Leptonic Scalars and Collider Signatures in a UV-complete Model

We study the non-standard interactions of neutrinos with light leptonic scalars ($\phi$) in a global $(B-L)$-conserved ultraviolet (UV)-complete model. The model utilizes Type-II seesaw motivated neutrino interactions with an $SU(2)_L$-triplet scalar, along with an additional singlet in the scalar sector. This UV-completion leads to an enriched spectrum and consequently new observable signatures. We examine the low-energy lepton flavor violation constraints, as well as the perturbativity and unitarity constraints on the model parameters. Then we lay out a search strategy for the unique signature of the model resulting from the leptonic scalars at the hadron colliders via the processes $H^{\pm\pm} \to W^\pm W^\pm \phi$ and $H^\pm \to W^\pm \phi$ for both small and large leptonic Yukawa coupling cases. We find that via these associated production processes at the HL-LHC, the prospects of doubly-charged scalar $H^{\pm\pm}$ can reach up to 800 (500) GeV and 1.1 (0.8) TeV at the $2\sigma \ (5\sigma)$ significance for small and large Yukawa couplings, respectively. A future 100 TeV hadron collider will further increase the mass reaches up to 3.8 (2.6) TeV and 4 (2.7) TeV, at the $2\sigma \ (5\sigma)$ significance, respectively. We also demonstrate that the mass of $\phi$ can be determined at about 10% accuracy at the LHC for the large Yukawa coupling case even though it escapes as missing energy from the detectors.


Introduction
Explanation of tiny but non-zero masses of neutrinos, as confirmed in various experiments over the past two decades [1], requires new physics beyond the Standard Model (SM). In addition to the origin of their masses and mixing, neutrinos pose many more unanswered questions. For example, we still do not know whether the neutrino masses are of Diractype or Majorana-type; see Ref. [2] for a recent review. We would also like to understand whether the neutrino sector contains new interactions beyond those allowed by the SM gauge structure, i.e. the so-called non-standard interactions (NSIs); see Ref. [3] for a recent status report. Just like neutrinos, the origin of dark matter (DM) is also a puzzle and it is conceivable that these two puzzles could be somehow correlated at a fundamental level [4]. We also wonder whether the leptonic sector breaks CP-symmetry and whether it is responsible for the observed matter-antimatter asymmetry in the Universe [5]. In order to address these outstanding puzzles, construction of neutrino models and investigation of their predictions at various experiments are highly motivated. If neutrinos are Majorana particles, lepton number L, which is an accidental global symmetry of the SM Lagrangian, must be broken either at tree-level or loop-level. On the other hand, if neutrinos are Dirac particles, lepton number (or some non-anomalous symmetry that contains L, such as B − L) remains a good symmetry of the Lagrangian. We will focus on this latter case, assuming that B − L is conserved even in presence of higher-dimensional operators. Thus, any new, additional degrees of freedom must be charged appropriately under global B − L [6]. In a recent paper [7], motivated by certain observational considerations at the LHC and beyond, we considered the possibility that Dirac neutrinos could exhibit NSIs with a new (light) scalar field φ which has a B−L charge of +2 but is a singlet under the SM gauge group. These were dubbed as "leptonic scalars", which can only couple to right-handed neutrinos (ν R ) (or left-handed anti-neutrinos) like ν T R Cν R φ at the renormalizable level. Then the question arises as to how these leptonic scalars couple to the SM fields. At the dimension-6 level, we can write an effective coupling of the form 1 Λ 2 (LH)(LH)φ , (1.1) where L and H are the SM lepton and Higgs doublets, respectively, and Λ is the new physics scale. After electroweak (EW) symmetry breaking, the operator (1.1) yields flavordependent NSIs of neutrinos with the leptonic scalar of the form λ αβ φν α ν β . Furthermore, at energy scales below the mass of φ, this leads to an effective non-standard neutrino self-interaction, which could have observable cosmological consequences [8][9][10][11][12][13]. Our goal in this paper is to find an ultraviolet (UV)-completion of the operator (1.1) and to test the model at the ongoing LHC and future 100 TeV colliders, such as the Future Circular Collider (FCC-hh) at CERN [14] and the Super Proton-Proton Collider (SPPC) in China [15]. To be concrete, we adopt a Type-II seesaw motivated neutrino mass model [16][17][18][19][20][21], which can also account for the baryon asymmetry in the Universe [22]. In our model the neutral component of the triplet scalar field ∆ does not acquire a vacuum expectation value (VEV), which keeps the custodial symmetry intact. The lepton number is not broken and the neutrinos are Dirac-type in this model. We also add a SM-singlet complex scalar field Φ, which gives rise to the leptonic scalar φ in the model. Beyond the NSIs between the active neutrinos and the leptonic scalar, the particle spectrum and new interactions in this model lead to rich phenomenology and consequently new observable signatures. In some other UV-complete models, the effective interactions of φ with the SM neutrinos stemming from Eq. (1.1) might also be relevant to DM phenomenology [10,12,[23][24][25].
In this paper we will show that the distinguishing features of the signatures of our UV-complete model compared to the standard Type-II seesaw model is due to the new sources of missing energy carried away by φ, which would help the model to be detected at the ongoing LHC and future higher-energy colliders. After taking into account the current limits from the low-energy lepton flavor violating (LFV) constraints (cf. Table 2 and Fig. 2) and the theoretical limits from perturbativity and unitarity (see Fig. 3), we consider three scenarios with respectively small, large and intermediate Yukawa couplings of the leptonic scalar φ. In all these scenarios, φ can be produced either from the doublycharged scalar H ±± → W ± W ± φ or from the singly-charged scalar H ± → W ± φ − channels which are unique and absent in the standard Type-II seesaw. As the leptonic scalar φ decays exclusively into neutrinos, these new channels will lead to same-sign dilepton plus missing transverse energy plus jets signal at the hadron colliders. Detailed cut-based analysis is carried out for both scenarios, and the technique of Boosted Decision Tree (BDT) [26] is also utilized to improve the observational significance (see Tables 4 and 6). We find that the mass of doubly-charged scalars in the small and large Yukawa coupling scenarios can be probed up to respectively 800 GeV and 1.1 TeV at the 2σ significance, corresponding to a 95% confidence level, in the new channels at the high-luminosity LHC (HL-LHC) with integrated luminosity of 3 ab −1 , and can be improved up to 3.8 TeV and 4 TeV respectively at future 100 TeV colliders with luminosity of 30 ab −1 . This can be further improved in the intermediate Yukawa coupling case, with the help of increasing leptonic decay channel of the doubly-charged scalar. We also show that since in the large Yukawa coupling case, the missing energy is completely from the leptonic scalar in the associate production channel pp → H ±± H ∓ , its mass can be determined with an accuracy of about 10% at the HL-LHC.
The rest of the paper is organized as follows. In Section 2, we present the model details and lay out relevant experimental and theoretical constraints, including the key parameters and resultant main decay channels of H ±± and H ± in Section 2.1, the current LFV constraints on H ±± in Section 2.2, and the high-energy limits from perturbativity and unitarity in Section 2.3. In Section 3, we discuss our search strategy at the LHC and future 100 TeV hadron colliders, presenting the small Yukawa coupling case in Section 3.1, large Yukawa coupling scenario in Section 3.2, and the intermediate Yukawa coupling case in Section 3.3. We show the discovery potential by utilizing the cut-based analysis and the BDT techniques, and obtain the prospect for determining the mass of φ in the large Yukawa coupling case even though the scalar φ escapes from the detectors as missing energy. The main results are summarized in Section 4. For the sake of completeness, the complete set of Feynman rules for the model are listed in Appendix A. The functions G and F for some three-body decays are given in Appendix B. The renormalization group equations (RGEs) for the couplings are detailed in Appendix C. The perturbativity limits are analytically derived in Appendix D, and the unitarity limits are described in Appendix E.

