Doubly-charged Higgs Boson at Future Electron-Proton Collider

TL;DR

This paper investigates the potential to discover a doubly-charged Higgs boson at the future FCC-eh electron-proton collider, analyzing its production, decay channels, and sensitivity to its mass and Yukawa couplings.

Contribution

It provides a detailed analysis of the discovery prospects for the doubly-charged Higgs at FCC-eh, including signal channels, background suppression, and sensitivity to model parameters.

Findings

Doubly-charged Higgs around 1 TeV detectable with 200 fb^{-1}

Masses up to 2 TeV probed with few years of data

Background-free signal in mu mu mode with 50 fb^{-1}

Abstract

We explore the discovery prospect of the doubly-charged component of an $SU(2)_L$-triplet scalar at the future $e^- p$ collider FCC-eh, proposed to operate with an electron beam energy of 60 GeV and a proton beam energy of 50 TeV. We consider the associated production of the doubly-charged Higgs boson along with leptons and jet(s), and its subsequent prompt decay to same-sign lepton pair. This occurs for ${\cal O}(1)$ Yukawa coupling of the scalar triplet with charged leptons, which is expected for reasonably small vacuum expectation values of the neutral component of the triplet field that governs the neutrino mass generation in the type-II seesaw. We present our analysis for two different final states, $3l+\ge1j$ and an inclusive $\ge2l+\ge1j$ channel. Considering its decay to electrons only, we find that the doubly-charged Higgs with mass around a TeV could be observed at $3\sigma$ confidence level with $\mathcal{O}(200)\, \rm{fb}^{-1}$ of integrated luminosity, while masses up to 2 TeV could be probed within a few years of data accumulation. The signal proposed here becomes essentially background-free, if it is triggered in the $\mu\mu$ mode and a $5\sigma$ discovery is achievable in this channel for a TeV-scale doubly-charged Higgs with an integrated luminosity as low as $\mathcal{O} (50)\, \rm{fb}^{-1}$. We also highlight the sensitivity of FCC-eh to the Yukawa coupling responsible for production of the doubly-charged Higgs boson as a function of its mass in both the $ee$ and $\mu\mu$ channels.

Figures

Tables

Table 1. Table 1: Particle content and their corresponding charges under the SM gauge group. The U ​ ( 1 ) Y 𝑈 subscript 1 𝑌 U(1)_{Y} hypercharges in the last row correspond to Y / 2 𝑌 2 Y/2 . Here i = 1 , 2 , 3 𝑖 1 2 3 i=1,2,3 stands for the family index.

Gauge Group $SU{\left(3\right)}_c$ $SU{\left(2\right)}_L$ $U{\left(1\right)}_Y$

Baryon Fields $Q_{L,i}={(u_{L,i},d_{L,i})}^T$ $u_{R,i}$ $d_{R,i}$ $3$ $3$ $3$ $2$ $1$ $1$ $1/6$ $2/3$ $-1/3$

Lepton Fields $L_i={({\nu }_{L,i},e_{L,i})}^T$  $e_{R,i}$ $N_{R,i}$ $1$ $1$ 1 $2$ $1$ 1 $-1/2$ $-1$ 0

Scalar Fields $\varphi$ $\Delta$   $1$ $1$   $2$ $3$   $1/2$ $1$

Table 2. Table 2: Representative benchmark points used in our analysis for studying the doubly-charged scalar production at the FCC-eh.

Benchmark Points	$v_{\Delta }$ [GeV]	$Y_{ee}^{\Delta }$	$\mu$ [GeV]	$M_{H^{--}}$ [GeV]	$M_{H^{-}}$ [GeV]	${\Gamma }_{H^{--}}$ [GeV]
BP1	10-10	0.31	$2\times {10}^{-9}$	908.6	916.9	3.5
BP2	10-10	0.39	$3\times {10}^{-9}$	1119.6	1126.2	6.8
BP3	10-10	0.45	$4\times {10}^{-9}$	1296.7	1302.5	10.4

Table 3. Table 3: Cut-flow table of the cross-section for the relevant SM background channels for the cuts A0 - A3 as mentioned in the text at the FCC-eh collider with 3.46 TeV center-of-mass energy. We assume l = e 𝑙 𝑒 l=e or μ 𝜇 \mu .

SM Backgrounds for FCC-eh Channels $e^{-}$ $p$ $\to$ $e^{-}$ $j$ $\gamma$ $Z$ ($\to l^{+}{l}^{-}$) $e^{-}$ $p$ $\to$ $j{e}^{-}{l}^{+}{l}^{-}$ $e^{-}$ $p$ $\to$ $e^{-}$ $jj$ Total Background

Effective Cross-section after applying cuts (fb)       A0           A1          A2       A3 0.12 0.03 0.02 - 53.13 12.57 3.63 0.08 $104.08\times {10}^3$ 0.2 - - 0.08

Table 4. Table 4: Cut-flow table of cross-section for signal for the cuts A0 - A3 as mentioned in the text at the FCC-eh collider with 3.46 TeV center-of-mass energy. Here we assume BR ​ ( H ± ± → e ± ​ e ± ) = 100 % BR → superscript 𝐻 plus-or-minus absent plus-or-minus superscript 𝑒 plus-or-minus superscript 𝑒 plus-or-minus percent 100 {\rm BR}(H^{\pm\pm}\to e^{\pm}e^{\pm})=100\% .

Signal for FCC-eh Process Signal BP I 3 $l$ + $\ge 1j$ BP1 I 3 $l$ + $\ge 1j$ BP2 I 3 $l$ + $\ge 1j$ BP3

Effective Cross-section after applying cuts (fb)         A0           A1      A2      A3 0.21 0.12 0.11 0.09 0.13 0.07 0.07 0.06 0.08 0.05 0.04 0.04

Table 5. Table 5: Cut-flow table of the cross-section for the relevant SM background channels for the cuts B0 - B3 as mentioned in the text at the FCC-eh collider with 3.46 TeV center-of-mass energy. We assume l = e 𝑙 𝑒 l=e or μ 𝜇 \mu .

SM Backgrounds for FCC-eh Channels e- p $\to$ $e^{-}{l}^{-}j\overline{\nu }$ $e^{-}$ $p$ $\to$ $e^{-}$ $j$ $\gamma$ e- p $\to$ $jlll$ $e^{-}$ $p$ $\to$ $e^{+}{e}^{-}j\nu$ $e^{-}$ $p$ $\to$ $e^{-}$ $j$   $j$ Total Background

Effective Cross-section after applying cuts (fb)         B0           B1      B2      B3 659.2 447.5 56.7 1.24 $2.24\times {10}^3$ 0.06 - - 53.13 41.2 40.4 0.1 346.0 0.22 0.04 $6.8\times {10}^{-4}$ $104.08\times {10}^3$ 327.18 37.0 0.37 1.71

Table 6. Table 6: Cut-flow table of cross-section for signal for the cuts B0 - B3 as mentioned in the text at the FCC-eh collider with 3.46 TeV center-of-mass energy. Here we consider BR ​ ( H ± ± → e ± ​ e ± ) = 100 % BR → superscript 𝐻 plus-or-minus absent plus-or-minus superscript 𝑒 plus-or-minus superscript 𝑒 plus-or-minus percent 100 {\rm BR}(H^{\pm\pm}\to e^{\pm}e^{\pm})=100\% .

Signal for FCC-eh Process Signal BP I + II $\ge$ 2 $l$ + $\ge$1 $j$ BP1 I + II $\ge$2 $l$ + $\ge$1 $j$ BP2 I + II $\ge$2 $l$ + $\ge$1 $j$ BP3

Effective Cross-section after applying cuts (fb)         B0           B1      B2      B3 0.36 0.29 0.24 0.21 0.22 0.18 0.14 0.13 0.13 0.11 0.09 0.08

Table 7. Table 7: Signal cross-section after all the cuts for e + + 2 ​ μ − + ≥ 1 ​ j superscript 𝑒 limit-from 2 superscript 𝜇 1 𝑗 e^{+}+2\mu^{-}+\geq 1j ( A0-A3 ) and 2 ​ μ − + ≥ 1 ​ j + ≥ 0 ​ l limit-from 2 superscript 𝜇 limit-from 1 𝑗 0 𝑙 2\mu^{-}+\geq 1j+\geq 0l ( B0-B3 ) mentioned in the text. Here we consider BR ​ ( H ± ± → e ± ​ e ± ) = BR ​ ( H ± ± → μ ± ​ μ ± ) = 50 % BR → superscript 𝐻 plus-or-minus absent plus-or-minus superscript 𝑒 plus-or-minus superscript 𝑒 plus-or-minus BR → superscript 𝐻 plus-or-minus absent plus-or-minus superscript 𝜇 plus-or-minus superscript 𝜇 plus-or-minus percent 50 {\rm BR}(H^{\pm\pm}\to e^{\pm}e^{\pm})={\rm BR}(H^{\pm\pm}\to\mu^{\pm}\mu^{\pm})=50\% .

