix all the couplings f L involved in the decay of H ±± L . Requiring at least one pair of displaced same-sign dileptons is to be reconstructed at colliders, the dominant SM backgrounds are from the low-mass Drell-Yan processes pp → e + e − , µ + µ − , with the charges of the electron or muon misidentified (and the electron misidentified as a muon or vice versa) [105] (see also Refs. [106,107]). However, these contributions are most substantial for small values of dilepton mass M H ±± L m , and the dileptons from Drell-Yan processes tend to be back-to-back at colliders, which could be easily distinguished from the four-body process pp → H ++ Thus the backgrounds are expected to be smaller than the assumed number of events (10 and 100) below. A jet might also be mis-identified as a lepton, with an energy-dependent fake rate 2 × 10 −3 for the lepton energy p T ( ) M H ±± L /2 M Z /4 [108]. To mimic the two leptons from the same vertex, we need two jets both misidentified, and the rate is even smaller. For simplicity, we have neglected the SM backgrounds for all the prospects below. Following a recent ATLAS analysis for displaced dilepton searches, which includes a SM background estimation for same-charge displaced dimuon vertices [105], we found that the backgrounds would not have substantial effects on our estimates of the signal sensitivities.
Requiring that the decay length 1 mm < bcτ 0 (H ±± L ) < 1 m, we have estimated the numbers of DV events at the HL-LHC at 14 TeV with an integrated luminosity of 3000 fb −1 and the ILC 1 TeV with 1 ab −1 luminosity, which are shown respectively as the solid and dashed contours in the plots of Fig. 9. The prospects at future 100 TeV collider FCChh are presented as the dot-dashed lines in Fig. 9, with a higher luminosity of 30 ab −1 and the decay length of 1 mm < bcτ 0 (H ±± L ) < 3 m. Here we have considered only the Drell- → e ± e ± , e ± µ ± , µ ± µ ± , which are the most promising channels with almost no SM backgrounds. The K-factors for the higher-order QCD corrections at HL-LHC depends on the doubly-charged scalar mass and could be even larger at the 100 TeV collider; for simplicity, we take a universal conservative K-factor of 1.2 [109] for both HL-LHC and FCC-hh. The higher-order electroweak corrections at lepton colliders like ILC are comparatively less important and are neglected here. In this sense, the DV prospects presented throughout this work are rather conservative; when the higher-order corrections are fully taken into consideration, the DV sensitivities might actually be enhanced to some extent.
At HL-LHC and FCC-hh we take the nominal cuts on the displaced leptons p T ( ) > 25 GeV and |η( )| < 2.5 and ∆φ( ) > 0.4, implemented by using CalcHEP [110]; at ILC we set an lower momentum cut p T ( ) > 10 GeV and keep other cuts the same as above. For simplicity, we have assumed naïvely the efficiency factor to be one for all these different decay channels of ee, eµ and µµ. To be concrete, we adopt the central values of neutrino data in Table 1 and assume the lightest neutrino mass m 0 = 0 for both the NH (left) and IH (right) cases. As in Fig. 8, the effects of larger neutrino masses on these DV prospects are minimal. The photon fusion process is not important for a relatively light doubly-charged scalar [34] and is not considered here. However, at lepton colliders, the laser photon fusion could largely enhance the production cross sections [37], and hence, the DV prospects.
At the HL-LHC, with an integrated luminosity of 3000 fb −1 , a large parameter space in the type-II seesaw can be probed in the searches of displaced same-sign dilepton pairs, spanning over m Z /2 < M H ±± L 250 GeV for the scalar mass and 10 −10 |f L | 10 −5.5 for the Yukawa couplings, which corresponds to a VEV of 10 4 eV v L 10 8 eV, as depicted in Fig. 9. With a higher energy and larger luminosity at future 100 TeV collider FCC-hh, H ±± L is likely to be more boosted and a much larger parameter space can be reached, up to M H ±± L ∼ 500 GeV and broader f L ranges. At the ILC, the center-of-mass energy is lower than at LHC, and the production cross section of H ±± L is smaller, thus in Fig. 9 the mass reach at ILC is weaker than that at HL-LHC and FCC-hh. Comparing the contours in the left and right panels of Fig. 9, we see that the DV signals have only a weak dependence on the neutrino data, as the most relevant quantity is the total width of H ±± L in Eq. (2.13). For larger neutrino masses with m 0 > 0, as long as they are within the cosmological bound i m i < 0.23 eV [99], the total width of H ±± L and the contours in Fig. 9 would not change too much.
It is worth noting that the diboson decay H ±± L → W ±( * ) W ±( * ) could also induce DVs at high energy colliders, and the searches of the displaced W decay products are largely complementary to the dilepton DV signals discussed above, in the sense that they are sensitive to different ranges of the VEV v L (or equivalently different ranges of Yukawa couplings f L ), as implied by the BR contours in Fig. 1. For pair produced H ±± L in the Drell-Yan process, we have in total four (off-shell) W boson, i.e. pp → H ++ H −− → 4W ( * ) . With the W boson decaying either hadronically or leptonically, we can have different sorts of DV signals involving a large number of jets (j) or charged leptons ( ) and neutrinos ( / E T ), such as 8j, 6j ν, 4j2 2ν, 2j3 3ν and 4 4ν. The data analysis with multiple jets and / E T is more challenging than the pure, all visible leptonic channels above, and the prospects are expected to be weaker.