The model
In this section, we present a global (B − L)-conserved UV-complete model of a leptonic scalar, which is motivated by the well-known Type-II seesaw model [16][17][18][19][20][21]. The enlarged particle content of the model includes a leptonic complex scalar Φ, which is a singlet under the SM gauge groups and carries a B −L charge of +2. The model contains also an SU (2) L triplet scalar ∆ with hypercharge +1 and B − L charge +2: The allowed Yukawa interactions in the model are given by where α, β = e, µ, τ are the lepton flavor indices, C is the charge-conjugation operator, σ 2 is the second Pauli matrix, y ν are the SM-like Yukawa couplings of the neutrinos, Y αβ are the new leptonic Yukawa couplings of the triplet that govern the heavy scalar phenomenology, andỹ ν are the Yukawa couplings of the leptonic scalar Φ to the righthanded neutrinos. In a (B − L)-conserved theory where ∆ and Φ do not acquire any VEV, neutrinos are Dirac fermions and non-zero neutrino masses can be generated after the EW symmetry breaking from the first term of the Yukawa Lagrangian given in Eq. (2.2), just like the other fermions in the SM. However, one requires y ν 10 −12 in order to satisfy the absolute neutrino mass constraints [27,28]. The kinetic and potential terms of the scalar sector are given by where the covariant derivatives are given by with g L and g Y respectively the gauge couplings for the SM gauge groups SU (2) L and U (1) Y , and σ a (a = 1, 2, 3) the Pauli matrices. The most general renormalizable potential involving the scalar fields of the model is given by where all the mass parameters m 2 H , M 2 ∆ , M 2 Φ and the quartic couplings λ and λ i are assumed to be real. The scalar ∆ in our model carries the same SU (3) C × SU (2) L × U (1) Y charges (1,3,1) as in the Type-II seesaw model. However, the presence of a (B −L)-charged Φ and the B −L conservation in our model have important phenomenological consequences associated with the triplet ∆, which is different from that in the Type-II seesaw scenario. In the Type-II seesaw model, the EW symmetry breaking induces a non-vanishing VEV for the triplet ∆ via the cubic term H T iσ 2 ∆ † H. However, due to the B − L conservation such a cubic term does not exist in our model, and as a result the triplet ∆ does not develop a VEV in our model. As we will see in Section 3, this leads to very interesting signatures  at the LHC and future 100 TeV colliders, which are key to distinguish our model from the  Type-II seesaw. After the EW symmetry breaking, the Higgs doublet H develops a VEV v = ( √ 2G F ) −1/2 with G F being the Fermi constant, and the mass matrix of the CP-even neutral components in the {h, δ 0r , Φ r } basis (here X r refers to the real component of the field X) is As the singlet and triplet scalars do not have VEVs, the component h from the SM doublet H does not mix with other neutral scalars, as can be seen from Eq. (2.7). Then h can be readily identified as the 125 GeV Higgs boson observed at the LHC [29,30], and the quartic coupling λ can be identified as the SM quartic coupling. The two remaining physical CPeven scalar eigenstates are from mixing of the components Φ r and δ 0r of the leptonic fields Φ and ∆ with B − L charge of +2, and thus are both physical leptonic scalars. Denoting H 1 as the lighter one and H 2 as the heavier one, they can be obtained by the following rotation where the mixing angle θ is given by 9) and the two eigenvalue masses are Similarly, the two CP-odd leptonic scalars (A 1 , A 2 ) from the imaginary components Φ i , δ 0i have exactly the same masses as the CP-even scalars, i.e.
For the sake of illustration, we choose to work in the regime where the leptonic scalars (A 1 , H 1 ) are in the mass range M h /2 < M H 1 ,A 1 O(100) GeV. The lower mass bound is to avoid the invisible decay of the SM Higgs h → H 1 H 1 , A 1 A 1 → νννν, while the upper bound is mainly motivated from our previous collider study [7], where the sensitivity in the vector boson fusion (VBF) channel was found to drop exponentially beyond 100 GeV or so. In order to keep the two leptonic scalars (A 1 , H 1 ) light, we choose the simplest scenario

Vertices
Couplings There is also a pair of heavy leptonic scalars H 2 and A 2 , which can either decay into neutrinos or cascade decay into gauge bosons and lighter scalars. For simplicity, we just assume (H 2 , A 2 ) to be heavier than the EW scale such that they are not relevant for our consideration here, and a detailed collider study of their phenomenology is deferred to future work. Finally, it is trivial to get the masses of the singly-and doubly-charged scalars, which are respectively given by Depending on the sign of λ 4 , H ± can be lighter or heavier than H ±± .  Table 1 are relevant for the pair production H ++ H −− and the associated production H ±± H ∓ of the doubly-charged scalar at hadron colliders, as in the Type-II seesaw case [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49]. The remaining couplings in Table 1 are relevant to the decays of H ± and H ±± . For the singlycharged scalar H ± , besides the leptonic final states, it can decay into a light neutral scalar H 1 or A 1 and a W boson, which is absent in the Type-II seesaw model. The corresponding partial decay widths are respectively

