1 Introduction

The neutrino oscillation experiments have confirmed that neutrinos have non-zero masses [1, 2], which is concrete evidence for new physics beyond the standard model (SM). With great effort made by the neutrino oscillation experiments in the last two decades, the picture of three neutrino oscillations has become more and more precise [3]. Currently, one major issue of the neutrino oscillation is the neutrino mass hierarchy, where the sign of \(\Delta m_{31}^2=m_{\nu _3}^2-m_{\nu _1}^2\) is undetermined. The combined analysis of Super-Kamiokande [4], T2K [5] and NO\(\nu \)A [6] experiments shows possible hints for normal ordering at present [3]. This hierarchy problem may be solved by future oscillation experiments such as DUNE [7] and JUNO [8]. The other two quantities that need to be measured are the octant of the atmospheric mixing angle \(\theta _{23}\) and the CP violation phase \(\delta _\text {CP}\) [9].

Because the oscillation experiments are only sensitive to the squared mass differences of neutrinos, they can not tell the absolute neutrino mass scale either. Meanwhile, the massive neutrinos could lead to sizable impacts on the cosmic microwave background and the large scale structure [10], which has set an upper bound on the sum of light neutrino masses as \(\sum m_\nu <0.12\) eV [11]. Therefore, neutrino masses are below the eV scale. Although the cosmological limit is model-dependent [12], it is more stringent than the direct kinematic measurement [13].

Besides the above unknowns, there are still many questions about neutrinos to be resolved. The crucial one is the Dirac or Majorana nature of the neutrino. Provided the neutrinos are Majorana type, then the lepton number is violated, which can be further confirmed by neutrinoless double beta decay [14]. Since no positive lepton number violation signature has been observed until now, the Dirac type neutrino is also possible. Then we have to consider the origin of the sub-eV scale Dirac neutrino mass. Of course, the Yukawa interaction with SM Higgs doublet as \(y_\nu {\bar{L}}{\tilde{\Phi }}_1\nu _R\) can generate the tiny neutrino mass with feeble coupling \(y_\nu \sim 10^{-12}\), which seems unnatural small compared with other SM fermions. The natural generation of Dirac neutrino mass is also studied extensively [15,16,17,18,19,20,21,22,23,24]. In this paper, we consider the neutrinophilic two Higgs doublet model (\(\nu \)2HDM) [25, 26], where a new Higgs doublet \(\Phi _2\) with a tiny vacuum expectation value (VEV) generates the Dirac neutrino mass.

In canonical seesaw models for Majorana neutrino, there is a strong connection between the branching ratio of heavy new particles and the neutrino oscillation parameters [27, 28]. Once these new particles are discovered at colliders, the measurement of corresponding branching ratios provides an alternative pathway to probe neutrino mixing parameters [29, 30]. In the \(\nu \)2HDM, the Dirac neutrino mass matrix is proportional to the product of the neutrinophilic Yukawa coupling \(y_\nu \) and the VEV \(v_2\). Since the decays of the new charged scalar \(H^\pm \) are also determined by the coupling \(y_\nu \), the branching ratios of \(H^\pm \) are expected to depend on the neutrino oscillation parameters [25, 26, 31]. In this paper, we study the capability of future colliders such as the CLIC [32] and FCC-hh [33] to probe the Dirac neutrino properties with dilepton signature. This signature is hopeful to unravel the neutrino mass hierarchy, the atmospheric octant, and the neutrino mass scale by precise measurements of BR\((H^\pm \rightarrow \ell ^\pm \nu )\).

The paper is organized as follows. In Sect. 2, we briefly review the \(\nu \)2HDM for Dirac neutrino mass and relevant constraints. The branching ratio of charged scalar \(H^\pm \) and its relation with the neutrino oscillation parameters are discussed in Sect. 3. The dilepton signature at the 3 TeV CLIC and 100 TeV FCC-hh are studied in Sects. 4.1 and 4.2 respectively. The conclusion is given in Sect. 5.

2 The \(\nu \)2HDM model

Besides the SM Higgs doublet \(\Phi _1\), this model further introduces one Higgs doublet \(\Phi _2\) and three right-handed neutrinos \(\nu _{R}\). To forbid the direct Yukawa interaction between SM Higgs \(\Phi _1\) and right-handed neutrino \(\nu _R\), a global U(1) symmetry is also imposed. Under this U(1) symmetry, both the new Higgs doublet \(\Phi _2\) and right-handed neutrino \(\nu _R\) carry charge \(+1\), while all SM fields remain uncharged. The Majorana mass term of right-handed neutrino \(M_R \overline{\nu ^c_R}\nu _R\) is also forbidden by the U(1) symmetry. The two Higgs doublets can be expressed in the form of

$$\begin{aligned} \Phi _i=\left( \begin{array}{c} \phi _i^+\\ (v_i+\phi _i^{0,r}+i \phi _i^{0,i})/\sqrt{2} \end{array}\right) , \end{aligned}$$
(1)

with \(v_i\) the corresponding VEV. The scalar potential under the global U(1) symmetry is

$$\begin{aligned} V= & {} m_{11}^2\Phi _1^\dag \Phi _1+m_{22}^2\Phi _2^\dag \Phi _2-\left[ m_{12}^2\Phi _1^\dag \Phi _2+\text {h.c.}\right] \nonumber \\{} & {} +\frac{\lambda _1}{2}\left( \Phi _1^\dag \Phi _1\right) ^2 +\frac{\lambda _2}{2}\left( \Phi _2^\dag \Phi _2\right) ^2\nonumber \\{} & {} +\frac{\lambda _3}{2}\left( \Phi _1^\dag \Phi _1\right) \left( \Phi _2^\dag \Phi _2\right) \nonumber \\{} & {} +\frac{\lambda _4}{2}\left( \Phi _1^\dag \Phi _2\right) \left( \Phi _2^\dag \Phi _1\right) , \end{aligned}$$
(2)