In the low-energy high-intensity experiments, H ±± L could only be produced off-shell, and the high precision measurements can be used to set limits on the effective cutoff scales Λ eff ∼ M H ±± L /|f L |, as shown in Table 2 and Fig. 2. At the high-energy colliders, the doublycharged scalar H ±± L can be produced on-shell, and the prompt decays and DV signals are respectively sensitive to relatively large and small Yukawa couplings |f L |. These highenergy and high-intensity experiments are largely complementary to each other; this can be clearly seen in the summary plot Fig. 10. Here we have collectively presented the lower limit of M H ±± L m Z /2 from Z boson width (gray), the most stringent LFV constraints in Table 2 and Fig. 2 from µ → eγ for NH and µ → eee for IH (brown), the same-sign dilepton pair constraints in Fig. 6 from LEP (pink), LHC 7 TeV (blue), 8 TeV (red) and 13 TeV (purple), and the HSCP searches by CMS in Fig. 8 (bright yellow). The dashed green, pink and blue curves in Fig. 10 correspond respectively to the DV prospects at ILC 1 TeV, HL-LHC and 100 TeV in Fig. 9, all assuming 100 signal events. The left and right panels are respectively for the NH and IH cases with the lightest neutrino mass m 0 = 0. For all the dilepton limits, we have left out the regions with cτ 0 (H ±± L ) > 0.1 mm as in Fig. 6, as in this region, the doubly-charged scalar H ±± L is very likely to be long-lived and the prompt same-sign dilepton limits are not applicable, or at least weakened [39,40]. At future high-energy hadron and lepton colliders, a light H ±± L will be highly boosted, thus there is a small region with |f L | ∼ 10 −6 to 10 −5 where the DV sensitivity regions in Fig. 10 overlap with the current limits from LEP and LHC, where H ±± L is less boosted.  Fig. 6).

Right-handed doubly-charged scalar in the LRSM
The LRSM [18][19][20], which provides a natural embedding of the type-II seesaw, contains two SU (2) triplets -∆ L and ∆ R -that transform nontrivially under SU (2) L and SU (2) R , respectively. In the limit of small mixing between all the components of ∆ L and ∆ R , the LH triplet ∆ L can be identified as that in the type-II seesaw in Eq. (2.1). The RH triplet is the counterpart of ∆ L under parity, and it couples to the RH lepton doublets ψ R = (N, R ) T via the Yukawa interactions, analogous to Eq. (2.7) for the LH sector: with N α the heavy RHNs, and α, β = e, µ, τ the lepton flavor indices.
with m D the Dirac mass matrix. For simplicity we assume here the type-I seesaw contribution is small, or in other words the LRSM is in the type-II dominance regime for neutrino mass generation, and the heavy and light neutrino masses are related via m ν /M N v L /v R [55,57,[116][117][118]. In this case, the RHN masses are proportional to that of the active neutrinos, rescaled by the VEV ratio v R /v L , and the RHN mixing matrix U R is identical to the LH PMNS matrix U in Eq. (2.9).

Decay Length
The RH doubly-charged scalar H ±± R decays predominantly to a pair of same-sign charged RH leptons: H ±± R → ± α ± β , and a pair of same-sign (off-shell) heavy W R bosons: . 3 The widths for the leptonic and bosonic channels are quite similar to those for the H ±± The current FCNC constraints from K and B meson oscillation data require that the W R boson is beyond roughly 3 TeV [119,120] for the gauge coupling g R = g L . The 13 TeV LHC searches yield a similar mass bound from the same-sign dilepton channel pp → W R → N → ± ± jj [124], depending on the RHN mass [125]. Thus in the diboson decay mode of a TeV-scale (or lighter) doubly-charged scalar, both W R 's can only be off-shell, which decay further into the SM fermions and heavy RHNs (if lighter than H ±± R ). The decay width of H ±± R → W ± * R W ± * R → ff f f (where the fermions f run over all the SM quarks, charged leptons and heavy RHNs) can be found in Appendix A. In the type-II seesaw dominance of LRSM, dictated by the parity symmetry f L = f R , the couplings f R are also related to the active neutrino masses and mixing angles, as f L is in the pure type-II seesaw. The BRs BR(H ±± depend also on the heavy W R mass and the v R scale, cf. Eq. (3.5). To be concrete, we set v R = 5 √ 2 TeV and the gauge coupling g R = g L (unless otherwise specified) which leads to M W R = g R v R / √ 2 3.3 TeV, consistent with the LHC and low-energy constraints. Some representative BR(H ±± R → ± α ± β ) = 1%, 10%, 50%, 90%, 99% are shown in Fig. 11, with the left and right panels respectively for the NH and IH of neutrino spectrum with the lightest neutrino mass m 0 = 0. As in the type-II seesaw, the total width and decay lifetime of H ±± R are not very sensitive to the lightest neutrino mass m 0 . The singly-charged scalar from ∆R is eaten by the heavy WR boson after symmetry breaking, so there is no cascade decay to singly charged scalars unlike in the ∆L case [22]. Moreover, the heavy neutral CP-even and odd scalars from the bidoublet are required to be at least 10-20 TeV from the flavor changing neutral current (FCNC) constraints [119,120]. So in the RH scalar sector, the only other long-lived candidate, apart from the doubly-charged scalar, is the real part of the neutral component of the triplet Re(∆ 0 R ), which has been studied in Refs. [121][122][123]. shown in Fig. 11. For an RH doubly-charged scalar mass m Z /2 < M H ±± R 200 GeV, the proper decay length could reach from 1 mm up to 1 meter if |f R | 10 −7 . Unlike the LH case in Fig. 1, for fixed M W R and v R , the width Γ(H ±± R → W ± R W ± R ) depends only on the doubly-charged scalar mass M H ±± R , thus for sufficiently small |f R | the partial width ) and the lifetime contours in Fig. 11 tend to be flat in the downward direction.