Key parameters and decay channels of
where the function As in the standard Type-II seesaw, the singly-charged scalar H ± can decay into a heavy scalar H 2 or A 2 and a W boson. However, the mass splitting between the triplet scalar components is severely constrained by the EW precision data (EWPT), in terms of the oblique S and T parameters [50,51]: depending on the triplet scalar masses, it is required that the mass splitting ∆M 50 GeV [37,46,52,53]. Therefore the W boson is always off-shell, i.e. H ± → W ± * H 2 , W ± * A 2 (the corresponding interaction can be found in Table 9), and the corresponding widths are given by where the function G(x, y) is explicitly given in Appendix B. This channel is highly suppressed by the off-shell W * boson, and will be neglected in the following sections. In our model, the doubly-charged scalar H ±± can decay into same-sign dilepton pairs and the three-body final state W ± W ± H 1 and W ± W ± A 1 . The partial widths are given respectively by where S αβ = 1/2 (1) for α = β (α = β) is a symmetry factor, and the dimensionless lengthy function F is put in Appendix B, which is a function of m 2 12 and m 2 23 . The phase space is integrated over the Dalitz plot where the ranges for m 2 12 and m 2 23 are respectively There is also a two-body bosonic channel, with the partial width with the function G(x, y) defined in Appendix B. As for the singly-charged scalar in Eq. (2.17), this channel is highly suppressed by the off-shell W boson, and will be neglected in the following analysis. Since the masses and decay properties of H 1 and A 1 are the same in our model, we henceforth collectively use φ to denote both the leptonic scalars In the standard Type-II seesaw, there is also the cascade decay channel for the doublycharged scalar [34,37]: In a large region of parameter space, the dilepton channels H ±± → ± ± and diboson channel H ±± → W ± W ± are highly suppressed respectively by the small Yukawa couplings Y αβ and the small VEV v ∆ of the triplet, and the doubly-charged scalar H ±± decays mostly via the cascade channel above. Similarly, in the standard Type-II seesaw model the singly-charged scalar H ± can decay into ± ν and hW ± , ZW ± , tb, which are respectively proportional to the couplings Y αβ and v ∆ [34]. When both Y αβ and v ∆ are relatively small, the decay of H ± will be dominated by where the W boson is again off-shell as a result of the EWPT limit on the triplet scalar mass splitting. As in the doubly-charged scalar case, the decay H ± → H 2 W ± * with a light H 2 in the Type-II seesaw is very similar to the channel H ± → W ± φ in our model, except for the off-shell W boson. Therefore, the new decay channels H ±± → W ± W ± φ and H ± → W ± φ make our model very different from the standard Type-II seesaw in the following aspects, which can be used to distinguish the two models at the high-energy colliders: • The W ± W ± φ final state from the H ±± decay is absent in the standard Type-II seesaw model, where the W bosons in the decays in Eqs. (2.19) and (2.22) are off-shell.
• Another distinguishing feature of this model is that the decays H ±± → W ± W ± φ and H ± → W ± φ does not necessarily correspond to the compressed mass gaps among different particle states of the triplet ∆, whereas in the standard Type-II seesaw 300 400 500 600 700 800 900 1000 300 400 500 600 700 800 900 1000 Depending on the value of the Yukawa couplings Y αβ , there are two distinct scenarios for the decays of H ±± and H ± : • Large Yukawa coupling scenario with Y αβ ∼ O(1). In this case the leptonic channels H ±± → ± ± and H ± → ± ν dominate, which are from the Yukawa interactions Y αβ .
For simplicity, we will not consider the intermediate scenarios, where the branching fractions (BRs) of bosonic and fermionic decay channels above are comparable. The Wdominated final states for small Yukawa couplings Y αβ depend on the scalar mixing angle sin θ, which in turn depends on λ 8 as shown in Eq. (2.9), where we find that λ 8 needs to be O(1) in order to have a sizable sin θ. The decay branching fractions of H ±± and H ± are shown respectively in the upper and lower panels of Fig. 1 as a function of their masses. The left and right panels are respectively for the large and small Yukawa coupling scenarios. As shown in the bottom left panel, if the Yukawa couplings are of order one, the dominant decay channels of H ± will be ± ν, but the bosonic channel W ± φ is still feasible in the high mass regime with a branching fraction around 10%. For small Yukawa couplings of order O(10 −2 ), the singly-charged scalar H ± decays predominantly into W ± φ, as demonstrated in the bottom right panel. On the other hand, as shown in the top left panel, the doubly-charged scalar H ±± will decay mostly to ± ± if the Yukawa couplings are large, while the W ± W ± φ channel is dominant for small Yukawa couplings although a crossover happens for low M H ±± , as shown in the top right panel.

LFV constraints
There exist numerous constraints on the charged Higgs sector from the low-energy flavor data, such as those from the LFV decays α → β γ δ , α → β γ [1,58], anomalous electron [59] and muon [60,61] magnetic moments, muonium oscillation [62], and the LEP e + e − → + − data [63]. Following Ref. [43], the updated LFV limits on the Yukawa couplings Y αβ are collected in Table 2, and the most stringent ones are shown in Fig. 2, as a function of the doubly-charged scalar mass M H ±± . We see that the products involving two flavor transitions are highly constrained, while the bounds on an individual coupling are much weaker, especially for the tau flavor.
It should be noted that the contributions of H ±± to the electron and muon g − 2 are always negative [64]. Therefore, the recent measurement of muon g − 2 at Fermilab [61] cannot be interpreted as the effect of H ±± in our model. On the other hand, we can use the reported measurement of Ref. [61] which is 4.2σ larger than the SM prediction [65], to set limits on the H ±± parameter space. We will use a conservative 5σ bound, i.e. require that the magnitude of the new contribution to (g − 2) µ from H ±± must not exceed 0.8 × 59 × 10 −11 . The corresponding limit on the Yukawa coupling Y µβ is shown by the purple shaded region in Fig. 2 and also in Table 2. Note that if a light scalar has an LFV coupling h µτ to muon and tau, it could be a viable candidate to explain the muon g − 2 anomaly, while satisfying all current constraints [66][67][68][69][70][71][72]. Such neutral scalar interpretations of muon g − 2 anomaly can be definitively tested at a future muon collider [73][74][75][76][77]. The doubly-charged scalar H ±± can induce leptonic decays of SM Z and Higgs boson at 1-loop level. With the coupling Y αβ , the corresponding partial widths are respectively [78,79] where M Z is the Z boson mass, m γ is the mass for the charged lepton γ , the factor of λ 1 v in Eq. (2.25) is from the trilinear scalar coupling hH ++ H −− in Table 7, and the loop Table 2. Upper limits on the Yukawa couplings |Y αβ | 2 (or |Y † αγ Y βγ |) from the current experimental limits on the LFV branching fractions of α → β γ δ , α → β γ [1,58], anomalous electron [59] and muon [60,61] magnetic moments, muonium oscillation [62], and LEP e + e − → + − data [63]. See also Fig. 2.    data [1] can only exclude |Y αβ | 2 1 for M H ±± = 1 TeV, and the corresponding limits are much weaker than those in Table 2 and Fig. 2.
Similarly, given the coupling Y αβ , the couplings of the leptonic scalar φ with neutrinos induce the tree-level invisible decays Z → ν α ν β φ, h → ν α ν β φ and the leptonic decay W → α ν β φ. However, the limits from current precision EW and Higgs data are at most Y αβ O(1) [6,7], and therefore, are not shown in Table 2 and Fig. 2.

