Production of the Doubly Charged Higgs Boson in Association with the SM Gauge Bosons and/or Other HTM Scalars at Hadron Colliders

Abstract

We investigate an extension of the Standard Model with one additional triplet of scalar bosons. Altogether, the model contains four Higgs bosons. We analyze the associated production of the doubly charged scalar with the Standard Model gauge bosons and the remaining Higgs bosons of the model, which are: the light (SM) and heavy neutral scalars and a singly charged scalar. We estimate, in the context of the present (HL–LHC) and future (FCC–hh) hadron colliders, the most promising processes in which a single produced doubly charged Higgs boson is involved.

1. Introduction

Experimental evidence of the Higgs boson’s existence, found at the Large Hadron Collider (LHC) [1,2], was a quantum leap in particle physics history. Nevertheless, questions related to the minimal Higgs sector are still open. From this point of view, extensions of the Standard Model can be realized in two ways: directly, by introducing additional multiplets, or indirectly, by extending the gauge group (e.g., left right symmetric models [3,4,5,6,7,8]). In the first scenario, many Beyond the Standard Model (BSM) theories assume the existence of additional Higgs bosons [9,10]. In the context of hadron colliders, doubly charged Higgs bosons [11,12,13] are especially attractive propositions; they lead to a production of same-charged leptons. In various BSM theories, they can occur at a significantly high rate. Thus, the Higgs sector with the Higgs triplet can lead to a discovery of new physics effects. In a general outline, triplet representations depend on the hypercharge $Y\equiv 2(Q-T_3)$ [13,14,15,16]. In this paper, we will focus on the Higgs Triplet Model (HTM), which extends the standard Higgs sector by adding one $SU\left(2\right)$ scalar triplet ($\Delta$) with hypercharge $Y=2$ [10,13]. The importance of investigation processes involving $H^{++}$ is due to two facts. First, the triplet vacuum expectation value (VEV) is very small, whereas $H^{++}$ mass can be at the level of a few hundred GeV. Such masses of $H^{++}$ scalars can be probed at the hadron colliders. Second, processes involving $H^{++}$ lead to the lepton flavor violation (LFV) as well as lepton number violating (LNV) processes, which can also be probed at hadron colliders. In addition, the model provides tiny masses of neutrinos by generating Majorana mass terms via the Type-II seesaw mechanism. Therefore, neutrino oscillation experiments put severe constraints on both the VEV $v_{\Delta }$ of the triplet scalar and Yukawa couplings.

The doubly charged Higgs bosons can be produced in pairs or as single events associated with gauge bosons and/or singly charged and neutral scalars. Amid the most important production channels at hadron colliders, it is worth mentioning the Drell-Yan processes via s-channel ${\gamma }^{*}/Z^{*}$ or $W^{*}$ exchange [17,18,19], the vector boson fusion process [20,21] or the gluon fusion process [22,23,24]. Pair production of doubly charged scalar and its signature as multi-leptons has been widely studied in the literature [25,26,27,28,29,30,31,32,33,34,35,36,37,38].

Searches of the charged scalars at the LHC show no significant excess from the SM backgrounds, thus lower bounds are put on their masses. The ATLAS collaboration set the lower bound on the mass of doubly charged scalar in the range of 770–870 GeV, assuming 100% branching ratios of leptonic modes [39] with $\sqrt{s}=13$ TeV. Recently, ATLAS has searched the decay of doubly charged scalar into gauge bosons with multileptons in final states and established the lower bound on doubly charged scalar as 350 GeV (pair production) and 230 GeV [40]. The CMS collaboration has also provided lower bounds that vary between 396–712 GeV for a pair production (assuming 100% branching ratios of leptons) and 479–746 GeV for an associated production, depending on the decay channel bounds [41] with $\sqrt{s}=13$ TeV. In this paper, we analyze the associated production of the doubly charged scalar with other scalars of the HTM model and/or SM gauge bosons. All relevant phenomenological constraints on the non-standard neutral and charged Higgs scalar parameters connected with HTM are taken into account. Among those worth mentioning are neutrino oscillations, low energy experiments, the bound on the $\rho$ parameter, limits on HTM contributions to ${(g-2)}_{\mu }$ and lepton LFV processes, as well as limits on HTM parameters coming from collider $e^{+}{e}^{-}$ and $e^{-}{e}^{-}$ scatterings. We will present results obtained for HL–LHC $\sqrt{s}=14$ TeV [42,43] and FCC-hh $\sqrt{s}=100$ TeV [44,45,46,47,48] proton-proton collision energies.