All the relevant production channels of H ±± R at hadron and lepton colliders can be found in Refs. [22] and [37] respectively. It could be produced at hadron colliders in the scalar portal from couplings with the SM Higgs and other heavy scalars in the LRSM, or in the gauge portal from interacting with the SM photon and Z boson (and the heavy Z R boson). The pair production of H ±± R in the Drell-Yan process turns out to be much larger than that in the SM Higgs portal, as the latter is suppressed by the loop-induced effective hgg coupling (g here being the gluon) [22]. The associated production of H ±± R with the W R boson is suppressed by the W R mass and can be neglected for a doubly-charged scalar with mass M H ±± R 700 GeV for g R = g L . Similarly, at lepton colliders, the dominant pairproduction channel is either Drell-Yan or photon fusion, depending on the doubly-charged scalar mass [37]. For the sake of DV searches at future hadron and lepton colliders, we consider in this paper only the Drell-Yan production of H ±± R in the LRSM.

Low and high-energy constraints
Similar to the H ±± L case in Eq. (2.21), the H ±± R also contributes to the Z boson width and is constrained to have mass M H ±± R > m Z /2 from the precision Z-pole data, irrespective of how it decays or whether it is long-lived. Note that as a singlet under the SM gauge group SU (2) L , the coupling of H ±± R to the SM Z boson is only due to the hypercharge, proportional to −2 sin 2 θ w , and does not depend on the RH gauge coupling g R [22]. The  Table 2 (third column) and Fig. 2.

High-energy collider constraints
As the coupling of H ±± R to the SM Z boson is proportional to −2 sin 2 θ w , smaller than that of H ±± L which is (1 − 2 sin 2 θ w ), the Drell-Yan production cross sections of H ±± R at lepton and hadron colliders are thus significantly smaller than that of H ±± L , roughly 1.3 times smaller at LEP and 2.4 times smaller at Tevatron and LHC. The same-sign dilepton searches of doubly-charged scalars in Section 2.2.3 apply also to the H ±± R case, i.e. those in LEP [39][40][41], Tevatron [42][43][44][45] [46,47]. In some of the data analysis, the doubly-charged scalar is assumed to be an LH triplet; the production cross sections therein have to be rescaled accordingly, with the theoretical predictions multiplied by a factor of 1/1.3 at LEP and 1/2.4 at Tevatron and LHC. All the same-sign dilepton limits on H ±± R are collected in Fig. 12, in the six different flavor channels: ee (upper left), µµ (upper right), eµ (middle left), eτ (middle right), µτ (lower left) and τ τ (lower right). As a result of the smaller couplings of H ±± R to the Z boson, the dilepton limits are to some extent weaker than those for H ±± L in Fig. 4.
With the parity relation f L = f R , the Yukawa coupling matrix f R is also related to the active neutrino data in Table 1, as in the pure type-II seesaw. Analogous to Fig. 5, the dilepton limits also depend on the lightest neutrino mass m 0 , which are collected in Fig. 13 for both the NH and IH neutrino spectra, in the limit of Γ(H ±±  Fig. 11), and therefore, in the regions with cτ 0 (H ±± R ) > 1 mm (conservative) and 0.1 mm (aggressive), shaded respectively in darker and lighter gray in Figs. 14 and 15, the prompt same-sign dilepton limits are not applicable.
As in Fig. 8 for the H ±± L in the type-II seesaw, the HSCP limits from Ref.
[102] can be applied to the H ±± R case. To be conservative, we again use only the "trackeronly" data in Ref.
[102] to constrain the couplings of H ±± R , with the decay length range of 43 mm < bcτ 0 (H ±± L ) < 1100 mm [103]. Setting again the RH scale v R = 5 √ 2 TeV, g R = g L and rescaling the production cross section in Ref.