where the soft term \(m_{12}^2\Phi _1^\dag \Phi _2\) breaks the global U(1) symmetry explicitly. Because a vanishing \(m_{12}^2\) would restore the global U(1) symmetry, it is naturally small [34]. The electroweak symmetry breaking is triggered spontaneously by \(m_{11}^2<0\), while \(m_{22}^2+(\lambda _3+\lambda _4)v_1^2/2>0\) is required to prevent the spontaneous breaking of U(1) symmetry. The soft term \(m_{12}^2\Phi _1^\dag \Phi _2\) induces a tiny VEV for \(\Phi _2\) as

$$\begin{aligned} v_2 = \frac{m_{12}^2 v_1}{m_{22}^2+(\lambda _3+\lambda _4)v_1^2/2}. \end{aligned}$$
(3)

Then \(v_2\sim \) eV can be obtained with \(m_{12}^2\sim \text {MeV}^2\). The relation \(v_2\ll v_1\) is stable under radiative corrections, because the corrections are only logarithmically sensitive to the cutoff [35, 36]. To ensure the stability of the potential at large field values, one needs

$$\begin{aligned} \lambda _{1,2}>0,\lambda _3+\sqrt{\lambda _1\lambda _2}>0,\lambda _3+\lambda _4+\sqrt{\lambda _1\lambda _2}>0. \end{aligned}$$
(4)

The physical Higgs bosons are given by [37]

$$\begin{aligned}{} & {} H^+=\phi ^+_2\cos \beta -\phi ^+_1\sin \beta ,~ A=\phi ^{0,i}_2\cos \beta -\phi ^{0,i}_1\sin \beta , \nonumber \\ \end{aligned}$$
(5)
$$\begin{aligned}{} & {} H=\phi ^{0,r}_2\cos \alpha -\phi ^{0,r}_1\sin \alpha ,~ h=\phi ^{0,r}_1\cos \alpha +\phi ^{0,r}_2\sin \alpha ,\nonumber \\ \end{aligned}$$
(6)

with the mixing angles \(\tan \beta =v_2/v_1\) and \(\tan 2\alpha \simeq 2 v_2/v_1\). Since the mixing angles are heavily suppressed by the tiny VEV \(v_2\), the new Higgs bosons are almost the neutrinophilic doublet. Masses of the scalars are

$$\begin{aligned}{} & {} M_{H^+}^2\simeq m_{22}^2\!+\frac{1}{2}\lambda _3v_1^2,~M_A^2\simeq M_H^2\simeq M_{H^+}^2\!+\frac{1}{2}\lambda _4v_1^2,\nonumber \\{} & {} M_h^2 \simeq \lambda _1 v_1^2, \end{aligned}$$
(7)

where terms of order \(m_{12}^2\) and \(v_2^2\) are neglected. The charged scalar can be pair produced at colliders and leads to the dilepton signature \(\ell ^+\ell ^-+P_T^{miss}\) [25, 26]. The direct searches for slepton at LHC in the same dilepton signature have excluded \(M_{H^+}\lesssim 700\) GeV [38, 39].

Under the global U(1) symmetry, the allowed new Yukawa interaction is

$$\begin{aligned} {\mathcal {L}}_Y=-y_\nu {\bar{L}}{\tilde{\Phi }}_2\nu _R + \text {h.c.}, \end{aligned}$$
(8)

which induces the tiny neutrino mass as \(m_\nu =y_\nu v_2/\sqrt{2}\). The Yukawa couplings of the new scalars in the neutrino mass eigenstate are

$$\begin{aligned} {\mathcal {L}}_Y= & {} -\frac{m_{\nu _i}}{v_2} H {\bar{\nu }}_i \nu _i + \frac{m_{\nu _i}}{v_2} A {\bar{\nu }}_i \gamma _5\nu _i\nonumber \\{} & {} - \frac{\sqrt{2} m_{\nu _i}}{v_2}\left( U_{\ell i}^{*} H^+ {\bar{\nu }}_i \ell _L + \text {h.c.}\right) , \end{aligned}$$
(9)

where U is the Pontecorvo-Maki-Nakagawa-Sakata*** (PMNS) matrix. The PMNS matrix can be written explicitly as [40]

$$\begin{aligned} U&= \left( \begin{array}{ccc} c_{12} c_{13} &{} s_{12} c_{13} &{} s_{13} e^{i\delta }\\ -s_{12}c_{23}-c_{12}s_{23}s_{13}e^{-i\delta } &{} c_{12}c_{23}-s_{12}s_{23}s_{13} e^{-i\delta } &{} s_{23}c_{13}\\ s_{12}s_{23}-c_{12}c_{23}s_{13}e^{-i\delta } &{} -c_{12}s_{23}-s_{12}c_{23}s_{13}e^{-i\delta } &{} c_{23}c_{13} \end{array} \right) \end{aligned}$$
(10)

where \(c_{ij}\equiv \cos \theta _{ij}\) and \(s_{ij}\equiv \sin \theta _{ij}\) for short, and \(\delta \) is the CP violation phase. In this paper, we consider the latest results for three neutrino oscillations [3].

The relatively large Yukawa coupling of charged scalar could induce the lepton flavor violation process [41]. Currently, the most stringent constraint comes from the MEG collaboration, which is BR\((\mu \rightarrow e\gamma )<4.2\times 10^{-13}\) at 90% CL [42]. Its branching ratio is calculated as [43]

$$\begin{aligned} \text {BR}(\mu \rightarrow e\gamma )=\frac{\alpha }{24\pi } \left( \frac{v_1}{v_2}\right) ^4 \frac{|m_{\nu _j}^2U_{ej}U_{\mu j}|^2}{M_{H^+}^4}. \end{aligned}$$
(11)

The experimental upper limit sets a lower bound on \(v_2\) as

$$\begin{aligned} v_2 > rsim 15 \text {eV} \left( \frac{m_\nu }{0.05\text {eV}}\right) \left( \frac{100\,\text {GeV}}{M_{H^+}}\right) . \end{aligned}$$
(12)

Under the direct constraints from the collider, \(v_2 > rsim {\mathcal {O}}(\text {eV})\) is allowed for \(M_{H^+}\sim {\mathcal {O}}(\text {TeV})\).