High-energy behavior: perturbativity and unitarity limits
Since larger values of λ 8 and Y αβ play important roles for the hadron collider signal of this model, let us first check the largest values of these couplings which can be accommodated at the EW scale without becoming non-perturbative at a higher energy scale. For the purpose of illustration, we set just one Yukawa coupling Y µµ to be non-vanishing, with all other Yukawa couplings Y αβ (αβ = µµ) to be zero. This choice is compatible with the current limits in Table 2, as the products of the Yukawa couplings must be small due to the existing LFV limits, while a single coupling (Y µµ in our case) can be as large as To implement the perturbativity limits from the high-energy scale, we use the RGEs in Appendix C for all the gauge, scalar and Yukawa couplings given in Eqs. (2.4), (2.5), (2.2) and (2.6). From the RGEs, we find that λ 8 depends on Y µµ at one-loop level, since both λ 8 and Y µµ are associated with the interaction terms which involve the triplet scalars. The dependence of perturbativity limits on λ 8 on the Yukawa coupling Y µµ (v) at the EW scale is shown in the left panel of Fig. 3, with perturbativity up to Planck scale M Pl and the grand unified theory (GUT) scale M GUT for the purple and red lines, and up to the 100 TeV and 10 TeV scales for the orange and pink lines, respectively. Comparing these lines, we can see that the perturbativity limits on λ 8 are very sensitive to the value of Y µµ at the EW scale. To have a perturbative λ 8 at the 10 TeV (100 TeV) scale, it is required that the coupling Y µµ (v) 1.6 (1.3). For a perturbative theory up to the GUT or Planck scale, the coupling Y µµ needs to be even smaller, i.e. Y µµ (v) 0.67. The perturbativity limits on λ 8 and Y µµ at the EW scale as function of the scale 10 TeV < µ < M Pl are shown in the right panel of Fig. 2. For the quartic coupling λ 8 , the solid and dashed lines correspond respectively to the cases of Y µµ set at the perturbative limit and Y µµ = 0 at the EW scale. As shown in both the two panels of Fig. 2, the quartic coupling λ 8 can be as large as 4 (2.7), with perturbativity holding up to 10 TeV (100 TeV). With the requirement of perturbativity up to the Planck (GUT) scale, we have λ 8 0.48 (0.58) at the EW scale.
The high-energy behavior of λ 8 , Y µµ and other couplings can be understood analytically from the solutions of RGEs for these couplings. As a rough approximation, let us first see the analytical solution of Y µµ without including the contributions from the gauge couplings , it is trivial to get the analytical solution of α µ at scale µ from Eq. (C.14) as It is clear from the above equation that the coupling Y µµ is not asymptotically free and will blow up when the scale parameter approaches the value of .

(2.27)
With an initial value of Y µµ (v) = 1.5 at the EW scale, we can get the critical value of t c 4.39, which corresponds to an energy scale of µ 20 TeV. The full analytic solution of Y µµ including the gauge coupling contributions is shown in Appendix D. Following the running of gauge couplings, and taking g L (M Z ) = 0.65100, g Y (M Z ) = 0.357254 [80][81][82][83][84], we find that in this case t c = 4.67, which corresponds to µ 26 TeV. The contribution of Y µµ to the evolution of λ 8 can be obtained from the following analytical solution of the RGE for λ 8 (see Appendix D for more details) where E 8 depends on Y µµ as well as the couplings g L, Y and the top-quark Yukawa coupling y t and is given in Eq. (D.12). As soon as Y µµ turns non-perturbative, the exponential becomes very large and λ 8 also becomes non-perturbative. We have also checked the unitarity constraints on Y µµ and λ 8 , and the details are given in Appendix E. It is found that the unitarity constraints are much weaker λ 8 < 10.0, compared to the perturbativity constraints obtained here.

Collider signatures
In this section we analyze the striking signatures of this model at the LHC and future 100 TeV hadron colliders. We consider both the pair production and the associated production  channels: The production cross sections in the two channels for the doubly-charged scalar coming from an SU (2) L -triplet ∆ at the 14 TeV LHC and future 100 TeV colliders have been estimated in Refs. [44,85], which are reproduced in Fig. 4. As shown in Section 2.1, the final states associated with these production processes depend on the decay branching fractions of H ±± and H ± . Our model predicts novel decay processes where the light leptonic scalars φ = H 1 , A 1 will escape from detection and lead to missing momentum. This can be used to distinguish our model from the standard Type-II seesaw.
In this paper, we will focus on these novel channels. The prospects of the small Yukawa coupling scenario at future hadron colliders are investigated in Section 3.1, the large Yukawa coupling case is analyzed in Section 3.2, and the intermediate Yukawa coupling case is considered in Section 3.3.