[102] to that of H ±± R at √ s = 13 TeV, the shaded orange and blue regions in Fig. 16 are excluded respectively for the NH and IH cases, with the lightest neutrino mass m 0 = 0. This corresponds to the Yukawa coupling range |f L | 10 −7 for the mass range 100 GeV < M H ±± R 155 GeV. Again, as in Fig. 8, the HSCP exclusion regions in Fig. 16 are not sensitive to the lightest neutrino mass m 0 for both the NH and IH scenarios.

Neutrinoless double beta decay
In the LRSM, with the heavy W R and H ±± R bosons, the RHNs N i and the RH interactions, there are additional diagrams contributing to 0νββ, cf. Fig. 17, as compared to the pure type-II seesaw case, cf. Fig. 3, which could be important, depending on the heavy particle masses and the couplings and mixings involved. In the type-II dominance, neglecting the heavy-light neutrino mixing (responsible for the type-I seesaw) and the small W − W R  Table 1, and the colorful bands are due to the 3σ uncertainties. The gray shaded region is excluded by the cosmological constraint on the sum of light neutrino masses i m i < 0.23 eV [99]. mixing, the 0νββ half-life is given by [53,57,58,[126][127][128][129][130][131] Table 1, and the colorful bands are due to the 3σ uncertainties. The darker and lighter gray regions correspond respectively to the proper decay length cτ 0 (H ±± R ) > 1 mm and 0.1 mm; within these regions the prompt dilepton limits are not applicable.
where the first term on the RHS is the LH contribution and same as in Eq. (2.18), whereas the second term is the RH contribution with M N being the corresponding NME and where m p is the proton mass, M N i the mass eigenvalues for the three heavy RHNs, and we have applied the fact that the mixing matrix of RHNs, U R = U under parity. Note that there is essentially no gauge dependence on the gauge coupling g R , as the W R boson couples to the fermions and H ±± R with the strength g 2 R , which is completely canceled out by the g R dependence in the W R propagator [72]. The H ±±   Figure 17. Additional Feynman diagram in the LRSM (in addition to Fig. 3) for the parton-level 0νββ, induced respectively by the heavy RHNs N i (left) and the RH doubly-charged scalar H ±± R (right), which correspond respectively to the amplitudes η N and η R DCS in Eq. (3.6). implies and therefore, To have a large η R DCS , the VEV ratio v L /v R has to be small, say v L ∼ eV and v R ∼ few TeV, such that the Yukawa couplings f R = f L m ν /v L are sizable or equivalently the RHN masses M N i are large in Eq. (3.8). This is a natural scenario one needs in the type-II seesaw dominance of LRSM to generate the tiny neutrino masses.
Comparing the two RH terms in Eq. (3.6), we find that to have a large H ±± R contribution to 0νββ we require the mass ratio M 2 to be large, such that the η N term is suppressed by the RHN masses, as compared to η R DCS . To be specific, we define the ratio When both ratios (M N /M ν )R DCS > 1 and |R DCS | > 1 (3.12) the 0νββ process in the LRSM is dominated by the H ±± R contribution and we can set meaningful limits on M H ±± R from the null results in current 0νββ searches, such as EXO-200 [133], KamLAND-Zen [134], GERDA [135], MAJORANA DEMONSTRATOR [136], CUORE [137] and NEMO-3 [138]. In particular, (M N /M ν )R DCS does not depend on any of the neutrino data in Table 1 but is only subject to the uncertainties of the NMEs M ν and M N . In addition, if the RHNs are too heavy (or equivalently the ratio (v L /v R ) is small for fixed value of v R ) and the doubly-charged scalar H ±± R is sufficiently light in Eq. (3.8), then the half life T 0ν 1/2 might be too small to be allowed by the current limits. To set limits on H ±± R from 0νββ, we use the most stringent half-life limits of 1.07×10 26 yrs for 136 Xe from KamLAND-Zen [134] and 8.0 × 10 25 yrs for 76 Ge from GERDA [135], both at the 90% CL. We assume the RH scale v R = 5 √ 2 TeV as above, and adopt the NMEs and the phase space factor G = 5.77 × 10 −15 yr −1 for 76 Ge and 3.56 × 10 −14 yr −1 for 136 Xe from Ref. [139]. We vary the neutrino oscillation parameters in Table 1 within their 3σ ranges and the lightest neutrino mass m 0 ∈ [0, 0.05] eV. Our results are shown in Fig. 18 for both NH (left) and IH (right) cases. All the gray points are excluded by either KamLAND-Zen [134] or GERDA [135] limit, which implies an upper bound on the combination 8.3 × 10 −7 GeV −2 for IH , (3.15) as shown by the long-dashed red lines in Fig. 18. Note that the dependence on the doublycharged scalar mass and the Yukawa coupling is different from the LFV limits in Table 1.