3 Branching ratio of \(H^\pm \)

Fig. 1
figure 1

Branching ratio of charged scalar \(H^+\) as a function of lightest neutrino mass. The shaded region is excluded by cosmological limit \(\sum m_\nu < 0.12\) eV

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Because there are strong correlations between the neutrino oscillation parameters and the branching ratio of charged scalar \(H^\pm \), we perform a detailed analysis in this section. Similar results have been obtained in previous studies [25, 31], which are reproduced with latest neutrino oscillation data in this paper. For eV scale \(v_2\), the partial decay widths as \(H^+\rightarrow tb,W^+h,W^+Z\) are heavily suppressed by the mixing angle \(\tan \beta =v_2/v_1\sim 10^{-11}\), thus they can be neglected. Meanwhile, the cascade decay \(H^+\rightarrow W^+H/W^+A\) can be kinematically forbidden provided \(M_{H,A}\ge M_{H^+}\), which is realized by requiring \(\lambda _4>0\). In this way, \(H^+\rightarrow \ell ^+\nu \) is the dominant decay mode. The partial decay width is calculated as [26]

$$\begin{aligned} \Gamma (H^+\rightarrow \ell ^+ \nu )=\frac{M_{H^+} \langle m_\nu ^2 \rangle _\ell }{8\pi v_2^2}, \end{aligned}$$
(13)

where the neutrino final states are summed over. The reweighted neutrino mass squared is defined by

$$\begin{aligned} \langle m_\nu ^2 \rangle _\ell =\sum _i m_{\nu _i}^2 |U_{\ell i}|^2. \end{aligned}$$
(14)

For a rough estimation, we can use the Tri-Bi-Maximal mixing pattern [44], which leads to

$$\begin{aligned} \langle m_\nu ^2 \rangle _e\simeq & {} \frac{2}{3}m_{\nu _1}^2+\frac{1}{3}m_{\nu _2}^2 \end{aligned}$$
(15)
$$\begin{aligned} \langle m_\nu ^2 \rangle _\mu =\langle m_\nu ^2 \rangle _\tau\simeq & {} \frac{1}{6}m_{\nu _1}^2+\frac{1}{3}m_{\nu _2}^2+\frac{1}{2} m_{\nu _3}^2 \end{aligned}$$
(16)

Since the absolute neutrino mass and hierarchy are not determined, the variable \(\langle m_\nu ^2 \rangle _\ell \) also varies. If we further assume the lightest neutrino is massless, then we can derive that

$$\begin{aligned} \langle m_\nu ^2 \rangle _e^\text {NH}\simeq \frac{1}{3} \Delta m_\text {SOL}^2&, \quad&\langle m_\nu ^2 \rangle _\mu ^\text {NH} =\langle m_\nu ^2 \rangle _\tau ^\text {NH}\simeq \frac{1}{2}\Delta m_\text {ATM}^2 ~,\nonumber \\ \end{aligned}$$
(17)
$$\begin{aligned} \langle m_\nu ^2 \rangle _e^\text {IH}\simeq \Delta m_\text {ATM}^2~&, \quad&\langle m_\nu ^2 \rangle _\mu ^\text {IH} =\langle m_\nu ^2 \rangle _\tau ^\text {IH}\simeq \frac{1}{2} \Delta m_\text {ATM}^2 ~,\nonumber \\ \end{aligned}$$
(18)

where \(\Delta m_\text {SOL}^2\simeq 7.4\times 10^{-5}\text {eV}^2\) and \(\Delta m_\text {ATM}^2\simeq 2.5\times 10^{-3}\text {eV}^2\). These rough estimations predict BR\((H^+\rightarrow e^+\nu )\sim \Delta m_\text {SOL}^2/(3\Delta m_\text {ATM}^2)\simeq 0.01\), BR\((H^+\rightarrow \mu ^+\nu ) =\)BR\((H^+\rightarrow \tau ^+\nu )\simeq 0.5\) for normal hierarchy (NH), and BR\((H^+\rightarrow e^+\nu )\simeq 0.5\), BR\((H^+\rightarrow \mu ^+\nu )=\)BR\((H^+\rightarrow \tau ^+\nu )\simeq 0.25\) for inverted hierarchy (IH) respectively. In Fig. 1, we show the predicted branching ratio by varying the neutrino oscillation parameters in the \(3\sigma \) range. The rough estimations are obviously consistent with the explicit calculation when \(m_{\nu _\text {lightest}}\lesssim 0.01\) eV.

Fig. 2
figure 2

Branching ratio of charged scalar \(H^+\) as a function of \(\sin ^2\theta _{23}\) for the normal hierarchy (left panel) and inverted hierarchy (right panel). The lightest neutrino mass is fixed to zero

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The neutrino mass hierarchy can be easily distinguished by the measurement of BR\((H^+)\). Because the sum of neutrino masses is constrained as \(\sum m_\nu <0.12\) eV, the lightest neutrino mass should be less than 0.03 eV (0.015 eV) for NH (IH). Under this constraint, it is clear in Fig. 1 that BR\((H^+\rightarrow e^+\nu )\) should be less than 0.2 for NH, and decreases as \(m_{\nu _\text {lightest}}=m_{\nu _1}\) becomes smaller. When \(m_{\nu _\text {lightest}}<0.01\) eV, BR\((H^+\rightarrow e^+\nu )\simeq 0.03\) is nearly a constant. So if the lightest neutrino mass is in the range of 0.01 eV \(<m_{\nu _\text {lightest}}<0.03\) eV, we can probe it by precise measurement of BR\((H^+\rightarrow e^+\nu )\). Meanwhile, we have BR\((H^+\rightarrow e^+\nu )\simeq 0.48\) for IH under the cosmological constraint, and precise measurement of BR\((H^+\rightarrow e^+\nu )\) is hard to tell the absolute value of the lightest neutrino mass.