2. The Higgs Triplet Model Phenomenological Constraints

To extend the Standard Model, one additional $SU{\left(2\right)}_L$ triplet $\Delta$ is introduced. Depending on the triplet hypercharge Y, this triplet contains neutral, singly and/or doubly charged scalar fields. In our case, since we are interested in the doubly charged Higgs production, we chose the $Y=2$ and convention $Q=\frac{1}{2}Y+T_3$ (where $T_3$ is the third component of the izospin). So the scalar sector contains one doublet and the mentioned $Y=2$ triplet:

$$\begin{array}{ccc}\hfill \Phi =\frac{1}{\sqrt{2}}\left(\begin{array}{c}\sqrt{2}{w}_{\Phi }^{+}\\ v_{\Phi }+h_{\Phi }+i{z}_{\Phi }\end{array}\right),& & \Delta =\frac{1}{\sqrt{2}}\left(\begin{array}{cc}{w}_{\Delta }^{+}& \sqrt{2}{\delta }^{++}\\ v_{\Delta }+h_{\Delta }+i{z}_{\Delta }& -w_{\Delta }^{+}\end{array}\right).\hfill \end{array}$$

(1)

The linear combination of neutral, charged and doubly charged fields of the above multiplets creates the physical states h (associated with the SM Higgs boson and mass around 125 GeV), H (heavy neutral boson), and singly $H^{\pm }$ and doubly charged $H^{\pm \pm }$ scalar particles (see Appendix A.1) and one pseudoscalar.

The entire Lagrangian can be found in [49] and is built from its kinetic potential and Yukawa parts. The scalar potential and the exact formulas for the scalar particles’ masses can be found in the Appendix A.1.

In this work, we assume that the non-standard scalar particles’ masses are degenerated. This choice is justified since the constraints from the unitarity, T-parameter and $h\to \gamma \gamma$ [50,51] process limit the mass gap between the $M_H$, $M_{H^{\pm }}$ and $M_{H^{\pm \pm }}$ to a few dozens of GeV [52,53].

Other constraints on the model parameters come from the low and high energy processes after the Yukawa part of the Lagrangian:

$${\mathcal{L}}_Y^{\Delta }=\frac{1}{2}{y}_{\ell {\ell }^{\prime }}{L}_{\ell }^T{C}^{-1}i{\sigma }_2\Delta L_{{\ell }^{\prime }}+\mathrm{h}.\mathrm{c}.$$

(2)

This leads to the $H^{\pm \pm }$ − $l_i$ − $l_j$ and $H^{\pm }$ − $l_i$ − ${\nu }_j$ vertices (see Appendix A.2). Therefore, this interaction provides contributions to several lepton flavor violating processes, for example, the ${\ell }_i\to {\ell }_j\gamma$, ${\ell }_i\to {\ell }_j{\ell }_k{\ell }_l$, $\mu \to e$ conversion. Besides, there are also contributions to a few standard model lepton scatterings and ${(g-2)}_{\mu }$ that are mediated by heavy charged scalars. These phenomenological constraints in the context of HTM have been analysed in detail in the literature [37]. Studies in [10,36,53,54] point out that $H^{\pm \pm }$ production strongly depends on neutrino oscillations, LHC and $e^{+}{e}^{-}$ SM processes and low-energy lepton flavour violating data. It appears that the VEV of $v_{\Delta }$ is very small. Due to neutrino oscillation data, the $v_{\Delta }$ minimal value is constrained to be at the (sub)electronvolt level, and due to theoretical constraints combined with low-energy neutral and charged processes, the $v_{\Delta }$ upper limit cannot exceed a gigaelectronovolt value. In this work, we take $v_{\Delta }=50$ eV and $v_{\Delta }=0.5$ GeV as reference values.

3. Results

Let us first analyse the production $2\to 2$ and $2\to 3$ processes at hadron colliders where a single $H^{\pm \pm }$ boson and SM gauge bosons or other Higgs scalars of the HTM model are simultaneously produced. We use the CTEQ6L1 Parton Distribution Functions (PDF) [55,56] and basic MadGraph cuts. In Table 1, we present production rates for eleven non-vanishing processes. Other processes with two neutral bosons production, such as, for example, $pp\to H^{\pm \pm }Zh$, are forbidden by the charge conservation principle. For the reader’s convenience, the most promising processes are given in blue. Corresponding Feynman diagrams, which contribute to the processes shown in Table 1, are presented in the Appendix A.3. In this work, calculations have been performed using the MadGraph [57] and Pythia [58,59] programs. The UFO files were generated using FeynRules [60] and were built on our model file [37,53,61], based on the default Standard Model implementation.