For heavier H ±± R and/or smaller coupling |(f R ) ee |, the contribution of H ±± R is suppressed [cf. Eq. (3.8)] and the 0νββ decay is dominated by the light and heavy neutrino terms in Eq. (3.6). In this case, the KamLAND-Zen and GERDA limits are no longer applicable to H ±± R , which is indicated by the short-dashed red lines in Fig. 18.

Displaced vertex prospects
As shown in Fig. 11, for sufficiently light H ±± R and sufficiently small Yukawa coupling |f R |, the decay length cτ 0 (H ±± R ) could reach up to 1 m. Requiring again a decay length of 1 mm < bcτ 0 (H ±± R ) < 1 (3) m at the LHC and ILC (future 100 TeV collider FCC-hh) and adopting the same setups as in Section 2.3, we predict the number of DV events for H ±± R → e ± e ± , e ± µ ± , µ ± µ ± at the HL-LHC 14 TeV with a luminosity of 3000 fb −1 , ILC 1 TeV with a luminosity of 1 ab −1 and FCC-hh 100 TeV with a luminosity of 30 ab −1 , which  → e ± e ± , e ± µ ± , µ ± µ ± in the LRSM, at HL-LHC 14 TeV with an integrated luminosity of 3000 fb −1 (solid contours), the 100 TeV collider FCC-hh with a luminosity of 30 ab −1 (dot-dashed contours) and ILC 1 TeV with 1 ab −1 (dashed contours). The red and blue contours respectively correspond to 10 and 100 DV events, as functions of M H ±± R and the largest Yukawa coupling |f R | max , for the neutrino spectra of NH (left) and IH (right) with lightest neutrino mass m 0 = 0. are depicted as the solid, dashed and dot-dashed curves in Fig. 19 respectively. The red and blue curves are the contours for respectively 10 and 100 DV events, for both NH with m 1 = 0 (left) and IH with m 3 = 0 (right). For extremely small couplings f R 10 −10 , with a fixed RH scale v R = 5 √ 2 TeV, the RHNs are expected to be lighter than the keV scale, i.e. M N = √ 2f R v R keV, and are tightly constrained by the cosmological data [140,141]. Thus we impose a lower cut of 10 −10 on the Yukawa coupling |f R | in Fig. 19. As shown in range can be probed at future 100 TeV colliders. Limited by the smaller center-of-mass energy, the sensitivities at ILC 1 TeV is relatively weaker. As in the pure type-II seesaw case, the total width of H ±± R and the DV prospects are not so sensitive to the lightest neutrino mass m 0 for either NH or IH case.
As in the case of H ±± L in type-II seesaw, the searches of DV same-sign dilepton signals from H ±± R are sensitive to relatively small Yukawa couplings |f R |, and are largely complementary to the low and high-energy constraints in Section 3.2. Similar to Fig. 10, we have collected the most important constraints and DV prospects for H ±± R in Fig. 20, which includes the most stringent LFV limit from µ → eγ (µ → eee) in Table 2 and Fig. 2 (brown), the 0νββ constraints in Fig. 18 (gray), the prompt dilepton constraints at LEP (pink), LHC 7 TeV (blue), 8 TeV (red) and 13 TeV (purple) in Fig. 14, the HSCP limits in Fig. 16 (bright yellow). The DV prospects with at least 100 events are shown for the HL-LHC (dashed pink), ILC 1 TeV (dashed green) and 100 TeV FCC-hh (dashed blue). The left and right panels are respectively for the NH and IH cases with the lightest neutrino mass m 0 = 0. We have spared the regions with cτ 0 (H ±± L ) > 0.1 mm from all the prompt dilepton search limits from LEP and LHC (cf. Fig. 14). In Fig. 20 we have re-interpreted the 0νββ limits in Eq. (3.15) as functions of the largest Yukawa coupling |f R | max . In the case of NH with m 1 = 0, the element |(m ν ) ee | is roughly 17 times smaller than the largest element |(m ν ) µµ |, thus the 0νββ limit in the left panel of Fig. 20 is much weaker than the IH case in the right panel wherein |(m ν ) ee | is the largest element.

Right-handed doubly-charged scalar in the LRSM with parity violation
If parity is not completely restored in the LRSM at the TeV scale, the Yukawa couplings f L and f R might be unequal. In addition, the minimization conditions of the scalar potential require that v L ∼ v 2 EW /v R [119]. This implies that for TeV scale v R and heavy RHNs we have v L ∼ O(MeV), which gives an unacceptably large type-II seesaw contribution ∼ f L v L to the light neutrino masses if f L = f R ∼ O (1). Without large cancellation of the type-I and type-II contributions, one natural solution is to eliminate the left-handed triplet ∆ L from the low-energy scale, e.g. in a LRSM with D-parity breaking [142]. Then the neutrino masses are generated via the type-I seesaw m ν −m D M −1 N m T D [111][112][113][114][115]. Without the parity symmetry, the couplings f R of H ±± R to the charged leptons are no longer directly related to the low-energy neutrino oscillation data, and all these elements can be considered as free parameters, though they are intimately connected to the heavy RHN masses through M N = √ 2f R v R . For illustration purpose and comparison to the parity-conserving case in Section 3, in this section we study a benchmark scenario with only one coupling (f R ) ee sizable in the Yukawa sector, and all other elements (f R ) αβ (αβ = ee) negligible. 5 The total width is then which can be readily evaluated as in Section 3.1.