Signal	$\sigma$ (fb)
	BP1	BP2	BP3
$e^{+}+2{\mu }^{-}+\ge 1j$	0.06	0.04	0.02
$2{\mu }^{-}+\ge 1j+\ge 0l$	0.12	0.08	0.05

Table 8. Table 8: Required integrated luminosity for achieving 3 ​ σ 3 𝜎 3\,\sigma and 5 ​ σ 5 𝜎 5\,\sigma excess of the 3 ​ l 3 𝑙 3\,l ( ≥ 2 ​ l absent 2 𝑙 \geq 2l ) signal, where the signal contains at least two electrons ( e − ​ e − superscript 𝑒 superscript 𝑒 e^{-}e^{-} ). Here we consider BR ​ ( H ± ± → e ± ​ e ± ) = 100 % BR → superscript 𝐻 plus-or-minus absent plus-or-minus superscript 𝑒 plus-or-minus superscript 𝑒 plus-or-minus percent 100 {\rm BR}(H^{\pm\pm}\to e^{\pm}e^{\pm})=100\% .

Benchmark Points BP1 BP2 BP3

$\int ℒ𝑑t$ (${\mathrm{fb}}^{-1}$) $𝒮=3\sigma$ $𝒮=5\sigma$ 118 (354) 328 (982) 245 (941) 681 (2615) 520 (2034) 1444 (5650)

Equations

$$\Delta =\left(\begin{array}{cc}\frac{{\delta }^{+}}{\sqrt{2}}& {\delta }^{++}\\ {\delta }^0& -\frac{{\delta }^{+}}{\sqrt{2}}\end{array}\right),$$

$$\mathcal{L}={\mathcal{L}}_{\mathrm{SM}}+{\mathcal{L}}_{\mathrm{kin}}^{N_R}+\mathrm{Tr}\left[(D_{\mu }\Delta {)}^{†}(D^{\mu }\Delta )\right]-\mathcal{V}(\varphi ,\Delta )+{\mathcal{L}}_{\mathrm{Yuk}},$$

$$D_{\mu }\Delta ={\partial }_{\mu }\Delta +\frac{ig}{2}\left[{\sigma }^i{W}_{\mu }^i,\Delta \right]+i{g}^{\prime }{B}_{\mu }\Delta ,$$

$$-{\mathcal{L}}_{\mathrm{Yuk}}=Y_{ij}^{\Delta }{L}_i^T{C}^{-1}{\sigma }_2\Delta L_j+Y_{ij}^D{\overline{L}}_i\tilde{\varphi }{N}_{R,j}+\frac{1}{2}{M}_{R,ij}{N}_{R,i}^T{C}^{-1}{N}_{R,j}+H.c.,$$

$$⟨\varphi ⟩=\left(\begin{array}{c}0\\ \frac{v}{\sqrt{2}}\end{array}\right),⟨\Delta ⟩=\left(\begin{array}{cc}0& 0\\ \frac{v_{\Delta }}{\sqrt{2}}& 0\end{array}\right).$$

$$M_{\nu }=M_L-M_D^T{M}_R^{-1}{M}_D,$$

$$\mathcal{V}(\varphi ,\Delta )$$

$$\varphi =\left(\begin{array}{c}{\varphi }^{+}\\ \frac{v+h_0+i{\xi }_1}{\sqrt{2}}\end{array}\right),\Delta =\left(\begin{array}{cc}\frac{{\delta }^{+}}{\sqrt{2}}& {\delta }^{++}\\ \frac{v_{\Delta }+{\delta }_0+i{\xi }_2}{\sqrt{2}}& -\frac{{\delta }^{+}}{\sqrt{2}}\end{array}\right).$$

$${\mu }_{\Delta }^2$$

$${\mu }_{\varphi }^2$$

$$M_{H^{\pm \pm }}^2=\frac{\sqrt{2}\mu v^2-{\lambda }_4{v}^2{v}_{\Delta }-2{\lambda }_3{v}_{\Delta }^3}{2v_{\Delta }},$$

$$M_{\pm }^2=\left(\sqrt{2}\mu -\frac{{\lambda }_4{v}_{\Delta }}{2}\right)\left(\begin{array}{cc}{v}_{\Delta }& -\frac{v}{\sqrt{2}}\\ -\frac{v}{\sqrt{2}}& \frac{v^2}{2v_{\Delta }}\end{array}\right).$$

$$\left(\begin{array}{c}{G}^{\pm }\\ H^{\pm }\end{array}\right)=\left(\begin{array}{cc}\cos {\beta }^{\prime }& \sin {\beta }^{\prime }\\ -\sin {\beta }^{\prime }& \cos {\beta }^{\prime }\end{array}\right)\left(\begin{array}{c}{\varphi }^{\pm }\\ {\delta }^{\pm }\end{array}\right).$$

$$M_{H^{\pm }}^2$$

$$M_{H^{\pm \pm }}^2\simeq M_{H^{\pm }}^2-\frac{1}{2}{\lambda }_4{v}^2.$$

$$\displaystyle M^{2}_{\rm even}\$$

$$\displaystyle M^{2}_{\rm odd}\$$

$$\left(\begin{array}{c}h\\ H^0\end{array}\right)=\left(\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{array}\right)\left(\begin{array}{c}{h}_0\\ {\delta }_0\end{array}\right)\mathrm{and}\left(\begin{array}{c}{G}^0\\ A^0\end{array}\right)=\left(\begin{array}{cc}\cos \beta & \sin \beta \\ -\sin \beta & \cos \beta \end{array}\right)\left(\begin{array}{c}{\xi }_1\\ {\xi }_2\end{array}\right).$$

$$\tan 2\alpha =\frac{2B}{A-C},\mathrm{and}\tan \beta =\frac{2v_{\Delta }}{v}=\sqrt{2}\tan {\beta }^{\prime }.$$

$$M_h^2$$

$$M_{H^0}^2$$

$$M_{A^0}^2$$

$$M_{G^0}^2$$

$${\lambda }_1$$

$${\lambda }_2$$

$${\lambda }_3$$

$${\lambda }_4$$

$$\lambda$$

$$\mu$$

$$Y_{\mu e}^{\Delta }{Y}_{ee}^{\Delta }<2.38\times 10^{-5}$$

$$\frac{|Y_{ee}^{\Delta }{|}^2}{M_{H^{\pm \pm }}^2}<1.2\times 10^{-7}.$$

$$\frac{|(Y^{\Delta }{)}_{ee}^{†}{Y}_{\mu \mu }^{\Delta }|}{M_{H^{\pm \pm }}^2}<1.2\times 10^{-7}.$$

$$\mathcal{S}=\sqrt{2\left[\left(s+b\right)\ln \left(1+\frac{s}{b}\right)-s\right]},$$