Small Yukawa coupling scenario
One typical choice of parameter is that the Yukawa coupling Y αβ 10 −2 to satisfy all the low-energy experimental limits in Section 2.2. Note that this choice of Y αβ would result in an effective ν α ν β φ coupling λ αβ of order 10 −3 , which is too small to probe in the VBF channel discussed in Ref. [7], but accessible in our UV-complete model due to the additional interactions, as shown below. In particular, under this choice of small Yukawa coupling, the doubly-charged scalar H ±± will mostly decay to two W bosons and a light neutral leptonic scalar φ = H 1 , A 1 ; cf. the top right panel of Fig. 1. With two same-sign W bosons decaying leptonically and the other two decaying hadronically, the final state of our signal features two same-sign leptons (e or µ) plus jets and large missing transverse momentum in the pair production channel, i.e.
Similarly, we also have the associated production pp → H ±± H ∓ with H ∓ → W ∓ φ which also has the same final states. However, due to the presence of less number of W 's, the contribution from the associated production is small to our signal. We use FeynRules [86] to define the fields and the Lagrangian of our model, then the resulting UFO model file is fed into MadGraph5 aMC@NLO [87] to generate the Monte Carlo events where the decay of vector bosons is achieved by the Madspin [88] module integrated within MadGraph5. Next-to-leading order corrections are included by a k-factor of 1.25 [89] for our signal process. The leading SM backgrounds come from W Z and W W productions and the sub-leading ones from W W W and ttW processes are also considered. We use MadGraph5 to generate the background events, and the leading ones are generated with two extra jets to properly account for the jet multiplicity in the final states. The events from the hard processes are showered with Pythia8 [90] and the jets are clustered using Fastjet [91] with the anti-k T algorithm [92] and the cone radius ∆R = 0.4. All the signal and background events are smeared to simulate the detector effect by our own code using Delphes CMS PhaseII cards [93].
Electrons (muons) are selected by requiring that p T > 10 GeV and |η| < 2.47 (2.5), jets are required to have p T > 20 GeV and |η| < 3. We adopt the b-tagging formula from the in unit of GeV) [93]. We apply some pre-selection cuts before launching the carefully designed analysis below. First, all events should have exactly two same-sign leptons and the number of jets should be at least 3: N jet ≥ 3. Finally we veto any event with b-tagged jet: N b-jet = 0.

Cut-based analysis
The same-sign W pair signal from H ±± → W ± W ± has been searched for at the LHC by the ATLAS collaboration [56,57]. In the searches of same-sign dilepton plus jets plus missing energy, the most stringent lower limit on doubly-charged scalar mass is 350 GeV [57]. As a case study, we first consider the scenario of M H ±± = 400 GeV, which satisfies the current direct LHC constraints. The kinematic variables we use to distinguish the signal from backgrounds are the missing transverse energy E miss T , the effective mass M eff defined as scalar sum of transverse momenta of all reconstructed leptons, jets, and missing energy, the separation ∆R between two leptons, the azimuthal angle ∆φ( , E miss T ) between the two lepton system and E miss T , the invariant mass of all jets M jets , and the cluster transverse mass from jets and E miss T defined as [94] M jets To enhance the signal-to-background ratio, the selection cuts we applied are as follows, and the corresponding cut-flows for the cross sections of signal and backgrounds are collected in Table 3.
• 0.3 < ∆R < 2.0. The lower limit of ∆R separates the leptons for isolation. The leptons in our signal emerge from the decay of two same-sign W bosons which are from the decay of H ±± . However, the leptons associated with the background processes emerge from the decays of W and Z bosons which are well separated. Therefore, the leptons in the signal tend to have smaller ∆R . The distributions of ∆R for the signal and backgrounds are presented in the top left panel of Fig. 5.
• E miss T > 110 GeV. One of the decay products emerging from H ±± is the light neutral scalar φ which decays only into neutrinos and appears to be invisible in the detector. Due to the existence of the massive φ along with the neutrinos from W boson decay, our signal tends to have larger missing transverse energy compared to the background processes (see the top right panel of Fig. 5 for distributions). Consequently, we choose a high E miss T threshold to distinguish the signal from backgrounds.
• M eff > 350 GeV. Borrowed from the SUSY searches [95,96], the effective mass M eff is a measure of the overall activity of the event. It provides a good discrimination especially for signals with energetic jets. The jets in our signal are from W decay while the jets associated with backgrounds are from the QCD productions, which makes the jets from the signal to be more energetic in general. This can be seen in the middle left panel of Fig. 5. Thus the effective mass associated with the signal is distributed at higher values.
• M jets T > 300 GeV. Since the decay products from H ±± contain invisible particles, we cannot fully reconstruct its mass. The transverse mass M jets T is an alternative option in this situation. We choose to reconstruct the transverse mass M jets T of H ±± using jets and E miss T in order to reproduce its mass peak as close as possible. From the distributions shown in the middle right panel of Fig. 5, we can see that the transverse mass for the signal peaks around 400 GeV while for backgrounds it peaks at a smaller value. Consequently, a large M jets T cut can help us to discriminate the signal from backgrounds.
• 150 GeV < M jets < 350 GeV. As mentioned above, the jets in the signal emerge from the hadronic decays of W boson while the jets associated with the main backgrounds are from QCD production. As a result, the invariant mass of all jets from backgrounds has a broader and flatter distribution, while the distribution for the signal is concentrated in the region between the two W boson mass threshold and the doubly-charged scalar mass, as shown in the bottom left panel of Fig. 5. This provides a good observable to distinguish the signal from backgrounds.
• ∆φ( , E miss T ) < 1.5. The contributions to E miss T associated with the signal are neutrinos and the light neutral scalar φ from the decay of H ±± . The signal decay products include also same-sign dileptons and, consequently, the azimuthal angle between the same-sign dilepton and E miss After all the cuts, it is found in Table 3 that the cross section for our signal is only a few times smaller than that for the SM backgrounds. To calculate the signal significance, we use the metric σ = S/ √ S + B where S and B are the numbers of events for signal and backgrounds respectively, and we have not included any systematic uncertainties in our analysis. The expected event yields at the HL-LHC after all the cuts above are shown in Table 4. It is clear that the significance can reach 5σ in the cut-based analysis, which implies a great potential for discovery of the signal H ±± → W ± W ± φ at the HL-LHC.