Low and high-energy constraints
In our case here with only one element (f R ) ee , most of the LFV constraints in Table 2, such as those from µ → eee and µ → eγ, can not be used to constrain the coupling (f R ) ee , as they depend also on other entries of the f R matrix like (f R ) eµ . Therefore, we focus on the collider and 0νββ constraints that are directly applicable to (f R ) ee , irrespective of other Yukawa elements.

Collider constraints
The heavy H ±± R in the t-channel could mediate the Bhabha scattering e + e − → e + e − and interfere with the SM diagrams. This alters both the total cross section and the differential distributions. If the Yukawa coupling (f R ) ee is of order one, H ±± R could be probed up to the TeV scale [41,83]. By Fierz transformations, the coupling (f R ) ee of H ±± R contributes to the effective contact four-fermion interaction where Λ eff M H ±± R /|(f R ) ee | corresponds to the effective cutoff scale. This is constrained by the LEP e + e − → e + e − data in Ref. [84], which turns out to be more stringent than those in Refs. [41,83], and requires that Λ eff > 5.2 TeV.  By mediating the Møller scattering e − e − → e − e − , the coupling (f R ) ee of H ±± R will also be constrained by the upcoming MOLLER data, which could probe the effective scale Λ = M H ±± R /|(f R ) ee | 5.3 TeV, slightly stronger than the current limit from LEP data above [72].
The H ±± R → e ± e ± limits from LEP, Tevatron and LHC are the same as that in the upper left panel of Fig. 12 for the parity-conserving case. As in Sections 2 and 3, for sufficiently small |(f R ) ee |, H ±± R decays into W R boson with a sizable branching fraction.
Thus the e ± e ± limits from LEP and LHC can be used to set an upper bound on the Yukawa coupling |(f R ) ee | as functions of the mass of H ±± R , as shown in Fig. 21. Following Figs. 6 and 15, within the darker and lighter gray regions the lifetime cτ 0 (H ±± R ) > 1 mm and 0.1 mm respectively, and the dilepton limits are not applicable. The Yukawa coupling (f R ) ee here is not directly related to the active neutrino data, thus the limits in Fig. 21 are free from the neutrino oscillation data uncertainties in Table 1.
Analogous to the parity-conserving case in Section 3, the HSCP limits from Ref. [102] can be applied to H ±± R in the parity-violating LRSM, which is shown in Fig. 22. For concreteness, we adopt again the RH scale v R = 5 √ 2 TeV, the gauge coupling g R = g L and use conservatively only the "tracker-only" data to set the limit. As H ±± R here decays only into e ± e ± and the W R bosons, H ±± R is a little longer-lived than in the parity-conserving case and the HSCP limit in Fig. 22 is slightly stronger than that in Fig. 16.

Neutrinoless double beta decay
In the parity-violating LRSM, the contribution of RHNs N i and H ±± R are roughly the same as in Eqs. (3.7) and (3.8). However, without the parity relation f L = f R , the RHN mixing matrix U R is in general different from the PMNS matrix U in Eq. (2.9), and the contribution of H ±± R to 0νββ can be re-written as explicit function of the coupling (f R ) ee , i.e.
Comparing the two terms above, with  could be comparable to the η ν term and thus get constrained by the limits from KamLAND-Zen [134] and GERDA [135].
As in Section 3.2.2, we vary the neutrino data in Table 1 within their 3σ ranges and the lightest neutrino mass m 0 ∈ [0, 0.05 eV] and the RH scale v R = 5 √ 2 TeV, and the results are shown in Fig. 23 for both the NH (left) and IH (right) cases. As in Fig. 18, all the gray points are excluded by the current limits from KamLAND-Zen [134] and GERDA [135] and the blue points are allowed. The 0νββ limits on the Yukawa coupling (f R ) ee turn out to be roughly the same as in the parity-conserving LRSM: 1.0 × 10 −6 GeV −2 for NH , 6.7 × 10 −7 GeV −2 for IH .