Table 1. Cross section for a production of a single $H^{\pm \pm }$ boson with associated SM gauge bosons and other HTM scalars at the $pp$ colliders in $2\to 2$ and $2\to 3$ processes. Cross sections are calculated for $\sqrt{s}=14$ TeV (100 TeV) and are given in pb. Charged scalar masses are degenerated, $M_{H^{\pm \pm }}=M_{H^{\pm }}=1000$ GeV.

Process		Cross Section [pb]	Process		Cross Section [pb]
$\left(I\right)$	$pp\to H^{\pm \pm }{W}^{\mp }$	∼${10}^{-22}$	$\left(II\right)$	$pp\to H^{\pm \pm }{H}^{\mp }$	$8.13\times {10}^{-5}$
		(∼${10}^{-20}$)			($8.78\times {10}^{-3}$)
$\left(III\right)$	$pp\to H^{\pm \pm }Z{W}^{\mp }$	∼${10}^{-12}$	$\left(IV\right)$	$pp\to H^{\pm \pm }Z{H}^{\mp }$	$6.29\times {10}^{-7}$
		($2.7\times {10}^{-9}$)			($1.56\times {10}^{-4}$)
$\left(V\right)$	$pp\to H^{\pm \pm }{W}^{\mp }{W}^{\mp }$	∼${10}^{-10}$	$\left(VI\right)$	$pp\to H^{\pm \pm }{W}^{\mp }h$	∼${10}^{-12}$
		($1.87\times {10}^{-9}$)			($3.47\times {10}^{-8}$)
$\left(VII\right)$	$pp\to H^{\pm \pm }{W}^{\mp }H$	$1.35\times {10}^{-6}$	$\left(VIII\right)$	$pp\to H^{\pm \pm }{W}^{\mp }{H}^{\mp }$	$2.88\times {10}^{-6}$
		($2.44\times {10}^{-4}$)			($6.81\times {10}^{-4}$)
$\left(IX\right)$	$pp\to H^{\pm \pm }h{H}^{\mp }$	$1.07\times {10}^{-7}$	$\left(X\right)$	$pp\to H^{\pm \pm }H{H}^{\mp }$	∼${10}^{-29}$
		($1.63\times {10}^{-6}$)			(∼${10}^{-26}$)
$\left(XI\right)$	$pp\to H^{\pm \pm }{H}^{\mp }{H}^{\mp }$	∼${10}^{-31}$
		(∼${10}^{-28}$)

The dominant process associated with a single $H^{\pm \pm }$ production is $pp\to H^{\pm \pm }{H}^{\mp }$. The expected cross sections for the given Higgs boson masses are $8.13\times {10}^{-5}$ for $\sqrt{s}=14\mathrm{TeV}$, and $8.78\times {10}^{-3}$ pb for $\sqrt{s}=100\mathrm{TeV}$. With assumed luminosity $L=4a{b}^{-1}=4000$$f{b}^{-1}$ for HL–LHC [43] and $L=25a{b}^{-1}=25000$$f{b}^{-1}$ for FCC–hh [47], it converts to a total number of ∼300 and ∼$2\times {10}^5$ events at HL–LHC and FCC–hh, respectively.

The following ‘blue’ processes in Table 1 are connected with $2\to 3$ processes. The processes that provide negligible cross sections can be understood by analysing individual Feynman vertices; see the Appendix A.3 and Table 2. As we can see in Table 2, many vertices are proportional to VEV $v_{\Delta }$, which, as explained in Section 1 and Section 2, is at the level of (sub)eV-GeV.

Table 2. Couplings for the non standard scalars: $H^{\pm }$ (charged, top table) and H (neutral, bottom table).