BDT improvement
In order to further control the backgrounds, we adopt the BDT technique. In particular, we use the XGBoost package [97] to build the BDT. In addition to the variables mentioned above, we also feed the BDT the following variables: • invariant mass M of same-sign dileptons;   • azimuthal angle ∆φ(j 1 , E miss T ) between leading jet and E miss T ; • separation ∆R 1 j 1 and ∆R 2 j 1 of leptons and leading jet; • minimum separation min∆R jj of two jets; • minimum separation min∆R j of leptons and jets; • minimum invariant mass minM jj of two jets.
Some of the distributions, such as those for minM jj , M , M T and min∆R jj , are shown in Fig. 6. We will see in the lower right panel of Fig. 7 that these distributions are also very important for discriminating the signal from backgrounds. The hyperparameters we used to train BDT are as follows: the learning rate is 0.1, the number of trees is 500, the maximum depth of each tree is 3, the fraction of events to train tree on is 0.6, the fraction of features to train tree on is 0.8, the minimum sum of instance weight needed in a child is 3, and the minimum loss reduction required to make a further partition on a leaf node of the tree is 0.2.
We split the data set into a training set and a testing set to make sure that there is no over-fitting. The BDT responses for our testing set are shown in the upper panel of Fig. 7. The BDT response close to 1 means the event is more signal-like while the response around 0 means the event is more background-like. We can see that our BDT classifier behaves quite good on the testing set. The receiver operating characteristic curve (ROC curve) of BDT and its feature importance are presented respectively in the lower left and right panels of Fig. 7. The feature importance is measured by "gain", which is defined as the average training loss reduction gained when using a feature for splitting. The importance plot shows the top 10 important variables in the BDT training. The observables used in the cut-based analysis rank among the top 10 by the BDT, where the most important one is the effective mass M eff , followed by M jets and E miss T . In addition, the BDT determines that the distributions minM jj , M , M T and min∆R jj shown in Fig. 6 are also very important.
We choose the BDT cut such that it maximizes the significance of signal. For M H ±± = 400 GeV, the event yields of signal and backgrounds after the BDT cut are reported in Table 4. We can see that the BDT can eliminate backgrounds significantly while keeping most of the signal. The significance can reach 10.36 with the help of BDT, which is improved remarkably in comparison to the cut-based method in Section 3.1.1. To explore the discovery potential of H ±± in the small Yukawa coupling scenario at the HL-LHC, we generate event samples for the signal process for M H ±± in the range from 300 GeV to 1.2 TeV with the step of 100 GeV. We build BDTs for different masses to discriminate the signal from the SM backgrounds and maximize the significance. The significance as a function of the doubly-charged scalar mass M H ±± is shown in Figure 8 as the solid line. It is found that we can reach M H ±± 800 GeV at the 2σ significance in the W ± W ± φ channel for the small Yukawa scenario at the HL-LHC.
At future 100 TeV hadron colliders such as FCC-hh and SPPC, the production cross section of H ±± can be largely enhanced, as shown in Fig. 4. Following the same BDT analysis as that at 14 TeV LHC, the significance of signal as a function of M H ±± is presented as the dashed line in Figure 8. Benefiting from the large cross section, the prospect of M H ±± can reach up to 3.8 TeV at the 2σ sensitivity at the 100 TeV collider.

Large Yukawa coupling scenario
Another case of interest in contrast to the previous one is the large Yukawa coupling scenario. According to the low-energy flavor limits in Table 2, most elements of the Yukawa coupling matrix Y αβ are bounded to be small while Y µµ can be of O(1) for TeV-scale H ±± . Note that the effective coupling between neutrinos and leptonic scalars (H 1 and A 1 ) in our model is of order λ αβ ∼ 2 √ 2 Y αβ sin θ (cf. Table 1); therefore, Y µµ ∼ O(1) could also be probed at hadron colliders via the VBF process discussed in our previous study [7]. For example, a Y µµ = 1.5 Yukawa coupling leads to an effective coupling λ µµ ∼ 0.58 which is within the 2σ LHC sensitivity in the VBF mode [7]. Although the Y τ τ coupling is the least constrained (cf. Table 2), final states involving taus at the hadron colliders are more difficult to analyze; therefore, we only focus on the muon final states and leave the tau signal for a future work.
After considering the constraints from perturbativity and unitarity in Section 2.3, we found that the Y µµ component can be as high as 1.5 as presented in Fig. 3. This is still consistent with the muon g − 2 bound given in Table 2 for a TeV-scale H ±± . In this scenario, the contributions from other Yukawa coupling elements are negligible, and the doubly-charged scalar H ±± decays predominately into a pair of same-sign muons, i.e. BR(H ±± → µ ± µ ± ) 100%. For large Y µµ the main decay channel for the singly-charged scalar will be H ± → µ ± ν. However, the H ± → W ± φ channel is still feasible and its BR varies from 10% to 20% depending on the mass of H ± , as shown in the lower left panel of Fig. 1. With the W boson decaying hadronically, the φ induced signal at the hadron collider emerges from the associated production channel as follows: i.e. same-sign muon pair plus two jets from W boson decay plus transverse missing energy from φ. We should mention here that the traditional 3-µ or 4-µ channels will still be the discovery mode for this scenario, but our choice of the final state and analysis is useful to determine the mass of leptonic scalar φ (H 1 /A 1 ) as will be shown in Section 3.2.2.