Displaced vertex prospects
With the total width in Eq. (4.1), it is straightforward to estimate the number of displaced e ± e ± events from H ±± R decay at the HL-LHC, ILC 1 TeV and 100 TeV FCC-hh. With the same setups for these colliders as in Section 3.3 (including the benchmark values of v R = 5 √ 2 TeV and g R = g L ) and taking only the leptonic decays H ±± R → e ± e ± , the prospects are collected in Fig. 24. Within the red and blue contours we can have respectively at least 10 and 100 DV events. Again, for extremely small couplings |(f R ) ee | 10 −10 , the RHNs are expected to be lighter than the keV scale and get constrained by the cosmological data [140,141]. Therefore we set a lower cut of 10 −10 on the Yukawa coupling |(f R ) ee | in Fig. 24. Though the dilepton partial width of H ±± R might be smaller than in the parityconserving case, as we have only the e ± e ± decay modes in the leptonic sector, the sensitivity contours in Fig. 24 do not differ too much from those in Fig. 19. All the shaded regions are excluded, which are derived from the e + e − → e + e − limit from LEP (orange), the 0νββ constraints for NH (brown) and IH (dashed brown) neutrino spectra, the MOLLER prospect (dashed purple), limit on M H ±± R from Z boson width (gray), the dilepton constraints from LEP (pink), LHC 7 TeV (blue), 8 TeV (red) and 13 TeV (purple), and the CMS HSCP limit (bright yellow). For all the dilepton limits at LEP and LHC, we have left out the regions with cτ 0 (H ±± R ) > 0.1 mm (cf. Fig. 21).
The summary plot for the H ±± R in the parity-violating LRSM is shown in Fig. 25. As in the case of H ±± L in type-II seesaw in Section 2 and H ±± R in the parity-conserving LRSM in Section 3, with the coupling (f R ) ee , the displaced e ± e ± pair from H ±± R decay in the parity-violating LRSM at future HL-LHC (dashed pink), ILC (dashed green) and FCC-hh (dashed blue) are largely complementary to the prompt e ± e ± limits from LEP (pink) and LHC (blue, red and purple) and the CMS HSCP data (bright yellow), as well as to the low-energy constraints from the LEP e + e − → e + e − data (brown) and 0νββ (solid and dashed gray). As in Fig. 20, we have spared the regions with cτ 0 (H ±± L ) > 0.1 mm from all the dilepton limits from LEP and LHC (cf. Fig. 21). The future sensitivity of MOLLER is also shown in this plot (dashed purple), which exceeds the current 0νββ and LEP constraints for high-mass H ±± R [72].

Conclusion
As one of the well-motivated solution to the tiny neutrino mass puzzle, the type-II seesaw and its left-right symmetric extensions with new scalar triplets offer a rich phenomenology at both low-and high-energy frontiers. As a class of almost background-free processes at high-energy colliders, the same-sign dilepton decays from the doubly-charged scalars H ±± L,R → ± α ± β undoubtedly provide a "smoking-gun" signal of triplet scalars beyond the SM. However, all the past searches at LEP, Tevatron and LHC 7 TeV, 8 TeV and 13 TeV have focused on the prompt H ±± L,R decays assuming relatively large Yukawa couplings 10 −7 to 10 −6 (see e.g. Figs. 6, 7, 14, 15 and 21). We point out that in a large parameter space where the Yukawa couplings f L,R are small, the doubly-charged scalars H ±± L,R could be long-lived at the high-energy colliders, and decay into a pair of displaced same-sign leptons (and potentially other final states in the bosonic channels), which offer a complementary probe of the type-II seesaw.
We have estimated the DV prospects at the HL-LHC, 100 TeV FCC-hh and 1 TeV ILC for the LH doubly-charged scalar H ±± L in the type-II seesaw, and the RH doublycharged scalar H ±± In the LRSM, if we have the parity symmetry at the TeV scale, then the Yukawa couplings f L = f R , and the leptonic decays of H ±± R are the same as of H ±± L in type-II seesaw, dictated by the active neutrino data. With the RH gauge interaction, the decay H ±± R → W ± * R W ± * R is possible, but highly suppressed by the W R mass. Therefore, for sufficiently small coupling f R , H ±± R is long-lived at the high-energy colliders, just like H ±± L . As shown in Fig. 19, in a large parameter space with 10 −10 |f R | 10 −6 and m Z /2 < M H ±± R 200 (400) GeV, we could detect the DV signals at the HL-LHC and ILC 1 TeV (FCC-hh). If parity is violated in the LRSM at the TeV scale, then f R could be different from f L and cannot be directly constrained by the light neutrino data. As a benchmark study of this case, we have investigated a sample texture of f R with only one non-vanishing element (f R ) ee . Although the dileptonic decay width of H ±± R in the parityviolating case is smaller than in the parity-conserving case, the life-time of H ±± R and the DV sensitivity region do not change significantly, as shown in Fig. 24.
In both type-II seesaw and its LRSM extension, the low-energy, high-precision/intensity constraints, such as those from the LFV decays α → β γ δ , α → β γ, electron and muon g − 2, muonium oscillation, LEP e + e − → + − data, and 0νββ searches, set severe constraints on the (LFV) couplings (f L,R ) αβ of the doubly-charged scalars. However, these limits only restrict f L,R 10 −3 for a 100 GeV doubly-charged scalar. The LFV constraints get weaker for heavier doubly-charged scalars (cf. Table 2 and Figs. 2,18,23). The displaced vertex signals, as discussed here, are also largely complementary to these low-energy constraints, as clearly shown in Figs. 10, 20 and 25. Therefore, the displaced vertex signatures of doubly-charged scalars offer a new avenue to probe the origin of the tiny neutrino masses at future colliders. the hospitality and local support, where part of the work was done. This work is supported by the U.S. Department of Energy under Grant No. DE-SC0017987.