The scanned result shows that BR(\(H^+\rightarrow \mu ^+\nu ,\tau ^+\nu \)) is sensitive to the oscillation parameters, while BR\((H^+\rightarrow e^+\nu )\) is not. Considering \(m_{\nu _\text {lightest}}=0\) and the fact that \(\sin ^2\theta _{13}\ll 1\), the factor \(\langle m_\nu ^2 \rangle _\ell \) can be approximately calculated as

$$\begin{aligned}&\langle m_\nu ^2 \rangle _e^\text {NH} \simeq \Delta m_\text {SOL}^2 s_{12}^2 + \Delta m_\text {ATM}^2 s_{13}^2, \qquad \nonumber \\ {}&\langle m_\nu ^2 \rangle _\mu ^\text {NH} \simeq \Delta m_\text {ATM}^2 s^2_{23}, \langle m_\nu ^2 \rangle _\tau ^\text {NH} \simeq \Delta m_\text {ATM}^2 c^2_{23}, \end{aligned}$$
(19)
$$\begin{aligned}&\langle m_\nu ^2 \rangle _e^\text {IH} \simeq \Delta m_\text {ATM}^2, \qquad \qquad \langle m_\nu ^2 \rangle _\mu ^\text {IH} \simeq \Delta m_\text {ATM}^2 c^2_{23}, \nonumber \\&\langle m_\nu ^2 \rangle _\tau ^\text {IH} \simeq \Delta m_\text {ATM}^2 s^2_{23}. \end{aligned}$$
(20)

In the approximation of \(\langle m_\nu ^2 \rangle _e^\text {NH}\), we have kept the \(\Delta m_\text {ATM}^2s_{13}^2\) term because \(\Delta m_\text {SOL}^2/\Delta m_\text {ATM}^2\sim s_{13}^2\). From Eqs. (19) and (20), we can derive that \(\sum \langle m_\nu ^2 \rangle _\ell ^\text {NH}\approx \Delta m_\text {ATM}^2\) and \(\sum \langle m_\nu ^2 \rangle _\ell ^\text {IH}\approx 2 \Delta m_\text {ATM}^2\). So for both mass hierarchies, BR\((H^+\rightarrow e^+\nu )\) is not sensitive to the oscillation parameters. The branching ratios of the charged scalar as a function of \(\sin ^2\theta _{23}\) are shown in Fig. 2. According to Eq. (19), BR(\(H^+\rightarrow \mu ^+\nu \)) increases while BR(\(H^+\rightarrow \tau ^+\nu \)) decreases as \(\sin ^2\theta _{23}\) increases for NH. The opposite is the case for IH. Therefore, the \(\theta _{23}\) octant could be determined by the branching ratio of charged scalar, e.g., if BR(\(H^+\rightarrow \mu ^+\nu \)) < BR(\(H^+\rightarrow \tau ^+\nu \)), we have \(\sin ^2\theta _{23}<0.5\) for NH while \(\sin ^2\theta _{23}>0.5\) for IH. It should be noted that DUNE [45] will measure \(\theta _{23}\) before CLIC or FCC-hh start. If the branching ratio pattern matches the one expected for the \(\theta _{23}\) measured in DUNE, then we would have strong evidence favoring this model.

4 Dilepton signature at colliders

Fig. 3
figure 3

Luminosity required at LHC for the normal hierarchy (left panel) and inverted hierarchy (right panel). Here, min (max) \(5\sigma \) or \(2\sigma \) refers to the minimum (maximum) required luminosity for \(5\sigma \) or \(2\sigma \) sensitivity at LHC, which is obtained with the largest (smallest) predicted branching ratio of BR\((H^\pm \rightarrow \mu ^+\nu ,e^+\nu )\). Meanwhile, bestfit \(5\sigma \) or \(2\sigma \) corresponds to the luminosity for \(5\sigma \) or \(2\sigma \) sensitivity with predicted BR\((H^\pm \rightarrow \mu ^+\nu ,e^+\nu )\) by using the best fit values of neutrino oscillation parameters in Ref. [3]. The lightest neutrino mass is fixed to zero during the calculation

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In this section, we study the striking dilepton signature from pair production of charged scalar at colliders. Currently, searches at LHC have excluded \(M_{H^+}\lesssim 700\) GeV [38, 39]. This limit is obtained by assuming BR\((H^+\rightarrow \ell ^+\nu )=1\) with \(\ell =e,\mu \). In the \(\nu \)2HDM, we have BR\((H^+\rightarrow \ell ^+\nu )\approx 0.5\) for NH and BR\((H^+\rightarrow \ell ^+\nu )\approx 0.75\) for IH. Detailed study on the dilepton signature in this model at LHC can be found in Ref. [26]. In this paper, we use the following cuts to study the sensitivity of LHC

$$\begin{aligned}{} & {} N_{\ell ^+}=N_{\ell ^-}=1,\quad N_j=0,\quad P_T^{\ell }>60~\text {GeV}, \nonumber \\{} & {} P_T^{miss}>100~\text {GeV},\quad M_{\ell ^+\ell ^-}>130~\text {GeV},\nonumber \\{} & {} M_{T2}>110~\text {GeV}, \end{aligned}$$
(21)

where \(M_{T2}\) is defined in ***Eqn.(37). The results are shown in Fig. 3. We find that \(M_{H^+}\lesssim 403\) GeV for NH and \(M_{H^+}\lesssim 464\) GeV for IH can be excluded by current experimental searches at 95% CL. With an integrated luminosity of 3000 fb\(^{-1}\), the projected future LHC result might exclude \(M_{H^+}\lesssim 760\) GeV for NH and \(M_{H^+}\lesssim 870\) GeV for IH at 95% CL.