Full text

††preprint: HRI-RECAPP-2019-001, IP/BBSR/2019-1

I Introduction

The observations of flavor-changing neutrinos in neutrino oscillation experiments have conclusively established that neutrinos have nonzero masses and mixing. Specifically, the solar and atmospheric mass square differences are measured to be $\Delta m^{2}_{12}\sim 7.6\times 10^{-5}$ $\rm{eV}^{2}$, and $|\Delta m^{2}_{13}|\sim 2.5\times 10^{-3}$ $\rm{eV}^{2}$, and the mixing angles are $\theta_{12}\sim 33^{\circ},\leavevmode\nobreak\ \theta_{23}\sim 42^{\circ}$, and $\theta_{13}\sim 8^{\circ}$ De Salas et al. (2018). These experimental observations can not be explained within the framework of the Standard Model (SM), and hence, some beyond the Standard Model (BSM) extension is required. So far, a number of BSM models have been proposed to explain small neutrino masses Mohapatra and Smirnov (2006). Among them, one of the most appealing frameworks for light neutrino mass generation is the so-called seesaw mechanism, where the lepton-number-violating (LNV) dimension-5 Weinberg operator $LLHH/\Lambda$ Weinberg (1979); Wilczek and Zee (1979) (where $L$ and $H$ are respectively the SM lepton and Higgs doublets, and $\Lambda$ is the LNV scale) generates Majorana masses of light neutrinos. There are three tree-level realizations of seesaw Ma (1998), namely, type-I Minkowski (1977); Mohapatra and Senjanović (1980); Yanagida (1979); Gell-Mann et al. (1979); Glashow (1980), type-II Konetschny and Kummer (1977); Schechter and Valle (1980); Magg and Wetterich (1980); Cheng and Li (1980); Lazarides et al. (1981); Mohapatra and Senjanović (1981), and type-III Foot et al. (1989) seesaw, depending on whether we add $SU(2)_{L}$-singlet fermions or $SU(2)_{L}$-triplet scalars or $SU(2)_{L}$-triplet fermions to the SM particle content, respectively. One can also have hybrid combinations, such as type-I+type-II seesaw Schechter and Valle (1980), which naturally occurs, for instance, in the left-right symmetric extension of the SM Mohapatra and Pati (1975a, b); Senjanović and Mohapatra (1975). There also exist several other variations of the minimal seesaw, such as inverse Mohapatra (1986); Nandi and Sarkar (1986); Mohapatra and Valle (1986), linear Malinsky et al. (2005) and extended Kang and Kim (2007); Gavela et al. (2009); Dev and Pilaftsis (2012); Law and McDonald (2013) seesaw, as well various radiative neutrino mass generation mechanisms Cai et al. (2017). If the scale of the dimension-5 Weinberg operator for neutrino mass generation is within a few TeV range, the seesaw mechanism can be tested by looking for LNV signatures in current and future colliders Atre et al. (2009); Deppisch et al. (2015); Golling et al. (2017); Cai et al. (2018); de Blas et al. (2018).

In this paper, we focus on the LNV signatures of seesaw at future $e^{-}p$ colliders, such as LHeC Abelleira Fernandez et al. (2012) and FCC-eh Acar et al. (2017). Although their center-of-mass energy is down with respect to the $pp$ option by a factor of $\sqrt{E_{p}/E_{e}}\sim 10\leavevmode\nobreak\ (30)$ for LHeC (FCC-eh), the $e^{-}p$ colliders are capable of exploring a broad range of BSM scenarios and characterizing BSM hints at $ee$ and $pp$ colliders Azuelos et al. (2018). There have been quite a few studies on the prospects of the $SU(2)_{L}$-singlet heavy neutrino production at future $e^{-}p$ colliders Liang et al. (2010); Blaksley et al. (2011); Duarte et al. (2015); Mondal and Rai (2016a); Lindner et al. (2016); Mondal and Rai (2016b); Antusch et al. (2017); Curtin et al. (2018); Mandal et al. (2018); Das et al. (2018); Li et al. (2018). Here we explore the prospects of the $SU(2)_{L}$-triplet scalar ($\Delta$), and in particular, the production and decay of the doubly-charged component ($H^{\pm\pm}$) of the triplet at a future $e^{-}p$ collider. For earlier studies of the doubly-charged Higgs production in $e^{+}p$ collision at HERA, see Refs. Accomando and Petrarca (1994); Aktas et al. (2006); Yue et al. (2007). The triplet Higgs acquires an induced vacuum expectation value (VEV) $v_{\Delta}$ and generates light neutrino masses at tree-level via the type-II seesaw mechanism Schechter and Valle (1980); Magg and Wetterich (1980); Cheng and Li (1980); Lazarides et al. (1981); Mohapatra and Senjanović (1981). We consider $v_{\Delta}$ to be small, i.e. $v_{\Delta}\sim\mathcal{O}(0.1)$ eV, in the same ballpark as the upper limit on the sum of neutrino masses from cosmology Ade et al. (2016). For this choice of the triplet VEV, the Yukawa couplings of the triplet scalar $\Delta$ with a SM lepton-pair are large: $Y^{\Delta}\sim\mathcal{O}(1)$. This is where the production of a doubly-charged scalar in $e^{-}p$ collision has a major difference when compared with its production in $pp$ collision. At the LHC, the dominant production modes for the doubly-charged Higgs are pair-production via the $s$-channel $\gamma/Z$ and associated production with a singly-charged Higgs via the $s$-channel $W$ exchange Grifols et al. (1989); Gunion et al. (1989); Muhlleitner and Spira (2003); Akeroyd and Aoki (2005); Han et al. (2007); Chao et al. (2008); Fileviez Perez et al. (2008); del Aguila and Aguilar-Saavedra (2009); Akeroyd and Chiang (2009); Akeroyd et al. (2010); Melfo et al. (2012); Alloul et al. (2013); Chun and Sharma (2014); del Águila and Chala (2014); Bambhaniya et al. (2014); Dev et al. (2016); Mitra et al. (2017); Ghosh et al. (2018); Li (2018); Borah et al. (2018); Antusch et al. (2018a); Crivellin et al. (2019); Bhupal Dev and Zhang (2018); Du et al. (2018). The pair-production through $\gamma\gamma$ fusion is also important for higher doubly-charged scalar masses Babu and Jana (2017), whereas the single-production through $WW$-fusion Huitu et al. (1997); Maalampi and Romanenko (2002); Kanemura et al. (2013); Englert et al. (2013); Dutta et al. (2014); Kanemura et al. (2014); Bambhaniya et al. (2015); Sirunyan et al. (2018) is suppressed by the smallness of the triplet VEV $v_{\Delta}$ and is not accessible to the LHC Sirunyan et al. (2018). Due to the hadrophobic nature of the triplet scalar, one can never exploit the large Yukawa $Y^{\Delta}$ to enhance the production cross-section of $\Delta$-fields at hadron colliders, although the size and structure of $Y^{\Delta}$ still play an important role in deciding their dominant decay modes. In contrast, at the $e^{-}p$ collider, the doubly-charged scalar can be singly produced in association with a lepton ($l$ or $\nu_{l}$) and a jet (see Fig. 1), with the cross-section directly proportional to the square of the Yukawa coupling to the electron flavor, $|Y^{\Delta}_{el}|^{2}$ (with $l=e,\mu,\tau$). Therefore, as we show in this paper, the Yukawa coupling $Y^{\Delta}$ can be directly measured at future $e^{-}p$ collider, as long as it is reasonably large $\sim{\cal O}(1)$ to give an observable signal-to-background ratio. This is largely complementary to the case of future lepton colliders, where the doubly-charged scalar can also be singly/pair-produced via the Yukawa coupling $Y^{\Delta}_{el}$ Mukhopadhyaya and Rai (2006); Chen et al. (2009); Rodejohann (2010); Yue et al. (2011); Rodejohann and Zhang (2011); Nomura et al. (2018); Agrawal et al. (2018); Bhupal Dev et al. (2018); Dev et al. (2018).

For large diagonal Yukawa couplings, the off-diagonal entries of $Y^{\Delta}$ are restricted from LFV processes, such as $\mu\to 3e$, $\mu\to e\gamma$, and $\mu-e$ conversion in nuclei Akeroyd et al. (2009); Dev et al. (2017, 2018); Bhupal Dev and Zhang (2018). To evade this conflict, we consider the scenario with large diagonal $Y^{\Delta}_{ii}$ (with $i=e,\mu$) and small off-diagonal $Y^{\Delta}_{ij}$ (with $i\neq j$) that are in agreement with LFV measurements and are also promising for collider searches of $H^{\pm\pm}$ at $e^{-}p$ collider. However, this choice can not reproduce the observed neutrino mixing angles, which require a highly non-diagonal PMNS mixing matrix. The conflict with the neutrino data can be evaded by considering variations in the pure type-II seesaw model. A simple possibility is to consider a hybrid seesaw model, that contains both the gauge-singlet, right-handed (RH) neutrinos and the triplet Higgs. The light neutrino mass is generated by both singlet neutrino (type-I) and the triplet Higgs (type-II) contributions. The phenomenology of the singlet neutrinos at the $e^{-}p$ collider will be the same as discussed before Mondal and Rai (2016a); Antusch et al. (2017), and our main interest here lies in the doubly-charged Higgs phenomenology.