Decay Channel	Coupling	Decay Channel	Coupling	Decay Channel	Coupling
$H^{\pm }\to W^{\pm }h$	∼$v_{\Delta }$	$H^{\pm }\to W^{\pm }\gamma$	∼$v_{\Delta }$	$H^{\pm }\to W^{\pm }Z$	∼$v_{\Delta }$
$H^{+}\to l_i^{+}{\overline{\nu }}_j$	∼$y_{{\ell }_i{\ell }_j}\sim \frac{1}{v_{\Delta }}$	$H^{+}\to l_i^{+}{\nu }_j$	∼$\frac{v_{\Delta }}{v_{\Phi }}$	$H^{\pm }\to q_i{\overline{q}}_j$	∼$\frac{\sqrt{2}}{v_{\varphi }}CKM$
$H\to hh$	∼$v_{\Delta }$	$H\to W^{+}{W}^{-}$	∼$v_{\Delta }$	$H\to ZZ$	∼$v_{\Delta }$
$H\to l^{+}{l}^{-}$	∼$v_{\Delta }$	$H\to q\overline{q}$	∼$v_{\Delta }$

In Figure 1, the $M_{H^{\pm \pm }}$ mass dependence of the $pp\to H^{\pm \pm }{H}^{\mp }$ cross section is given. As we can see, it depends strongly on the doubly charged Higgs boson mass. Note that in Table 1, results are given for $M_{H^{\pm \pm }}=M_{H^{\pm }}=1000$ GeV. However, as discussed in the Introduction and in [37], smaller masses of heavy Higgs bosons are still possible, so signal predictions given for processes in Table 1 can be even larger.

Figure 1. Cross section for $pp\to H^{\pm \pm }{H}^{\mp }$ process as a function of $M_{H^{\pm \pm }}$. In the calculations, $M_{H^{\pm }}$ is taken to be 1000 GeV.

Concerning the decay modes of the HTM Higgs bosons, in the case of doubly charged Higgs boson $H^{\pm \pm }$, we discussed the decay channels in detail in Section 5.2 of our previous work [37]. Here, we initiate a similar analysis for singly charged $H^{\pm }$ and heavy neutral scalar bosons H. First, possible decay modes of $H^{\pm }$ and H with corresponding strengths of interactions (couplings) are given in Table 2.

As one can find in Table 2, singly charged scalar $H^{\pm }$ coupling can break the lepton number ($H^{+}\to l_i^{+}{\nu }_j$) and lepton flavour number ($H^{+}\to l_i^{+}{\tilde{\nu }}_j$). These couplings also contribute to muon ${(g-2)}_{\mu }$, which was discussed in the Appendix of [37].

As we mentioned in Section 2, the triplet VEV $v_{\Delta }$ is bounded from below due to the LFV processes and from the top because of the $\rho$ parameter constrain. From Table 2 we can see that $H^{\pm }$ decay depends on $v_{\Delta }$, as well as the doubly charged scalar decay modes (see Figure 11 in [37]). Due to the assumed scalar’s mass degeneration, decays to the non-standard scalars are not considered, even though those might be significant as off-shell processes and are worth discussion in future studies.

Since $H^{\pm \pm }$ and $H^{\pm }$ decays depend on the triplet VEV, to examine different decay modes, we choose two values; $v_{\Delta }$ = 50 eV and $v_{\Delta }$ = 0.5 GeV. We assume degenerated mass scenario $M_{H^{\pm \pm }}=M_{H^{\pm }}=M_H$. For $v_{\Delta }$ = 50 eV, the doubly charged scalar $H^{\pm \pm }$ decays dominantly to a pair of leptons, and $H^{\pm }$ decays to a charged lepton and an (anti)neutrino. Branching ratios for particular flavours depend on the neutrino mass hierarchy and mixing angles. In the second case, $v_{\Delta }=0.5$ GeV, $H^{\pm \pm }$ decays to a gauge boson pair $W^{\pm }{W}^{\pm }$. Regarding the singly charged scalar $H^{\pm }$, the situation is a bit more complicated as shown in Figure 2. For clarity, we present the dominant decay channels for $H^{\pm \pm }$ and $H^{\pm }$ in Table 3. Other possible final states are negligible.

Figure 2. Branching ratios for $H^{\pm }$ (charged, a) and H (neutral, b) Higgs boson decays in HTM.

Table 3. Dominant decays of $H^{\pm }$ and $H^{\pm \pm }$ for different values of triplet VEV, $v_{\Delta }$ = 50 eV and $v_{\Delta }$ = 0.5 GeV.