In this paper, we choose \(M_{H^+}=800\) GeV as the benchmark point for the following collider study. As shown in Fig. 4, the production cross sections of \(H^+H^-\) at the 3 TeV CLIC and 100 TeV FCC-hh are much larger than it at the 14 TeV LHC for TeV scale \(H^\pm \). So we consider the dilepton signature at the former two colliders.

Fig. 4
figure 4

Cross section of \(H^+H^-\) at the 14 TeV LHC (green dashed) the 3 TeV CLIC (red solid) and the 100 TeV FCC-hh collider (blue dashed)

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To ascertain the promising region, we employ MadGraph5_aMC@NLO_v3.5.1 [46] to generate the background and signal events, which are all at the leading order. Fragmentation and hadronization are incorporated by Pythia_v8.311 [47]. The Delphes_v3.5.0 package [48] is used to simulate the detector response. The Delphes cards are those included in Delphes, i.e., parameters in delphes_card_CLICdet_Stage3.tcl for CLIC and FCChh.tcl for FCC-hh. Due to lower tagging efficiency, we do not consider the \(\tau \) final state in the following collider simulations.

4.1 The 3 TeV CLIC

The dilepton signature at the future electron-positron collider CLIC arises from the pair production of charged scalar as

$$\begin{aligned} e^+e^-\rightarrow H^+ H^- \rightarrow \ell ^+\nu _\ell + \ell ^- {\bar{\nu }}_\ell \rightarrow \ell ^+\ell ^-+P_T^{miss}, \end{aligned}$$
(22)

where \(\ell =e,\mu \) and the missing transverse momentum \(P_T^{miss}\) corresponds to the contribution of two light neutrinos. The dominant SM backgrounds are

$$\begin{aligned} e^+e^-\rightarrow \ell ^+\ell ^-,\ell ^+\ell ^-\nu _\ell {\bar{\nu }}_\ell ,W^+W^-\nu _\ell {\bar{\nu }}_\ell , \end{aligned}$$
(23)

where the \(\ell ^+\ell ^-\nu _\ell {\bar{\nu }}_\ell \) background includes the contribution of \(ZZ,W^+W^-\) with leptonic decay of \(Z,W^\pm \). Normalized distributions of some variables are shown in Fig. 5.

Fig. 5
figure 5

Normalized distribution of transverse momentum \(P_{T}^{\ell }\) (a), missing transverse momentum \(P_{T}^{miss}\) (b), dilepton invariant mass \(M_{\ell ^{+}\ell ^{-}}\) (c), vector sum of dilepton momentum \(P_{\ell ^{+}\ell ^{-}}\) (d), dilepton energy \(E_{\ell ^{+}\ell ^{-}}\) (e), polar angle \(\theta _{\ell ^{\pm }}\) (f), pseudorapidity \(\eta _{\ell ^{\pm }}\) (g), and dilepton azimuthal angle \(\alpha _{\ell ^{+}\ell ^{-}}\) (h) at the 3 TeV CLIC

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Table 1 Cut flow table for the dilepton signal at the 3 TeV CLIC and corresponding backgrounds. The cross sections are in unit of fb
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In this paper, we perform a simple cut based analysis. Firstly, we select events with two leptons of opposite charge

$$\begin{aligned} N_{\ell ^+}=N_{\ell ^-}=1, P_T^\ell >50~\text {GeV}. \end{aligned}$$
(24)

According to Fig. 5a, the cut on \(P_T^\ell \) aims to suppress the \(\ell ^+\ell ^-\nu _\ell {\bar{\nu }}_\ell \) and \(W^+W^-\nu _\ell {\bar{\nu }}_\ell \) backgrounds. The missing transverse momentum \(P_T^{miss}\) is another distinct feature. We observe that the signal tends to have a larger \(P_T^{miss}\), so we further require relatively large values as

$$\begin{aligned} P_T^{miss}>540~\text {GeV} \end{aligned}$$
(25)

This cut is quite efficient to eliminate the \(\ell ^+\ell ^-\) background. After applying these cuts, the dominant background is \(\ell ^+\ell ^-\nu _\ell {\bar{\nu }}_\ell \). In the distribution of invariant mass of dilepton \(M_{\ell ^+\ell ^-}\), the \(\ell ^+\ell ^-\nu _\ell {\bar{\nu }}_\ell \) background has a sharp peak around \(M_Z\), which indicates that the dilepton originates from \(Z\rightarrow \ell ^+\ell ^-\). One may perform a simple Z-veto in the invariant mass window between 80 GeV and 100 GeV to suppress the \(\ell ^+\ell ^-\nu _\ell {\bar{\nu }}_\ell \) background. However, the \(M_{\ell ^+\ell ^-}\) from \(W^+W^-\nu _\ell {\bar{\nu }}_\ell \) channel has a long tail structure. Based on panel (c) of Fig. 5, we then apply the cut

$$\begin{aligned} M_{\ell ^+\ell ^-}>710~\text {GeV}. \end{aligned}$$
(26)
Fig. 6
figure 6

Luminosity required at the 3 TeV CLIC for \(5\sigma \) discovery. Here \(M_{H^\pm }=800\) GeV

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Fig. 7
figure 7

Luminosity required at the 3 TeV CLIC for \(5\sigma \) discovery as a function of \(M_{H^+}\) for the normal hierarchy (left panel) and inverted hierarchy (right panel)

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Table 2 Cut flow table for the dilepton signal at 100 TeV FCC-hh and corresponding backgrounds. The cross sections are in unit of fb
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In the distribution of vector sum of dilepton momentum \(P_{\ell ^+\ell ^-}\equiv |\vec {P}_{\ell ^+}+\vec {P}_{\ell ^-}|\) as in panel (d) of Fig. 5, some events of the \(\ell ^+\ell ^-\nu _\ell {\bar{\nu }}_\ell \) background could lead to \(P_{\ell ^+\ell ^-}\) larger than the signal. To exclude such background samples, we adopt the cut

$$\begin{aligned} P_{\ell ^+\ell ^-}<980~\text {GeV}, \end{aligned}$$
(27)

which is very efficient to eliminate the \(\ell ^+\ell ^-\nu _\ell {\bar{\nu }}_\ell \) background.