The direct search constraints on the doubly-charged scalar from the LHC put a lower bound on $M_{H^{\pm\pm}}\gtrsim 800$ GeV at 95$\%$ C.L CMS (2017); Aaboud et al. (2018). For the dominant $H^{\pm\pm}$ production channels considered here, we find that the proposed LHeC with $\sqrt{s}=1.3$ TeV (or even the high-energy LHeC with $\sqrt{s}=1.8$ TeV) cannot probe the doubly-charged Higgs boson beyond its current LHC limit. Therefore, for the collider analysis, we consider the future circular collider-based $e^{-}p$ scenario (FCC-eh) with $\sqrt{s}=3.46$ TeV. We focus on two different signals: (i) $3l$ + $\geq 1j$, and (ii) an inclusive search for $H^{--}$ in the $\geq 2l$ + $\geq 1j$ final state, with at least two negatively-charged leptons of same flavor in each case. We consider a number of SM backgrounds that can mimic our signal. By choosing suitable cuts on the kinematic variables, we reduce the background significantly and achieve an observable signal-to-background ratio. We show that doubly-charged Higgs masses up to 2 TeV can be probed at $3\sigma$ confidence level with 500 fb*-1* integrated luminosity at FCC-eh.

The rest of the paper is organized as follows. In Section II, we review the basic relevant features of the triplet Higgs model. In Section III, we discuss the dominant production channels and the signals for doubly-charged scalar at $e^{-}p$ collider. Followed by this, we present a detailed event analysis using cut-based techniques for the signal and background at FCC-eh. Finally, we present our conclusions in Section IV.

II Triplet Higgs Model

We consider a hybrid type-I+type-II seesaw model, where the SM particle content is extended by an additional $SU(2)_{L}$-triplet scalar with hypercharge $Y=2$:

[TABLE]

and the SM gauge-singlet RH neutrinos $N_{R,i}$ (with $i=1,2,3$ being the family index). The complete particles spectrum is shown in Table 1, along with the corresponding charges under the SM gauge group. Note that the presence of both type-I and type-II seesaw components can be naturally motivated as originating from a left-right symmetric gauge group, or even higher gauge groups like Pati-Salam or $SO(10)$ at high scale.

The complete Lagrangian with the additional triplet scalar and RH neutrino is given by,

[TABLE]

where $\mathcal{L}_{\rm SM}$ is the complete Lagrangian for the SM particles and ${\cal L}_{\rm kin}^{N_{R}}=i\overline{N}_{R}\gamma^{\mu}\partial_{\mu}N_{R}$ is the kinetic term of the right-handed neutrinos. The covariant derivative for the triplet Higgs takes the following form:

[TABLE]

where $W^{i}_{\mu},\ B_{\mu}$ are the $SU(2)_{L}$ and $U(1)_{Y}$ gauge fields, with the corresponding gauge couplings $g,\ g^{\prime}$ respectively, and $\sigma^{i}$’s (with $i=1,2,3$) are the Pauli matrices.

The Yukawa term in Eq. (2) takes the following form:

[TABLE]

where $C$ is the Dirac charge conjugation matrix with respect to the Lorentz group, $\widetilde{\phi}\equiv i\sigma_{2}\phi^{*}$ and $Y^{D}$ is the Dirac Yukawa coupling for neutrinos. The 1st and 3rd term in Eq. (4) violate lepton number by two units and give rise to the light neutrino mass at tree-level via type-II and type-I seesaw mechanisms, respectively, after electroweak symmetry breaking. The neutral component of the Higgs doublet acquires a spontaneous VEV, while the neutral component of the Higgs triplet acquires an induced VEV:

[TABLE]

The Majorana neutrino mass matrix for the light neutrinos have the following form:

[TABLE]

where $M_{L}=\frac{v_{\Delta}}{\sqrt{2}}Y^{\Delta}$ and $M_{D}=\frac{v}{\sqrt{2}}Y^{D}$. In the limit $Y^{D}\to 0$, the light neutrino mass matrix becomes type-II dominant: $M_{\nu}\simeq\frac{v_{\Delta}}{\sqrt{2}}Y^{\Delta}$. Since the triplet VEV $v_{\Delta}$ governs the light neutrino mass in type-II seesaw, it is expected to be much smaller than the electroweak VEV $v$, i.e. $v_{\Delta}\ll v$. In fact, the electroweak precision data, and in particular, the $\rho$-parameter constraint requires that $v_{\Delta}\lesssim 2$ GeV Bhupal Dev and Zhang (2018). For the collider study of doubly-charged Higgs production and decay (see Section III), we consider low VEV of the triplet Higgs, $v_{\Delta}\sim 0.1$ eV, that generates large Yukawa $Y^{\Delta}\sim\mathcal{O}(1)$. In particular, we consider large diagonal elements of $Y^{\Delta}_{ii}$ (with $i=e,\mu$), and small off-diagonal Yukawa $Y^{\Delta}_{ij}$ (with $i\neq j$) in order to avoid the bounds from LFV processes mediated by the doubly-charged scalars Akeroyd et al. (2009); Barry and Rodejohann (2013); Bambhaniya et al. (2016); Dev et al. (2017); Bhupal Dev and Zhang (2018). Therefore, we need the type-I seesaw contribution in Eq. (6) to reproduce the observed pattern of the PMNS mixing matrix in the light neutrino sector. Apart from fitting the neutrino mass, the Dirac Yukawa couplings $Y^{D}$ play no other role in our subsequent discussion.

The scalar potential in Eq. (2) has the following form:

[TABLE]

In the above potential, one can introduce another term $\lambda_{5}(\phi^{\dagger}\Delta^{\dagger}\Delta\phi)$, but this quartic coupling $\lambda_{5}$ can be easily absorbed by redefining the couplings $\lambda_{1}$ and $\lambda_{4}$, by virtue of the relation $\phi^{\dagger}\Delta\Delta^{\dagger}\phi+\phi^{\dagger}\Delta^{\dagger}\Delta\phi=\phi^{\dagger}\phi{\rm Tr}\left[\Delta^{\dagger}\Delta\right]$, which is valid because $\Delta$ is a traceless $2\times 2$ matrix [cf. Eq. (1)].

We expand the doublet and triplet scalar fields around their VEVs to obtain 10 real-valued field components:

[TABLE]

We consider $\mu^{2}_{\phi}>0$ and $\mu^{2}_{\Delta}>0$ in Eq. (7). We also assume the parameter $\mu$ in the cubic term of Eq. (7) to be real. Upon minimization of the scalar potential with respect to the VEVs, we get the following relations:

[TABLE]

and the mass terms for the scalar fields. The mass term of the doubly-charged Higgs has the following form:

[TABLE]

where we have defined the mass basis of the doubly-charged Higgs as $\delta^{\pm\pm}=H^{\pm\pm}$.

The mass matrix of the singly-charged Higgs in the basis ($\phi^{\pm},\delta^{\pm}$) takes the following form:

[TABLE]

We define the mass basis for the singly-charged scalars as ($G^{\pm},H^{\pm}$), which are related to the Lagrangian fields by an orthogonal matrix parameterized by the mixing angle $\beta^{\prime}$ with $\tan\beta^{\prime}=\frac{\sqrt{2}\,v_{\Delta}}{v}$ in the following way:

[TABLE]

After diagonalization, we get the following eigenvalue for the physical singly-charged scalar:

[TABLE]

whereas $M^{2}_{G^{\pm}}=0$, corresponding to the Goldstone bosons $G^{\pm}$ responsible for the mass of the singly-charged gauge bosons $W^{\pm}$. In the limit $v_{\Delta}\ll v$, using Eq. (9), we can relate the doubly and singly-charged scalar masses in Eqs. (11) and (14) as follows:

[TABLE]

We assume the quartic coupling $\lambda_{4}\ll 1$, so that the cascade decay of $H^{\pm\pm}\to H^{\pm}W^{\pm(*)}$ is kinematically suppressed, as compared to the dilepton decay mode $H^{\pm\pm}\to l^{\pm}l^{\pm}$. The diboson channel $H^{\pm\pm}\to W^{\pm}W^{\pm}$ is always suppressed, as long as $v_{\Delta}\lesssim 0.1$ MeV Fileviez Perez et al. (2008).