Analysis and mass reaches
The signal samples are generated by using MadGraph5. Since the final state is similar to the small Yukawa coupling case, we use the same background samples as in Section 3.1. The muon and jet definitions are also kept unchanged. All the events are required to have two reconstructed same-sign muons and two jets without any b-tagged jet. In addition, to further control the backgrounds the following cuts are applied, and the corresponding cut-flows for the cross sections of signal and backgrounds are presented in Table 5.
in the signal is from the scalar φ = H 1 , A 1 , it tends to have a larger value than the backgrounds with a broader distribution, as shown in the upper panel of Fig. 9.
• ∆R jj < 2. The two jets in the signal are from the decay products of a very energetic W boson, so they tend to be more collimated than the backgrounds. With the distributions shown in the lower left panel of Fig. 9   • 700 GeV < M µ ± µ ± < 1100 GeV. Since the same-sign muon pair appears from the decay of the H ±± boson, their Breit-Wigner peak provides a strong discrimination against the SM backgrounds. This can be clearly seen in the lower right panel of Fig. 9.
As a result of very distinct topologies of the signal and backgrounds, the number of background events can be highly suppressed after the cuts, as reported in Table 5. The Table 6. Number of events in cut-based and BDT analysis for associated production H ±± H ∓ in the benchmark scenario (3.4) and the SM backgrounds at the HL-LHC with 3 ab −1 luminosity. The last column shows the significance of signal. Backgrounds that are essentially eliminated by our cuts are denoted by "−"s.  Table 6. In the cut-based analysis, the significance can reach σ = 3.67 for the benchmark scenario in Eq. (3.4).
As in the small Yukawa coupling case in Section 3.1, BDT can help us improve to some extent the sensitivity. In addition to the observables above in cut-and-count analysis, we also use the following observables: • transverse momenta p T, µ 1 and p T, µ 2 of the two muons; • effective mass M eff ; • invariant mass M jj of two jets; • total transverse momentum p T, jj of two jets; • transverse mass M T constructed from jets and E miss T ; • azimuthal angle ∆φ(µµ, E miss T ) between two muons and E miss T .
The BDT score distribution is presented in Fig. 10. As expected, the signal is well separated from the backgrounds. Therefore the BDT can eliminate almost all the background events while keeping most of the signal events. The expected numbers of signal and background events after optimal BDT cuts are collected in the last row of Table 6. With the help of BDT, the sensitivity can reach a higher value at σ = 4.35.
Since the backgrounds can be highly suppressed by the BDT analysis, the significance will be mainly determined by the cross section of signal, which in turn depends on the mass of H ±± . We generate our signal samples in the step of 100 GeV for M H ±± varying from 900 GeV to 1.5 TeV. The resultant significance at the HL-LHC as a function of M H ±± is shown in Fig. 11 as the solid line. It turns out H ±± can be probed up to 1.1 TeV at the 2σ sensitivity at the HL-LHC in the large Yukawa coupling scenario. At a future 100 TeV collider, the production cross section σ(pp → H ±± H ∓ ) can be enhanced by over one order of magnitude (see Fig. 4). The corresponding prospect of M H ±± can reach up to 4 TeV at the 2σ sensitivity, which is indicated by the dashed line in Fig. 11.

Mass determination of the leptonic scalar φ
For the associated production H ±± H ∓ in the large Yukawa coupling case, the only missing particles is φ = H 1 , A 1 , which provides a possibility to measure its mass. However, at the hadron colliders such as LHC, we can at most determine the transverse momentum of φ while its longitudinal momentum is completely lost. Therefore the usual method to determine a particle's mass is not applicable here. An alternative approach is to utilizes the transverse mass of a mother particle whose decay products contain a massive invisible daughter particle. To achieve this, we need to modify the definition of transverse mass in Eq. (3.3). In that equation, we do not consider the mass of the missing particles but simply assume the transverse energy of missing particles to be the same as the missing transverse momentum. The modified definition of missing transverse energy is wherem is the assumed mass of φ, and p T, miss is the missing transverse momentum. Thus the cluster transverse mass M T can be re-expressed as a function of the assumed massm: As shown in Refs. [109,110], the endpoint of M T distribution will increase with the assumed massm, and a kink will appear at the point ofm = m when the assumed massm is equal to the real mass m of the invisible daughter particle.
As an explicit example, we choose the scalar mass m φ = 89.28 GeV, and the masses of charged scalars are set as in Eq. (3.4). We calculate the transverse mass M T of the simulated events by Eq. (3.6) with different choices ofm, and then use package EdgeFinder [111] to find the endpoint of M T distribution for eachm choice. The result is shown in Fig. 12. By fitting the data points, a kink is found atm = (93.60 ± 11.43) GeV. Comparingm at the kink with the real mass m φ , we find that this method provides a great potential for measuring the mass of the invisible light scalar φ = H 1 , A 1 at the LHC.
We note that the fitting process may be associated with some uncertainties for both M T edges and m φ . To test the robustness of fitting result, we smear the M T edge according to the initial error bars from the EdgeFinder package in a Normal distribution. Using 100 points for trial, we find that the mass determination by the kink yields a resultm = (93.55 ± 11.41) GeV. Since the uncertainty range does not change, we can state that the kink-finding method leads to a rather reliable mass determination. It should be noted that it is difficult to apply the mass determination technique used here to the small Yukawa coupling scenario in Section 3.1, since in that case φ is from H ±± decay, which leads to the appearance of missing energy from both neutrinos from W boson decay and the invisible scalar φ.    For the completeness of our study, we also investigate the mass reach in the intermediate Yukawa coupling scenario. If the Yukawa coupling is of order O(10 −2 − 1), the branching fraction of leptonic channel H ±± → ± α ± β could be comparable to the bosonic channel H ±± → W ± W ± φ. Since these two channels make up all the doubly-charged scalar decay, once we fix the branching fraction of one channel, the other one could be easily obtained, thus we could scale the cross section of pair production pp → H ++ H −− accordingly to estimate the mass reach with different final states.
The first process we consider is the same as that in small Yukawa coupling scenario in Section 3.1, i.e. with both doubly-charged scalars decaying bosonically, and the samesign W bosons decaying leptonically. The final state would be a pair of same-sign leptons plus jets and large missing transverse energy: Since the branching fraction of the bosonic channel is no longer 100% for intermediate Yukawa couplings, the mass reach would be undermined by the rising branching fraction of the leptonic decay channel H ±± → ± ± . The significance of H ±± in this channel is shown in the top left panel of Fig. 13 as function of M H ±± , where the red and blue lines are respectively for the HL-LHC and future 100 TeV collider. As shown in this figure, the doubly-charged scalar can be probed at the 2σ C.L. with mass below 500 GeV (2.9 TeV) at the HL-LHC (future 100 TeV collider) for BR(H ±± → ± ± ) = 50%. As the leptonic BR decreases, the mass reach increases, as expected, up to the ones reported in Fig. 8 (corresponding to BR(H ±± → ± ± )=0).
When the leptonic branching fraction is large enough, it is more likely that one of the pair-produced H ±± decays leptonically and the other one decays bosonically. In this case, the final states with two or three charged leptons are of great interest. The two same-sign leptons can be used to reconstruct the Breit-Wigner peak of the mother doubly-changed scalar, making such signals almost background free. The corresponding significances of • In light of all the low-energy LFV constraints, the coupling Y µµ can be as large as O(1) for a TeV-scale H ±± while all other Yukawa couplings are more stringently constrained (see Fig. 2 and Table 2). Using RGEs, we have also determined the largest values of λ 8 and Y αβ at the EW scale in order to keep the theory perturbative all the way to the UV-complete scale, as shown in Fig. 3. It is remarkable that as a good approximation the perturbativity limits can be obtained analytically. We checked also the unitarity constraints for these couplings and found them to be much weaker compared to the perturbativity limits.
• Originating from the gauge couplings, H ±± and H ± can decay into the light leptonic The scalar φ provides additional sources of missing energy (along with the neutrinos from the decays of W when the leptonic final states are selected) since it decays only into neutrinos, i.e. φ → νν. These new decay channels H ±± → W ± W ± φ and H ± → W ± φ dominate for small Y αβ . For O(1) values of Y αβ , H ±± and H ± decay primarily into ± ± and ± ν respectively, while the decay H ± → W ± φ can still occur with a BR of 10% − 20% level, as shown in the left panels of Fig. 1, which is used for signal selection in this case.
• For our LHC analysis, we utilized the presence of the new source of missing energy from φ in the decays of H ±± and H ± , and the BDT analysis can improve significantly the signal significance, in particular for the small Yukawa coupling case. At the HL-LHC, we found that for small and large Y αβ , the 2σ (5σ) sensitivity reaches for H ±± are respectively 800 (500) GeV and 1.1 (0.8) TeV (see Tables 4 and 6), as denoted by the solid lines in Figs. 8 and 11. These prospects are well above the current LHC constraints.
• At a future 100 TeV collider, the production cross section of H ±± can be enhanced by over one order of magnitude in both pair production and associated production channels (see Fig. 4). Therefore the mass reaches of H ±± can be largely improved via the observation of φ induced signals. For the small and large Yukawa coupling cases, the mass M H ±± can reach up to 3.8 (2.6) TeV and 4 (2.7) TeV respectively at the 2σ (5σ) significance (see Tables 4 and 6 • In the large Yukawa coupling scenario, the missing transverse energy is completely from the invisible light scalar φ at the parton level in the pp → H ±± H ∓ → µ ± µ ± + 2j + E miss T channel, and the mass m φ can be determined with 10% accuracy at the LHC via the transverse mass distributions associated with jets and missing energy. This is demonstrated in Fig. 12 and + + − − channels. The corresponding prospects of H ±± depend largely on the leptonic branching fraction of H ±± and the search channels. For the purpose of studying the leptonic scalar φ in the final state, the intermediate Yukawa coupling case can be most beneficial, from combining the leptonic and bosonic decay channels. In this paper, we have focused on the light leptonic scalar case with mass M h /2 < M φ O(100 GeV). It should be noted that the analysis in this paper can be generalized to the cases with relatively heavier leptonic scalars φ, say with masses of few hundreds of GeV or even larger. Then the φ-induced signals will depend largely on the mass M φ . The light φ induced signal in this paper can also be compared with the searches of H ±± at future hadron colliders in the standard Type-II seesaw. For instance, the H ±± mass reach has been estimated in the standard Type-II scenario for the LHC and future 100 TeV colliders in Refs. [44,85]. In a large region of parameter space of Type-II seesaw, the bosonic decay channel H ±± → W ± W ± dominates, and the mass reach of H ±± is found to be 1.8 TeV at 5σ at the 100 TeV collider, which is smaller than our reach of ∼2.6 TeV in both the large and small Yukawa coupling scenarios (cf. the dashed line in Figs. 8 and 11). The better reach in our model is due to the extra source of missing energy via φ. This makes the signal in our model more easily distinguishable from the SM backgrounds.