$v_{\Delta }$ = 50 eV	$H^{\pm }$	→	$l_i^{-}{\nu }_j/l_i^{+}{\tilde{\nu }}_j$	$H^{\pm \pm }$	→	$l_i^{\pm }{l}_j^{\pm }$
$v_{\Delta }$ = 0.5 GeV	$H^{\pm }$	→	$W^{\pm }Z$	$H^{\pm \pm }$	→	$W^{\pm }{W}^{\pm }$
			$W^{\pm }h$
			$q_i{\overline{q}}_j$

The final state for the above processes depends on the SM and heavy scalar particles’ decays. For the purpose of this work, we chose the least complicated case—the $pp\to H^{\pm \pm }{H}^{\mp }$ process. As one can see in Table 3, decay modes depend on the triplet VEV. Moreover, in the same table we presented the simplified relationship between the scalar-leptons couplings and triplet and doublet VEVs. In the Table 4 we present the cross section for the intermediate states and corresponding background. In the Appendix A.2, we demonstrate more exact formulas. Those vertices depend on the PMNS matrix—on neutrino mixing parameters and mass hierarchy. Because decay to a $l_i^{\pm }{l}_j^{\pm }$ pair depends strongly on the neutrino parameters, we decided to follow the convention adopted in [37] and find the parameter space where decay to same flavour lepton is most probable. That means:

Table 4. Singly and doubly charged scalar production $pp\to H^{\pm \pm }{H}^{\mp }$ with primary decays to the SM particles for CM energy 14 TeV (100 TeV).

${\mathit{v}}_{\mathbf{\Delta }}$	Process:			Signal [pb]:	Background [pb]:
50 eV	$pp\to H^{\pm \pm }{H}^{\mp }$	→	$e^{\pm }{e}^{\pm }{e}^{\mp }{Ɇ}_T$	1.79 × ${10}^{-5}$	5.05 × ${10}^{-2}$
				(2.00 × ${10}^{-3}$)	(3.15 × ${10}^{-1}$)
			${\mu }^{\pm }{\mu }^{\pm }{\mu }^{\mp }{Ɇ}_T$	1.43 × ${10}^{-5}$	5.05 × ${10}^{-2}$
				(1.55 × ${10}^{-3}$)	(3.15 × ${10}^{-1}$)
0.5 GeV	$pp\to H^{\pm \pm }{H}^{\mp }$	→	$W^{\pm }{W}^{\pm }{W}^{\mp }Z$	3.62 × ${10}^{-5}$	7.00 × ${10}^{-4}$
				(3.93 × ${10}^{-3}$)	(1.60 × ${10}^{-2})$
			$W^{\pm }{W}^{\pm }{W}^{\mp }h$	3.53 × ${10}^{-5}$	6.19 × ${10}^{-5}$
				(3.84 × ${10}^{-3}$)	(1.12 × ${10}^{-3}$)
			$W^{\pm }{W}^{\pm }jj$	3.68 × ${10}^{-10}$	3.09 × ${10}^{-1}$
				(4.01 × ${10}^{-8}$)	(5.65)

• to maximize the $e^{\pm }{e}^{\pm }{e}^{\mp }$ signal: $m_{{\nu }_3}=0$ (inverted neutrino mass hierarchy), ${\alpha }_1=\frac{\pi }{2}$, ${\alpha }_2=\frac{\pi }{2}$,
• to maximize the ${\mu }^{\pm }{\mu }^{\pm }{\mu }^{\mp }$ signal: $m_{{\nu }_1}=0.015$ eV (normal neutrino mass hierarchy), ${\alpha }_1=0$, ${\alpha }_2=0$,

where ${\alpha }_1$, ${\alpha }_2$ are Majorana phases in the PMNS matrix (see Equation (A9)) and other neutrino parameters are taken from nu-fit.org (accessed on 2 July 2021) [62].

In Table 4, we present the above process with primary scalar particles and Standard Model particles. As one can see, for $v_{\Delta }=50$ eV, the possible final state is $l^{\pm }{l}^{\pm }{l}^{\mp }{Ɇ}_T$. In Table 5, we present the results after using the following cuts:

Table 5. Results from Table 4 for $v_{\Delta }=50$ eV, after cuts. Centre mass energy: $\sqrt{s}=14$ TeV.

Process:	Signal:		Background:
	Before Cuts [pb]:	After Cuts [fb]:	Before Cuts [pb]:	After Cuts [fb]:
$e^{\pm }{e}^{\pm }{e}^{\mp }{Ɇ}_T$	1.79 × ${10}^{-5}$	5.46 × ${10}^{-3}$	5.05 × ${10}^{-2}$	0.94
${\mu }^{\pm }{\mu }^{\pm }{\mu }^{\mp }{Ɇ}_T$	1.43 × ${10}^{-5}$	9.68 × ${10}^{-3}$	5.05 × ${10}^{-2}$	1.77

• Lepton identification criteria: transverse momentum $p_T\ge 20$ GeV, pseudorapidity $\left|\eta \right|<2.5$;
• Hard $p_T$ cuts: $p_T\left(l_1\right)>30$ GeV, $p_T\left(l_2\right)>30$ GeV, $p_T\left(l_3\right)>20$ GeV;
• Detector efficiency for electron (muon): 70% (90%);
• Lepton–lepton separation: $\Delta R_{ll}\ge 0.2$;
• Z-veto—the invariant mass of any same flavour and opposite charge lepton should satisfy the condition: $|m_{l_1{l}_2}-M_{{\mathrm{Z}}_1}|\ge 6{\Gamma }_{{\mathrm{Z}}_1}$.