Among the four neutral scalars, two of them are CP even and the remaining two are CP odd, as can be seen from Eq. (8). Assuming CP conservation, we can write the CP-even and CP-odd mass matrices as follows:

[TABLE]

We define the mass eigenbasis both for CP-even and CP-odd scalars, which are related to the Lagrangian fields in the following manner:

[TABLE]

Here $h$ represents the SM-like Higgs and $G^{0}$ represents the Goldstone boson associated with the SM $Z$-boson. The mixing angles related to the neutral fields depend on the VEVs and the quartic couplings in the following manner:

[TABLE]

The eigenvalues of the mass matrices (16) and (17) are given by

[TABLE]

We can easily write down the quartic couplings and the $\mu$-parameter appearing in Eq. (7) in terms of the above-mentioned physical masses:

[TABLE]

There are miscellaneous constraints on the quartic couplings from electroweak precision measurements, absence of tachyonic modes, boundedness of the potential, unitarity, vacuum stability and naturalness considerations Arhrib et al. (2011); Chun et al. (2012); Bhupal Dev et al. (2013); Kobakhidze and Spencer-Smith (2013); Chabab et al. (2016); Haba et al. (2016); Das and Santamaria (2016); Xu (2016); Dev et al. (2017); Ghosh et al. (2018), all of which have been taken into account in choosing the benchmark points in our subsequent collider analysis.

III Signatures at $e^{-}p$ Collider

In this section, we analyze the discovery prospects of a doubly-charged Higgs in the future $e^{-}p$ collider. We consider the FCC-eh design with beam energies $60$ GeV (for electron) and $50$ TeV (for proton), so the center-of-mass energy is $\sqrt{s}\simeq\sqrt{4E_{e}E_{p}}=3.46$ TeV and a TeV-scale doubly-charged Higgs can be produced on-shell, through associated production, along with a charged lepton or neutrino and an energetic jet. We consider the following two production modes (see Fig. 1):

• •

Process - I : $e^{-}p\ \rightarrow\ e^{+}\,j\,H^{--}$,   $H^{--}\rightarrow l^{-}\,l^{-}$  ,

• •

Process - II : $e^{-}p\ \rightarrow\ j\,\nu\,H^{--}$,   $H^{--}\rightarrow l^{-}\,l^{-}$ ,

where $l=e,\mu$. The diagrams in Fig. 1 clearly show the dominant production mode, depending on the choice of the triplet VEV. While the first three diagrams would be relevant for low values of the triplet VEV as it would lead to a larger Yukawa $Y^{\Delta}_{ee}$ coupling, they become less relevant when the VEV is chosen larger, in which case the vector-boson-fusion (VBF) diagram becomes dominant.111Thus, in the small VEV limit, our analysis equally applies to both $SU(2)_{L}$ and $SU(2)_{R}$ scalar triplets, such as in the left-right symmetric model. A polarized electron beam could, in principle, distinguish between the two contributions, by analyzing the angular distributions of the final-state leptons, for instance. We have only considered the unpolarized beam in this analysis and the effect of polarization is left for future study. In Process-I, the contribution of $Z$-exchange is much smaller, as compared to the photon exchange. As for the photon-mediated processes, depending on the photon virtuality, the proton may remain intact (elastic process) or dissociates (inelastic process). As discussed below, a selection cut of 40 GeV on the transverse momentum of the final state jet is imposed in our analysis. This implies large photon virtuality (compared to the mass of the hadronic final state), and hence, the inelastic process, where the interaction involves a quark inside the proton, gives the dominant contribution.

It is worthwhile to note that the production of a doubly-charged scalar at FCC-eh has a major difference when compared with its production at the LHC. At the LHC, one can never exploit the large Yukawa $Y^{\Delta}_{ee}$, although it still plays a very effective role in deciding the dominant decay modes for the doubly-charged Higgs. Notably, large $Y^{\Delta}$ at the LHC leads to the heavier-flavor charged lepton (tau) to be pair-produced more effectively from the $H^{\pm\pm}$ decay when the neutrino masses exhibit “normal hierarchy”, whereas for “inverted hierarchy” the electron and muon flavors will be pair-produced more copiously than the tau flavor. On the other hand, for $Y^{\Delta}\lesssim 10^{-7}$, either the decay of $H^{\pm\pm}\to l^{\pm}l^{\pm}$ is displaced Bhupal Dev and Zhang (2018); Antusch et al. (2018b), or the diboson decay modes, such as $H^{\pm\pm}\to W^{\pm}W^{\pm}$, take over the dilepton mode Kanemura et al. (2013), depending on the $H^{\pm\pm}$ mass. So $Y^{\Delta}_{ee}$ still has a secondary role in the phenomenology of doubly-charged Higgs at hadron colliders, but cannot be directly probed due to its hadrophobic nature. In this work we explore this region of parameter space where the triplet VEV is small, $v_{\Delta}\sim{\cal O}(0.1)$ eV, and $Y^{\Delta}_{ee}$ can be directly probed, as it plays the all important role in the doubly-charged scalar phenomenology at the FCC-eh.

It is also worth noting that we always produce the negatively-charged state at $e^{-}\,p$ colliders when produced singly. The decay of $H^{--}$ into two same-sign dileptons leads to the following signals, which we explore in detail:

• •

Signal - I : $3\,l$ + $\,\geq 1\,j$  (with at least two negatively-charged leptons of same flavor),

• •

Signal - II : $\geq 2\,l\,+\,\geq 1\,j$  (where at least two negatively-charged leptons are of same flavor).

Note that Signal-I with a trilepton final state comes from the subprocesses listed as Process-I in Fig. 1 while Signal-II, which has at least two charged leptons can get contributions from all the subprocesses in Fig. 1, i.e. Process-I and Process-II. Signal-II also has substantial MET due to the presence of neutrinos in the final state as shown in Process-II. Although a trigger upon the MET could reduce the contributions from subprocesses in Process-I which will have very little MET, it could prove useful in SM background suppression.

The production cross-sections for $e^{-}p\rightarrow e^{+}\,j\,H^{--}$ and $e^{-}p\rightarrow j\,\nu\,H^{--}$ depend on the Yukawa coupling strength $Y^{\Delta}_{ee}$ quadratically. So we must choose large diagonal entries for $Y^{\Delta}$, as mentioned before. The off-diagonal Yukawa couplings are then required to be quite small, so as to satisfy the LFV constraints from $\mu\to 3e$, $\mu\to e\gamma$, etc. The current bounds on the branching ratio of $\mu\to e\gamma$ and $\mu\to eee$ are $4.2\times 10^{-13}$ Baldini et al. (2016) and $10^{-12}$ Bellgardt et al. (1988), respectively. For our model, if the doubly-charged Higgs mass is around 1 TeV, then the corresponding bounds on the elements of $Y^{\Delta}_{\alpha}$ matrix are given by Akeroyd et al. (2009); Dev et al. (2017); Bhupal Dev and Zhang (2018)

[TABLE]

Note that the diagonal elements $Y^{\Delta}_{ii}$ cannot be arbitrarily large either. Specifically, a large $Y^{\Delta}_{ee}$ would significantly contribute to the Bhabha scattering $e^{+}e^{-}\to e^{+}e^{-}$ via the $t$-channel $H^{\pm\pm}$, altering both the total cross section and the differential distributions Abdallah et al. (2003); Abbiendi et al. (2003); Achard et al. (2003). Using the LEP data on the $e^{+}e^{-}\to e^{+}e^{-}$ cross-section measurements Abbiendi et al. (2003); Abdallah et al. (2006), we obtain a 90% CL upper limit Bhupal Dev et al. (2018)

[TABLE]

Thus, for a TeV-scale $H^{\pm\pm}$, we must have $|Y^{\Delta}_{ee}|<0.35$.222Note that the neutrinoless double beta decay limits are not relevant here, because the amplitude of the $H^{--}$-mediated process depends on the $H^{--}W^{+}W^{+}$ coupling, which is suppressed by the small triplet VEV $v_{\Delta}$. Similarly, both $Y^{\Delta}_{ee}$ and $Y^{\Delta}_{\mu\mu}$ contribute to the muonium-antimuonium oscillation, i.e. the LFV conversion of the bound states $(\mu^{+}e^{-})\leftrightarrow(\mu^{-}e^{+})$, whose oscillation probability has been constrained by the MACS experiment Willmann et al. (1999), and leads to the 90% CL upper limit Bhupal Dev et al. (2018)

[TABLE]

There exist other constraints involving the diagonal elements from $\mu\to e\gamma$ and electron/muon anomalous magnetic moment Bhupal Dev et al. (2018), which are however weaker than the two constraints mentioned above.