In panel (e) of Fig. 5, it is clear that the sum of dilepton energy \(E_{\ell ^+ \ell ^-}=E_{\ell ^+}+E_{\ell ^-}\) from the \(\ell ^+\ell ^-\) background is around 3000 GeV. Then the cut

$$\begin{aligned} E_{\ell ^+\ell ^-}<1900~\text {GeV} \end{aligned}$$
(28)

is able to suppress the \(\ell ^+\ell ^-\) background to a negligibly small level. Meanwhile, the \(\ell ^+\ell ^-\nu _\ell {\bar{\nu }}_\ell \) and \(W^+W^-\nu _\ell {\bar{\nu }}_\ell \) backgrounds need more cuts to suppress. In this paper, we consider the dilepton azimuthal angle \(\alpha _{\ell ^+\ell ^-}\), the pseudorapidity \(\eta _{\ell ^\pm }\), and lepton polar angle \(\theta _{\ell ^\pm }\). According to the distributions in panel (f)–(h) of Fig. 5, we then apply the cuts

$$\begin{aligned} 1.5<\alpha _{\ell ^+\ell ^-}<2.8,~ |\eta _{\ell ^\pm }|<1.3,~ 1<\theta _{\ell ^\pm }<2.3. \end{aligned}$$
(29)

These cuts are able to make the signal and backgrounds at the same order.

Fig. 8
figure 8

Normalized distribution of transverse momentum \(P_{T}^{\ell }\) (a), missing transverse momentum \(P_{T}^{miss}\) (b), dilepton invariant mass \(M_{\ell ^{+}\ell ^{-}}\) (c) and variable \(M_{T2}\) (d) at 100 TeV FCC-hh

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Fig. 9
figure 9

Luminosity required at the 100 TeV FCC-hh for \(5\sigma \) discovery. Here \(M_{H^\pm }=800\) GeV

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Fig. 10
figure 10

Luminosity required at the 100 TeV FCC-hh for \(5\sigma \) discovery as a function of \(M_{H^+}\) for the normal hierarchy (left panel) and inverted hierarchy (right panel)

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Results are summarized in Table 1. During this calculation, we have fixed the neutrino oscillation parameters to the best fit values with zero lightest neutrino mass. With an integrated luminosity of 5000 fb\(^{-1}\), the significance can reach \(10.8\sigma \) for NH and \(16.0\sigma \) for IH respectively. The significance is estimated as \(S/\sqrt{S+B}\), where S and B are the event number of signal and background.

Since BR\((H^\pm \rightarrow e^\pm \nu )\ll \)BR\((H^\pm \rightarrow \mu ^\pm \nu )\) for NH, the dilepton final state is actually dominant by the \(\mu ^+\mu ^-\) pair, and the \(e^+e^-,e^\pm \mu ^\mp \) pairs are hard to probe. Meanwhile, because of BR\((H^\pm \rightarrow e^\pm \nu )\simeq 2\times \)BR\((H^\pm \rightarrow \mu ^\pm \nu )\) for IH, the dilepton final state ratio is \(e^+e^-:e^\pm \mu ^\mp :\mu ^+\mu ^-\approx 4:4:1\). Therefore, the neutrino mass hierarchy can also be determined by the flavor structure of dilepton signature. Such a specific dilepton ratio is useful to distinguish this model from the other neutrino models with dilepton signature [49].

Since the branching ratio is affected by neutrino oscillation parameters, we show the required luminosity for \(5\sigma \) discovery in Fig. 6. From the discussion in Sect. 3, we are aware that the lightest neutrino mass and the mixing angle \(\theta _{23}\) have the dominant influence. The lower and upper lines are derived with maximum and minimum predicted values of BR\((H^\pm \rightarrow e^\pm \nu ,\mu ^\pm \nu )\) for both NH and IH respectively. The minimum (maximum) required luminosity for \(5\sigma \) discovery is about \(450~\text {fb}^{-1}\)(\(760~\text {fb}^{-1}\)) for NH and \(350~\text {fb}^{-1}\)(\(450~\text {fb}^{-1}\)) for IH, respectively. With relatively larger branching ratio into e and \(\mu \) final states, we find that the required luminosity of \(5\sigma \) discovery for IH is always smaller than that for NH under the constraint of the lightest neutrino mass. The biggest uncertainty comes from the mixing angle \(\theta _{23}\), which is clearly shown in right panel of Fig. 6. The luminosity of \(5\sigma \) discovery decreases as \(\sin ^2\theta _{23}\) increases for NH, while it increases for IH. Using these linear relations, the value of \(\sin ^2\theta _{23}\) can be improved once there is clear excess in the dilepton signature.