A Calculation of the four-body decays of doubly-charged scalars
In the type-II seesaw, if the doubly-charged scalar H ±± L from the scalar triplet ∆ L is light, which is directly relevant to the displaced vertex searches at future hadron and lepton colliders, it could decay into the SM quarks and leptons through two (off-shell) W -bosons, i.e.
with f = q, , ν running over all the quark and lepton flavors except for the top and bottom quarks. For simplicity we neglect the small quark mixings among different generations. Note that for the cases with identical particles in the final states f = f and f = f , there are two Feynman diagrams, which correspond respectively to the processes (W ±( * ) → ff )(W ±( * ) → f f ) and (W ±( * ) → ff )(W ±( * ) → f f ). The two diagrams interfere with each other, which is important for a light H ±± L [10]. Take the leptonic decays νν as an explicit example. Denoting the momenta of the charged leptons and neutrinos respectively as p 1 to p 4 , we obtain the reduced amplitude squared with ∆ ij = [(p i + p j ) 2 − m 2 W + im W Γ W ] −1 (Γ W being the W boson width). The Lorentz invariant four-body phase space dΠ 4 has only five independent kinematic variables, which can be chosen as the Cabibbo variables [143], i.e. the effective mass squared s 12 = (p 1 +p 2 ) 2 for the two charged leptons, the effective mass squared s 34 = (p 3 +p 4 ) 2 for the two neutrinos, the angle θ 12 of the momentum of the charged lepton in the dilepton rest frame with respect to the dilepton momentum in the rest frame of H ±± L , the angle θ 34 of the momentum of the neutrino ν in the dineutrino rest frame with respect to the dineutrino momentum in the rest frame of H ±± L , and the angle φ between the planes defined by the dilepton and dineutrino momenta. The ranges for these variables are respectively, with λ(a, b, c) ≡ a 2 + b 2 + c 2 − 2ab − 2ac − 2bc . (A.7) In the limit of M 2m W , to a good approximation, In an analogous way, one can calculate the decay widths into four quarks Γ(H ±± L → qq q q ) or quarks plus leptons Γ(H ±± L → qq ν). For concreteness, we define the same flavor (SF) and different flavor (DF) partial width units [10]: L → W ±( * ) W ±( * ) → e ± e ± ν e ν e ) , Γ DF ≡ Γ(H ±± L → W ±( * ) W ±( * ) → e ± µ ± ν e ν µ ) . (A.9) Then the leptonic, semileptonic and hadronic decay widths are respectively (where N C is the color factor) which sum up to the total width Γ total (H ±± L → W ±( * ) W ±( * ) ) (3 + 2N C )Γ SF + (3 + 5N C + 2N 2 C )Γ DF . (A.11) For the decay of RH doubly-charged scalar H ±± R in both the parity conserving and violating LRSMs: the calculation is almost the same as in the case of H ±± L , with the SM W boson replaced by the heavy W R boson. The calculations of the four-body decay here are done keeping in mind the displaced vertex searches at colliders; therefore, the Yukawa couplings f R are supposed to be very small. For the RH scale v R ∼ few TeV, the RHN masses m N = 2f R v R are expected to be much smaller than the masses of W R and H ±± R . Hence, we include here all the decay modes of W ± * R → qq , N (with q, q = u, d, s, c) in the limit of M N /M H ±± R 1.

B Formulas for the LFV decays
The partial width for the tree level three-body decay α → β γ δ is [54,144] with M ±± the doubly-charged scalar mass, f αβ the Yukawa couplings of doubly-charged scalar to the charged leptons, and G F the Fermi constant. All the formulas in this appendix apply to both the left-handed doubly-charged scalar H ±± L in type-II seesaw and the RH doubly-charged scalar H ±± R in the LRSM (therefore, we drop the subscripts "L" and "R"). At 1-loop level, the LFV couplings f αβ contribute to the two-body decays [51] BR( α → β γ) where α EM is the fine structure constant, and we have summed up all the diagrams involving a lepton γ running in the loop. In the type-II seesaw, it is equivalent to doing the summation γ (m ν ) T αγ (m ν ) βγ . Similarly, we can calculate the contributions of the doubly-charged scalar loops to the anomalous magnetic moments of electron and muon (with α = e, µ) [21,61,65,66]: where m α is the charged lepton mass and we have summed up again the loops involving all the three lepton flavors β = e, µ, τ . The muonium-antimuonium oscillation, i.e. the LFV conversion of the bound states (µ + e − ) ↔ (µ − e + ), can be induced by the effective four-fermion Lagrangian via exchanging the doubly-charged scalar [68]: with a = (α EM µ) −1 and µ = m e m µ /(m e + m µ ) the reduced mass of the muonium system. By performing a Fierz transformation, the effective coefficient is related to the couplings and doubly-charged scalar mass via These formulas have been used in the derivation of the LFV bounds in Table 2.