By choosing the diagonal elements $Y^{\Delta}_{ee,\mu\mu}\in[0.1,0.3]$ to be consistent with the bounds in  (23) and (24), and assuming $Y^{\Delta}_{\tau\tau}$ to be small (since it does not play any role in our subsequent collider analysis), we show the allowed values of the off-diagonal Yukawa coupling parameters $Y^{\Delta}_{e\mu,e\tau,\mu\tau}$ as a scatter plot in Fig. 2 which satisfy the LFV constraints shown in (22). Here we have chosen $M_{H^{\pm\pm}}=1$ TeV as a benchmark value. The vertical and horizontal shaded regions are excluded from the experimental constraints on the tau-LFV decay modes $\tau\to 3e$ and $\tau\to 3\mu$, respectively Amhis et al. (2017), for fixed $Y^{\Delta}_{ee,\mu\mu}=0.3$. It is clear that the off-diagonal Yukawa couplings are constrained to be small for our choice of large diagonal Yukawa couplings.

To simulate the signal events, we implement the model described in Section II in Feynrules Alloul et al. (2014) to create the UFO model files. The package Madgraph (v2.4.3) Alwall et al. (2014) is used to generate parton-level events using CTEQ6L1 Pumplin et al. (2002) parton distribution functions for the colliding proton beam. The showering and subsequent hadronization has been done with Pythia (v6.4) Sjostrand et al. (2006). Finally, we pass the generated events through Delphes (v3.4.1) de Favereau et al. (2014) for detector simulation. To reconstruct jets in the final state, we have used the anti-$k_{t}$ jet clustering algorithm Cacciari et al. (2008) with a radius parameter $R=0.4$ in Fastjet Cacciari et al. (2012) for jet formation.

We show the variation of the single production cross-section of the doubly-charged Higgs as a function of its mass in Fig. 3 for both Process-I and Process-II. Note that Process-I with a larger multiplicity of subprocesses as well as the enhanced coupling of the photon to the two units of charge of the doubly-charged scalar is significantly larger than the Process-II with $W$ boson exchange, where the second (VBF) subprocess hardly contributes to the production cross-section due to the very small VEV. The mass range of interest for the doubly-charged Higgs for our analysis is dictated by the limits from $\sqrt{s}=13$ TeV LHC which is about 800 GeV CMS (2017); Aaboud et al. (2018). As shown in Fig. 3, for $M_{H^{\pm\pm}}$ in between 500 GeV and 1 TeV, the cross-section varies between 20 fb and 2 fb. For larger mass values the cross-section falls rapidly, as the mass approaches the center-of-mass energy of $\sqrt{s}=3.46$ TeV for the FCC-eh.

In Table 2 we give the choice of our benchmark points which give us three different mass values for the doubly-charged Higgs boson. A closer look at Eq. (11) shows that for a given choice of triplet VEV and $\lambda_{4}$, the value of $\mu$ would dictate the mass values. For simplicity, we just vary $\mu$ to generate different values for the doubly-charged scalar mass, while making sure that the quartic coupling values are consistent with all the theoretical constraints Arhrib et al. (2011); Chun et al. (2012); Bhupal Dev et al. (2013); Kobakhidze and Spencer-Smith (2013); Chabab et al. (2016); Haba et al. (2016); Das and Santamaria (2016); Xu (2016); Dev et al. (2017); Ghosh et al. (2018). The first benchmark point we choose has a doubly-charged scalar mass close to the current LHC bound, which is kind of the most optimistic scenario for its discovery at FCC-eh. The $Y^{\Delta}_{ee}$ value chosen for each benchmark is the maximum value allowed by the LEP limit [cf. Eq. (23)] for the corresponding doubly-charged scalar mass. We also give the total decay width of the doubly-charged scalar in each case, which is simply given by $\Gamma_{H^{--}}\simeq\frac{|Y^{\Delta}_{ee}|^{2}}{8\pi}M_{H^{--}}$, assuming that it dominantly decays to the $ee$ final state only.

In what follows, we perform a detailed cut-based analysis for the signal and background for both Signal-I and Signal-II identified above.333Since we consider the on-shell production of the doubly-charged scalar, and its width/mass is less than 1% (see Table 2), the interference between signal and background is negligible.

III.1 Signal - I ($3\,l$ + $\,\geq 1\,j$) :

The trilepton signal containing at least a pair of same-sign, same-flavor charged leptons can come from Process-I only. The SM background for the signal can have several sources, depending on their rate of production. We list the dominant and relevant SM processes below:

The most dominant SM background for the above final state, at an $e^{-}\,p$ collider is the irreducible background given by $e^{-}p\rightarrow j\,e^{-}\,l^{+}\,l^{-}$. 2. 2.

A subdominant contribution can also come from the reducible, on-shell $Z$ boson production which then decays to a pair of charged leptons via the process $e^{-}p\rightarrow e^{-}jZ$. Note that the requirement of a same-sign, same-flavor pair of charged leptons in the final state ensures that only the $Z$ decay to $e^{-}\,e^{+}$ contributes. 3. 3.

A relatively weaker background contribution could also come from an $\alpha_{\rm em}$-suppressed process $e^{-}p\rightarrow e^{-}j\gamma Z$ where the $Z$ boson again decays leptonically and specifically to $e^{-}\,e^{+}$ only, while the photon is misidentified as a lepton. However, the photon misidentification rate is less than $10^{-4}$ for electron and $10^{-3}$ for muon Aad et al. (2011)444These numbers from Ref. Aad et al. (2011) include both photons and jets misidentified as leptons. So the photon-only contribution will be even smaller., so this background will be negligible, as we will see below (in Table 3). 4. 4.

One should also account for the possibility of contributions to the SM background where a jet may fake a charged lepton. Notwithstanding the fact that the fake-rate for such events would be significantly smaller, one begs the question whether a relatively sizable cross-section for jet-enriched final state leaves an imprint on the signal under consideration. For the estimation of fake-rate induced background, we have considered the process $e^{-}p\rightarrow e^{-}jj$ and assumed the rate for a jet faking a charged lepton as 0.1$\%$ ATL (2016).

To begin with, all the above channels for the background contribute substantially with basic acceptance cuts for the subprocess under consideration. However, the contributions to the actual signal topology starts showing the relative importance of the backgrounds that would eventually contribute to the final analysis. We begin the event generation by demanding:

1. A0.

The final state leptons and jets satisfy a minimum requirement for the transverse momenta and their pseudorapidity, given by $p_{T}^{j}>40$ GeV, $p_{T}^{l}>10$ GeV, $|\eta^{j,l}|\leq 4.5$.

In Table 3 and Table 4 we present the cut-flow estimates for both the SM background and the signal cross-section at the proposed FCC-eh machine. The A0 cuts are simply aimed at understanding the signal and background characteristics in a few kinematic variables which could be then utilized to device suitable cuts that help in improving the significance of the signal. Note that for the signal, the decay of the doubly-charged Higgs boson is an important variable. Since the off-diagonal elements of $Y^{\Delta}$ are assumed to be small, only the lepton-flavor-conserving modes $H^{\pm\pm}\to l^{\pm}l^{\pm}$ occur. As mentioned before, we shall neglect the $\tau$ decay mode and focus only on the $e^{-}$ and $\mu^{-}$ modes. We first consider the simple case where $H^{\pm\pm}$ decays to $e^{\pm}e^{\pm}$ with 100% branching ratio, i.e. only the $ee$ element of $Y^{\Delta}$ is large and all other elements small. Later in Section III.3, we will also consider a case where $H^{\pm\pm}$ decays to both $e^{\pm}e^{\pm}$ and $\mu^{\pm}\mu^{\pm}$, each with 50% branching ratio, i.e. both $Y^{\Delta}_{ee}$ and $Y^{\Delta}_{\mu\mu}$ are of same size. This would help in generalizing the result when exploring the sensitivity for heavier mass values of the doubly-charged Higgs. To ensure that we only select electron or muon-flavored leptons, we impose a further requirement:

1. A1.

The final state events must contain at least three charged leptons, among which we have $N(e^{+})=1$, $N(\mu^{+}$) = 0, and either a pair of $e^{-}$ or $\mu^{-}$ i.e., $N$($e^{-}$ or $\mu^{-}$) = 2 while the number of jets in the final state must be greater than or equal to one, i.e. $N(j)\geq 1$.