In Fig. 7, we show the required luminosity for \(5\sigma \) discovery as a function of \(M_{H^+}\). Here, we consider \(M_{H^+}>500\) GeV to satisfy current LHC dilepton searches with realistic branching ratio. The red lines are derived with best fit neutrino oscillation parameters, while the blue and green lines correspond to the maximum and minimum required values. Compared to IH, the NH is less promising for certain \(M_{H^+}\). With 5000 fb\(^{-1}\) integrated luminosity, the 3 TeV CLIC could discover \(M_{H^+}\lesssim 1220\) GeV for NH, and \(M_{H^+}\lesssim 1280\) GeV for IH. Based on the result in Fig. 4, it is obvious that when \(M_{H^+}\) is larger than 1300 GeV, the production cross section of \(H^+H^-\) at 3 TeV CLIC will be suppressed by the phase space. For heavier \(M_{H^+}\), it is expected more promising at the 100 TeV FCC-hh collider.

4.2 The 100 TeV FCC-hh

Following the analysis of dilepton signature at LHC [26], we then investigate the dilepton signature at the planned 100 TeV FCC-hh collider in this section. As already shown in Fig. 3, LHC could only discover \(M_{H^+}\lesssim 540\) GeV for NH and \(M_{H^+}\lesssim 610\) GeV for IH with an integrated luminosity of 3000 fb\(^{-1}\). Compared to the LHC, the higher luminosity and energy of the FCC-hh enable the detection of leptons with higher momenta, implying that we can use more stringent cuts on \(P_T^{\ell }\) to reduce background and enhance the significance. The signal process is

$$\begin{aligned} pp\rightarrow H^+H^-\rightarrow \ell ^+\nu _\ell + \ell ^- {\bar{\nu }}_\ell \rightarrow \ell ^+\ell ^-+P_T^{miss}. \end{aligned}$$
(30)

The dominant SM backgrounds come from

$$\begin{aligned} pp\rightarrow t{\bar{t}},W^+W^-,ZZ. \end{aligned}$$
(31)

There is also one possible background from \(pp\rightarrow \gamma ^*Z\rightarrow \ell ^+\ell ^-+\nu _\ell {\bar{\nu }}_\ell \). The initial cross section of the \(\gamma Z\) background (\(\simeq 95\) fb) is much smaller than the ZZ background (\(\simeq 2641\) fb). Meanwhile, the lepton momentum mediated by the off-shell \(\gamma \) is also less energetic than the ZZ background. Since the contribution of the ZZ background is tiny under the cuts in Table 2, we do not consider the \(\gamma Z\) background in the following study. Normalized distributions of some variables at the 100 TeV FCC-hh collider are shown in Fig. 8.

Similar to the analysis in previous Sect. 4.1 for CLIC, we first select events with two opposite sign charged leptons

$$\begin{aligned} N_{\ell ^+}=N_{\ell ^-}=1. \end{aligned}$$
(32)

The final states of \(H^+H^-\) are purely leptonic, whereas the final states of \(t{\bar{t}}\) usually contain several jets from cascade decay \(t\rightarrow b W\). During the simulation, we use the anti-\(k_t\) algorithm with a radius parameter of 0.4 to reconstruct jets [50]. In order to reduce the enormous \(t{\bar{t}}\) background, events are required to have no energetic jets, i.e.,

$$\begin{aligned} N_j =0. \end{aligned}$$
(33)

The leptons from direct decay of charged scalars are typically more energetic than the backgrounds, so a cut on the transverse momentum of leptons is also applied to suppress the SM backgrounds

$$\begin{aligned} P_T^{\ell }>100~\text {GeV} \end{aligned}$$
(34)

To obtain events with clear missing transverse momentum, we then require

$$\begin{aligned} P_T^{miss}>100~\text {GeV}. \end{aligned}$$
(35)

After implementing these simple selection cuts, the SM backgrounds exhibit a cross section of approximately a few fb at the 100 TeV FCC-hh collider, whereas the signal’s cross section is around \({\mathcal {O}}(0.01)\) fb for \(M_{H^+}=800\) GeV.

For the ZZ background, the opposite sign dilepton is mostly from direct \(Z\rightarrow \ell ^+\ell ^-\) decay. Then the ZZ background can be suppressed by requiring the invariant mass of dilepton

$$\begin{aligned} M_{\ell ^+\ell ^-}> 130~\text {GeV}, \end{aligned}$$
(36)

which is an efficient way to eliminate the ZZ background to a negligible level. Besides the different mass of the decay particle, the pure leptonic decay of \(W^+W^-\) channel has the same topology as the signal. Because there are two neutrinos in the final states, we can not directly reconstruct the mass of the charged scalar or W boson. Instead, the \(M_{T2}\) variable is one good option to probe the difference between \(M_{H^+}\) and \(M_{W}\), which is defined as [51, 52],

$$\begin{aligned}{} & {} M_{T2}\nonumber \\ {}{} & {} = \underset{{{\textbf {q}}}_{T,1} + {{\textbf {q}}}_{T,2} =~ {{\textbf {P}}}_T^{miss}}{\text {min}} \left\{ \text {max}\left[ M_{T} ({{\textbf {P}}}_{T}^{l_{1}},{{\textbf {q}}}_{T,1}),M_{T}({{\textbf {P}}}_{T}^{l_{2}},{{\textbf {q}}}_{T,2})\right] \right\} ,\nonumber \\ \end{aligned}$$
(37)

where \({{\textbf {P}}}_{T}^{l_{1}}\) and \({{\textbf {P}}}_{T}^{l_{2}}\) are the transverse momentum vectors of the two leptons, \({{\textbf {q}}}_{T,1}\) and \({{\textbf {q}}}_{T,2}\) are all possible combinations of two transverse momentum vectors that satisfy \({{\textbf {q}}}_{T,1} + {{\textbf {q}}}_{T,2} ={{\textbf {P}}}_T^{miss}\). The transverse mass \(M_T\) is defined as

$$\begin{aligned} M_T({{\textbf {P}}}_{T}^{l},{{\textbf {q}}}_{T}^{miss})=\sqrt{2(|{{\textbf {P}}}_{T}^{l}||{{\textbf {q}}}_{T}^{miss}|-{{\textbf {P}}}_{T}^{l}\cdot {{\textbf {q}}}_{T}^{miss})} \end{aligned}$$
(38)