Vertices
Couplings CP-odd scalars A 1 , A 2 ; the singly-charged scalars H ± ; and the doubly-charged scalars H ±± . The component h from the SU (2) L -doublet is identified with the 125 GeV SM Higgs boson. In our convention, H 1 is lighter than H 2 , and A 1 is lighter than A 2 . The trilinear and quartic scalar couplings are collected in Tables 7 and 8 respectively, the trilinear and quartic gauge couplings are presented in Tables 9 and 10 respectively, and the Yukawa couplings can be found in Table 11.

B The functions G and F
For the decays in Eq. (2.17), the function G(x, y) is given by For the decays in Eq. (2.19), the function F is defined as Table 9. Trilinear gauge couplings. Here p 1 , p 2 are the momenta of the first and second particles in the vertices.

D Analytical perturbativity limits
For the gauge couplings g i , it is trivial to get the analytical one-loop expressions for the couplings, which turn out to be with α 3 = g 2 S /4π, α 2 = g 2 L /4π, α 1 = g 2 1 /4π for the SU (3) c , SU (2) L and U (1) Y couplings respectively, and b 3 = −7, b 2 = −5/2, b 1 = 47/6 [cf. Eqs. (C.1)-(C. 3)]. For the SM topquark Yukawa coupling y t , let us first consider only the y 3 t and g 2 S y t terms on the RHS of Eq. (C.13), i.e.: To implement the running of g S , we rewrite the equation above to be in the form of Then we can obtain the analytical running of y t :

(D.4)
If we include also the g 2 L y t and g 2 1 y t terms in Eq. (C.13), it is straightforward to get the full analytical one-loop solution for y t : where the function E α (µ) = α In the one-loop RGE of Y µµ in Eq. (C.14), if we consider only the Y 3 µµ term on the RHS, it is trivial to obtain where α µ ≡ Y 2 µµ /4π. It is clear that the coupling Y µµ will blow up when the t parameter approaches the value of .

(D.8)
With an initial value of Y µµ (v) = 1.5, we can get the critical value of t c 4.39. As in Eq. (D.2), we can first include the gauge coupling g L , then In this case, the coupling g L becomes divergent when the parameter t c = 4.62. If we have all the terms on the RHS of Eq. (C.14), it turns out that In this case, the critical value t c = 4.67.

(E.15)
Finally, the sub-matrix for ( 1 To implement the unitarity bounds, we can set all the eigenvalues in Eqs. (E.5), (E.8), (E.12), (E.14) and (E.17) to be smaller than 8π. As a comparison to the perturbativity bounds, we set the quartic couplings to be the benchmark values, λ 1 = 0.1 , λ 4 = −1 , λ 2, 3, 5, 6, 7 = 0 , (E. 18) and check the unitarity bounds on λ 8 . It turns out for this specific benchmark scenario, only the following bounds are relevant to λ 8 : Among the four constraints, the most stringent one is from λ − 146 , which leads to which is much weaker than the perturbativity bound discussed in Section 2.3.