This requirement is good enough to suppress most of the SM background, especially the jet-faking-lepton channel as seen from Table 3. However, note that the doubly-charged Higgs signal strength is totally dependent on the production cross-section which depends on its mass. As LHC already puts strong limits, the mass range of our interest would entail small production cross-section even at the FCC-eh machine, evident in Fig. 3. Thus we need to find methods of suppressing the background further to ensure sensitivity to heavier mass values of the doubly-charged Higgs. The two very significant characteristics one expects is that the charged leptons coming from the decay of the heavy scalar would carry a significantly large transverse momentum and the invariant mass of the same-sign, same-flavor lepton pair would be concentrated in regions of very high mass, specific to the mass of the doubly-charged Higgs. In comparison, the SM background events should have leptons with relatively softer $p_{T}$, as well as a more continuum invariant mass distribution peaked at smaller values of $M_{ll}$.

In Fig. 4, we show the transverse momentum ($p_{T}$) distribution of the leading charged lepton, both for signal and background. Additionally, we also show the distribution of the invariant mass of the same-sign same-flavor leptons which shows expected differences between the signal and background events. We therefore impose the following cuts (A2 and A3), which turn out to be very much useful in reducing the backgrounds events.

1. A2.

We impose cuts on the transverse momentum of the leading and sub-leading charged lepton and also on the leading jet as: $p_{T}^{l_{i}}>50$ GeV ($i=1,2$) and $p^{j_{1}}_{T}>50$ GeV. 2. A3.

We also demand that the invariant mass $M_{ll}$ of same-sign lepton pair is large, with $M_{ll}>800\,\,{\rm GeV}$. This cut reduces the background significantly.

In fact, the only relevant SM background events that remain after cut A3 are from the process $e^{-}\,p\to e^{-}\,l^{+}l^{-}j$. The suppression in the signal cross-section for all the three benchmark points is about 50% for the event selection cuts, as can be seen in Table 4.

III.2 Signal - II ($\geq 2\,l\,+\,\geq 1\,j$) :

We now consider the multi-lepton channel where we have at least two charged leptons in the final state and both need to be of the same sign and flavor. This would correspond to the inclusive search of the signal for a singly-produced doubly-charged Higgs coming from both Process-I and Process-II. The combination of both the processes for the inclusive search would mean that one avoids kinematic selection of events that could potentially reduce contribution from any of the particular processes. This on the other hand would mean that there would be additional sources of background for the final state under consideration from the ones discussed in the previous subsection which are listed below.

The irreducible background coming from $e^{-}p\rightarrow e^{-}l^{-}j\bar{\nu_{l}}$. Again, for the same-sign, same-flavor requirement of the charged-lepton pair, the signal topology ensures that $l=e$. 2. 2.

To add to the fake-rate contribution, we must consider the process $e^{-}p\rightarrow e^{-}j\gamma$ where $\gamma$ can fake the charged lepton $e^{-}$. 3. 3.

In addition to fake rates, we note that charge mismeasurement can also lead to potential backgrounds which mimic the signal. We evaluate $e^{-}p\rightarrow e^{-}e^{+}j\,\nu$ process, where we consider a conservative $0.1\%$ Khachatryan et al. (2016) as the charge mismeasurement or charge-flip rate/probability in the detector. 4. 4.

We also estimate the background that arises due to the jet faking a lepton which as before is assumed to be 0.1$\%$ .

As before, B0 and B1 discussed below define the selection of events for the signal in consideration.

1. B0.

As in Signal-I, the final state leptons and jets must satisfy a minimum requirement for the transverse momenta and their pseudorapidity, given by $p_{T}^{j}>40$ GeV, $p_{T}^{l}>10$ GeV, $|\eta^{j,l}|\leq 4.5$. 2. B1.

We then demand that we select only those events which have at least 2 charged leptons and a minimum of 1 jet, which defines the signal as $\geq 2l+\geq 1j$.

Similar to the previous analysis, we again implement a few similar cuts on the events to suppress the background and improve signal sensitivity.

1. B2.

The events satisfying B0 and B1 are then required to contain two leptons of same sign and same flavor. The kinematic properties of these charged leptons is expected to be similar to what was observed in Fig. 4 and therefore we impose a cut on the $p_{T}$ of the leading and sub-leading charged lepton as $p_{T}^{l_{i}}>50$ GeV ($i=1,2$) while the leading jet must satisfy $p^{j_{1}}_{T}>50$ GeV. 2. B3.

To utilize the hardness of the invariant mass distribution of the same sign-same-flavor dilepton for the signal in suppressing the background, we further demand that $M_{ll}>800\,\,{\rm GeV}$ .

In Table 5 and Table 6 we show the cut-flow of the SM background and signal for the aforementioned cuts. As before, the most selective of all the cuts which improves the signal over the SM background is B3. However, although the ($\geq 2l+\geq 1j$) inclusive channel gets contributions from both the processes shown in Fig. 1, the signal-to-background ratio is still inferior to the trilepton channel. The reason for this is because of the size of the continuum background $e^{-}l^{-}\,j\,\bar{\nu}$.

III.3 $\mu\mu$ Channel

An important point which we should highlight here is the fact that so far we have only considered the electron flavor for the same-sign leptons in the decay of the doubly-charged Higgs, while in principle, it could decay to any of the lepton flavors depending on the Yukawa structure $Y^{\Delta}_{ij}$. However, at an $e^{-}\,p$ machine, the SM background with two same-sign negatively-charged leptons would be dominated by the electron pair. Thus, for a suitable choice of $Y^{\Delta}_{ee}$ and $Y^{\Delta}_{\mu\mu}$, we could essentially have the signal containing two same-sign muons in the final state which would be relatively background-free after the cuts we put. To put this fact in perspective, we list in Table 7 the signal events for both the final states in consideration, written explicitly as $e^{+}+2\mu^{-}+\geq 1j$ and $2\mu^{-}+\geq 1j+\geq 0l$, after all the cuts mentioned before. Here, we assume that both the Yukawa couplings $Y^{\Delta}_{ee}$ and $Y^{\Delta}_{\mu\mu}$ are non-zero and of equal strength with the values given by Table 2 for each benchmark point, while satisfying the constraint given in Eq. (24), such that the $H^{--}$ decay is now equally shared between the electron and muon modes, which means a 50% branching ratio for the $\mu\mu$ channel. The presence of a large $Y^{\Delta}_{ee}$, on the other hand, ensures a significant production cross-section section of the doubly-charged Higgs boson.

Note that the leading contribution for the SM background in the $\mu\mu$ channel would arise from the subprocesses: $e^{-}\,p\to\nu_{l}\,Z\,W^{-}\,j$ and $e^{-}\,p\to e^{-}\,Z\,W^{-}\,j$, with both the $W$ and $Z$ bosons decaying into the muon channels. The parton-level cross section for these processes is itself at the level of $\sim 10^{-2}$ fb and assuming that similar cut efficiencies for the background would apply as in the case of the $ee$ mode, the background numbers would get further suppressed by a factor of $\sim\mathcal{O}(10^{-3})$, rendering the background almost negligible for the $\mu\mu$ channel.

Therefore, the cross-sections for the signal events given in Table 7 imply a nearly background-free signal of $\sim 5-10$ events with an integrated luminosity of about 100 ${\rm fb}^{-1}$. Thus our predictions are more conservative when including only electron flavor in our analysis, which is still significantly robust in the trilepton channel for mass values of the doubly-charged scalar we have chosen. We show this by presenting the signal significance in the following subsection.