The \(M_{T2}\) variable theoretically predicts \(M_{T2}<M_{H^+}\) for the signal and \(M_{T2}<M_{W}\) for the \(W^+W^-\) background. Distributions of \(M_{T2}\) are shown in panel (d) of Fig. 8. We then apply the cut

$$\begin{aligned} M_{T2}>110~\text {GeV}. \end{aligned}$$
(39)

In Table 2, we summarize the efficiency for each cut, where the cross sections of signal are obtained by fixing the neutrino oscillation parameters to the best fit values with zero lightest neutrino mass. These cuts are able to suppress the SM backgrounds to zero, while keeping the cross section of signal at the order of 0.01 fb. With an integrated luminosity of 3000 fb\(^{-1}\), the significance can reach \(7.3\sigma \) for NH and \(9.6\sigma \) for IH respectively. The required luminosity for \(5\sigma \) discovery is about \(1400~\text {fb}^{-1}\) for NH and \(820~\text {fb}^{-1}\) for IH. Therefore, the future 100 TeV FCC-hh collider needs more integrated luminosity than the 3 TeV CLIC to discover the charged scalar with \(M_{H^+}=800\) GeV.

In Fig. 9, we show the impact of lightest neutrino mass and mixing angle \(\theta _{23}\) on the \(5\sigma \) discovery luminosity at 100 TeV FCC-hh. For NH, the maximum required luminosity is about \(1600~\text {fb}^{-1}\), and the minimum is about \(950~\text {fb}^{-1}\). For IH, the required luminosity varies from \(750~\text {fb}^{-1}\) to \(950~\text {fb}^{-1}\). The dependence of required luminosity on \(\sin ^2\theta _{23}\) for FCC-hh is the same as the CLIC.

Based on the cuts in Table 2, we then explore the feasibility of dilepton signature as a function of \(M_{H^+}\). The results are shown in Fig. 10. With an integrated luminosity of 30 ab\(^{-1}\), the future 100 TeV FCC-hh collider could probe \(M_{H^+}\lesssim 1810\) GeV for NH and \(M_{H^+}\lesssim 2060\) GeV for IH. Impacted by the uncertainties of neutrino oscillation parameters, the upper discovery value of \(M_{H^+}\) for NH would vary from 1600 GeV to 2000 GeV, while for IH, the impact is relatively smaller.

5 Conclusion

The Majorana or Dirac nature of neutrinos is still undetermined. If neutrinos are Dirac particles, the neutrinophilic doublet \(\Phi _2\) with eV scale VEV can naturally generate tiny neutrino masses. Such small VEV comes from the soft U(1) broken term \(m_{12}^2 \Phi ^\dag _1\Phi _2\). Under the U(1) charge assignment \(Q_{\Phi _2}=Q_{\nu _R}=1\), this global symmetry forbids both the Yukawa interaction of right-handed neutrinos to SM Higgs and the Majorana mass term of right-handed neutrinos, but allows the interaction between the new Higgs \(\Phi _2\) and right-handed neutrinos.

Because the generation of neutrino mass and decay of charged scalar both involve the same Yukawa coupling \(y_\nu \), the leptonic branching ratio of charged scalar \(H^\pm \) has a strong correlation with the neutrino oscillation parameters. The two parameters that can obviously affect the branching ratio are the lightest neutrino mass and the atmospheric mixing angle \(\theta _{23}\). When the lightest neutrino mass is below 0.01 eV, the NH predicts BR\((H^+\rightarrow e^+\nu )\ll \) BR\((H^+\rightarrow \mu ^+\nu )=\) BR\((H^+\rightarrow \tau ^+\nu )\simeq 0.5\), while the IH predicts BR\((H^+\rightarrow e^+\nu )/2\simeq \) BR\((H^+\rightarrow \mu ^+\nu )\simeq \) BR\((H^+\rightarrow \tau ^+\nu )\simeq 0.25\). Under the cosmological constraint \(\sum m_\nu < 0.12\) eV, we can probe the absolute lightest neutrino mass via precise measurement of BR\((H^+\rightarrow e^+\nu )\) for NH when \(m_{\nu _1} > rsim 0.01\) eV. However, BR\((H^+\rightarrow e^+\nu )\) is nearly a constant for IH. On the other hand, BR(\(H^+\rightarrow \mu ^+\nu \)) is greatly affected by the mixing angle \(\theta _{23}\). Therefore, the precise measurement of BR(\(H^+\rightarrow \mu ^+\nu \)) can indicate the value of \(\theta _{23}\).

The charged scalar can be pair produced at colliders, which then leads to the dilepton signature \(\ell ^+\ell ^-+P_T^{miss}\) with \(\ell =e,\mu \). Although current LHC has excluded \(M_{H^+}\lesssim 700\) GeV with BR(\(H^+\rightarrow \ell ^+\nu \))=1, we find the actual limits should be \(M_{H^+}\lesssim 403\) GeV for NH and \(M_{H^+}\lesssim 464\) GeV for IH with more realistic branching ratios, respectively. In this paper, we then perform a detailed simulation of this signature and corresponding background at the 3 TeV CLIC and the 100 TeV FCC-hh colliders. For fixed value of \(M_{H^+}\), the required \(5\sigma \) discovery luminosity of the NH scenario is always larger than the IH scenario. The 3 TeV CLIC could reach \(5\sigma \) discovery when \(M_{H^+}\lesssim 1220\) GeV for NH and \(M_{H^+}\lesssim 1280\) GeV for IH with 3 ab\(^{-1}\) data. Meanwhile, the future 100 TeV FCC-hh collider could probe \(M_{H^+}\lesssim 1810\) GeV for NH and \(M_{H^+}\lesssim 2060\) GeV for IH with 30 ab\(^{-1}\) data.