III.4 Signal Significance

In determining the statistical significance of the signal we have used the following general expression Cowan et al. (2011):

[TABLE]

where $s$ and $b$ denote the number of signal and background events (i.e. cross section times integrated luminosity) respectively, after all selection cuts are applied. Note that the formula (25) reduces to the more well-known form $\mathcal{S}=s/\sqrt{b}$ in the limit $s\ll b$. In Table 8 we show the required integrated luminosity for either $3\,l$ or $\geq 2l$ channel to be observed with $3\sigma$ and $5\sigma$ statistical significance, where we consider $l=e$ only. Owing to the better $s/b$ ratio for the trilepton channel, one is clearly sensitive to the new physics signal in the trilepton channel with a relatively smaller data set when compared to the inclusive channel. However, both the channels provide a robust discovery prospect of the doubly-charged Higgs boson at FCC-eh, provided we have a sizable Yukawa coupling to the electron flavor, $Y^{\Delta}_{ee}\sim{\cal O}(1)$. Now, if the signal is driven by the $\mu\mu$ channel too, with the decay probability of the doubly-charged Higgs boson being 50% in the muon mode, then the required integrated luminosity for discovery reduces drastically. In fact, assuming a conservative background of around $10^{-5}$ fb in the $\mu\mu$ channel as discussed above and using the trilepton signal data from Table 7, we find that for BP1 discovery at $5\sigma$ statistical significance, the required integrated luminosity is just 27 ${\rm fb}^{-1}$, while it is 43 ${\rm fb}^{-1}$ and 95 ${\rm fb}^{-1}$ for BP2 and BP3, respectively.

To highlight the sensitivity of the FCC-eh to a doubly-charged Higgs mass at a given integrated luminosity, we should know what values of the Yukawa coupling $Y^{\Delta}_{ee}$ yield the required $H^{--}$ production cross-section. This is shown in Fig. 5 where we plot the $2\sigma$ and $3\sigma$ reach for the doubly-charged Higgs boson in the trilepton (Signal-I) and the inclusive dilepton (Signal-II) channels, and show the range of the Yukawa coupling values which would achieve the signal significance for a fixed integrated luminosity of 3 ${\rm ab}^{-1}$. This is for the scenario where we demand that $H^{--}$ decays to $e^{-}e^{-}$ with 100 $\%$ branching ratio. The sensitivity curves are obtained using Eq. (25) with the signal and background values obtained from the cut-based analysis described above. Fig. 5 in turn also gives us information on the maximum allowed value of the triplet VEV (corresponding to the minimum value of $Y^{\Delta}_{ee}$ accessible), so as to observe a signal for the doubly-charged Higgs boson at the FCC-eh. The vertical line in Fig. 5 shows the current lower limit on the doubly-charged scalar mass from the $\sqrt{s}=13$ TeV LHC data, assuming 100% branching ratio of the $H^{--}$ decay to same-sign electron-pair CMS (2017); Aaboud et al. (2018). Strictly speaking, this bound is applicable only when the $H^{\pm\pm}\to e^{\pm}e^{\pm}$ decay is prompt, which requires $Y^{\Delta}_{ee}\gtrsim 10^{-7}$ (but practically indistinguishable from zero in the linear $Y^{\Delta}_{ee}$-scale of Fig. 5). The green-shaded region is excluded by LEP data on the Bhabha scattering process $e^{+}e^{-}\to e^{+}e^{-}$ Abbiendi et al. (2003); Abdallah et al. (2006) as given by Eq. (23). The direct search limits from HERA Aktas et al. (2006), LEP Abdallah et al. (2003); Abbiendi et al. (2003); Achard et al. (2003) and Tevatron Acosta et al. (2004); Abazov et al. (2008) are only relevant for smaller $M_{H^{\pm\pm}}$ well below the mass range shown here and are anyway superseded by the LHC limits. It is worth noting that even the high-luminosity phase of the $\sqrt{s}=14$ TeV LHC cannot improve the mass reach beyond 1 TeV Mitra et al. (2017), and therefore, the FCC-eh provides a unique opportunity to directly probe the Yukawa coupling of heavier doubly-charged scalars to electrons. Only the proton-proton mode of FCC (FCC-hh) will be able to achieve a higher sensitivity in $M_{H^{\pm\pm}}$ up to about 6 TeV Du et al. (2018), independent of $Y^{\Delta}_{ee}$, as long as $H^{\pm\pm}$ dominantly decays to same-sign lepton pairs. On the other hand, the future $e^{+}e^{-}$ colliders like ILC (or CLIC) might be able to cover a similar (or larger) parameter space in the direct (or indirect) search channel, as compared to the FCC-eh reach shown in Fig. 5, depending on the center-of-mass energy Bhupal Dev et al. (2018).

The more robust and striking channel of discovery would however be the $\mu^{-}\mu^{-}$ channel, as illustrated in Fig. 6, which is relatively background free, barring very low rates from associated $W^{-}\,Z$ productions. As a large production rate for the doubly-charged Higgs is a necessity, we ensure a 50% decay probability of the $H^{\pm\pm}$ in the $\mu\mu$ mode while respecting the bound of Eq. (24). We show the much improved sensitivity reach using the signal rates for $e^{+}+2\mu^{-}+\geq 1j$ final state, where a similar $2\sigma$ or $3\sigma$ range as in Fig. 5 can now be attained with a much smaller integrated luminosity of 100 fb*-1*. A similar sensitivity is also expected in the inclusive $2\mu^{-}+\geq 1j+\geq 0l$ channel. This reach can be further improved if the signal significance of both the $ee$ and $\mu\mu$ channels (observed independently) are combined together.

IV Conclusion

We have analyzed the discovery prospect of a doubly-charged Higgs at the newly proposed FCC-eh electron-proton collider, operating with beam energies $E_{e}=60$ GeV for electron beam and $E_{p}=50$ TeV for proton beam. As the LHC would have limited sensitivity to a doubly-charged Higgs mass, even with the high-luminosity phase, searching for heavier doubly-charged scalars would be a challenge. A type-II seesaw-based scenario for neutrino mass generation with very tiny triplet VEV would imply a substantially large Yukawa coupling of the scalar to the charged leptons. Neither the LHC nor an $e^{+}e^{-}$ machine could gain from this, unless the collisions are of $e^{-}e^{-}$ type Mukhopadhyaya and Rai (2006), where the production of doubly charged Higgs has direct dependency on Yukawa. However, the proposed FCC machine, where the collisions are between an electron and a proton beam (FCC-eh) would open up the possibility of looking for these heavier doubly-charged states through their single on-shell production in association with lepton and a jet.

We have studied two types of final states, namely, $3l+\geq 1j$ and inclusive $\geq 2l+\geq 1j$, to probe TeV-scale doubly-charged scalars at FCC-eh. We consider a few representative benchmark points for a hybrid type-I+type-II seesaw with a small triplet VEV $v_{\Delta}\sim{\cal O}(0.1)$ eV and large diagonal Yukawa couplings $Y^{\Delta}_{ee}\sim{\cal O}(1)$. The model is consistent with neutrino oscillation data, as well as, in agreement with the bounds from lepton flavor violating processes, such as, $\mu\to e\gamma$ and $\mu\to eee$. We find that a doubly-charged Higgs with mass around a TeV can be probed at $3\sigma$ significance in FCC-eh with $\mathcal{L}\sim 245\,\rm{fb}^{-1}$ in the $3l+\geq 1j$ ($l=e$) mode. Moreover, higher mass values, up to about 2 TeV, could also be probed within a few years’ of running of the FCC-eh. A more robust and striking channel would be observed with only $\mathcal{L}=100\,\rm{fb}^{-1}$ of data if the doubly-charged Higgs has a decay probability in the $\mu\mu$ mode, which is comparable, if not more than the $ee$ mode. This reach can be further improved if the signal significance of both the $ee$ and $\mu\mu$ channels (observed independently) are combined.

Acknowledgments

BD, MM and SKR thank the organizers of WHEPP-XIV at IIT, Kanpur where this work was initiated, and the organizers of SUSY 2017 at TIFR, Mumbai where part of this work was done, for the local hospitality. SK would like to thank Subhadeep Mondal for help with numerical aspects at the initial stage of the work. The work of BD is supported by the US Department of Energy under Grant No. DE-SC0017987. MM acknowledges the support of DST-INSPIRE research grant IFA14-PH-99. The work of SKR was partially supported by funding available from the Department of Atomic Energy, Government of India, for the Regional Centre for Accelerator-based Particle Physics (RECAPP), Harish-Chandra Research Institute. SK would like to acknowledge support from the Department of Atomic Energy (DAE) Neutrino Project under the plan project of Harish-Chandra Research Institute and the European Union’s Horizon 2020 research and innovation programme Invisibles Plus RISE under the Marie Sklodowska-Curie grant agreement No 690575.