The signal after the above cuts for $v_{\Delta }=50$ eV is still negligible in comparison to the background. However, we are planning to repeat the above studies for $0.5$ GeV, since that case looks much more promising.

4. Summary and Outlook

Observation of the Higgs boson opens up possibilities for other charged and neutral scalars to be detected at present and future colliders. In this paper, we studied the associated production of a doubly charged scalar within the Higgs triplet model at the $pp$ colliders and with a centre of mass energy of 14 (HL–LHC) and 100 TeV (FCC–hh). We analysed all the associated processes of $H^{\pm \pm }$ with gauge and scalar bosons and found that five of them are worth further study. We found that the largest cross section is for the $pp\to H^{\pm \pm }{H}^{\pm }$ process. Our preliminary studies show that the background signals for the considered processes are substantial, and a more detailed analysis of kinematic conditions and appropriate distributions or choice of final states, which can enhance new physics signals over the SM background, are needed and will be undertaken in the near future.

Further, we analyzed the dominant decay modes of singly charged and heavy neutral scalars within the model. The final state’s signature at the hadron colliders will depend strongly on HTM VEV; we discuss possible decay scenarios for $v_{\Delta }=50$ eV and $v_{\Delta }=0.5$ GeV. For $v_{\Delta }=50$ eV, the dominant decay modes are leptonic, while for $v_{\Delta }=0.5$ GeV the dominant modes involve SM gauge bosons. Scenarios with off-shell, non-standard scalar decays will also be compelling to explore.

We have examined the $v_{\Delta }=50$ eV. Even though applied cuts slightly improved the significance, the result is still very low compared to the background. We are planning to repeat the detailed studies with appropriate cuts for $v_{\Delta }=0.5$ GeV, which seems to be more attractive in comparison to the background.

As an outlook similar to the work in [37], for a case of doubly charged Higgs boson pair production at hadron colliders, we should consider how HTM signals—where a single $H^{\pm \pm }$ Higgs boson is involved—can be discriminated from other models, including $H^{\pm \pm }$ scalars, notably the Left–Right Symmetric Models [21,52,63,64,65,66].

Appendix A.1. The Scalar Potential of the Model

The scalar potential can be expressed as [49]:

$$\begin{array}{ccc}\hfill V& =& -m_{\Phi }^2\left({\Phi }^{†}\Phi \right)+\frac{\lambda }{4}{\left({\Phi }^{†}\Phi \right)}^2+M_{\Delta }^{2}Tr\left({\Delta }^{†}\Delta \right)+\left[\mu \left({\Phi }^{T}i{\sigma }_2{\Delta }^{†}\Phi \right)+\mathrm{h}.\mathrm{c}.\right]\hfill \\ & & +{\lambda }_1\left({\Phi }^{†}\Phi \right)Tr\left({\Delta }^{†}\Delta \right)+{\lambda }_2{\left[Tr\left({\Delta }^{†}\Delta \right)\right]}^2+{\lambda }_{3}Tr\left[{\left({\Delta }^{†}\Delta \right)}^2\right]+{\lambda }_4{\Phi }^{†}\Delta {\Delta }^{†}\Phi ,\hfill \end{array}$$

(A1)

where $\Phi$ indicates SM doublet, $\mu$, $\lambda$ and ${\lambda }_i$$(i=1,\dots ,4)$ are real dimensionless couplings. The $m_{\Phi }^2$ and $M_{\Delta }^2$ can be expressed as functions of model parameters together with triplet and doublet VEV’s ($v_{\Delta }$, $v_{\Phi }$):

$$m_{\Phi }^2=\frac{\lambda }{4}{v}_{\Phi }^2+\frac{({\lambda }_1+{\lambda }_4)}{2}{v}_{\Delta }^2-\sqrt{2}\mu v_{\Delta },$$

(A2)

$$M_{\Delta }^2=-({\lambda }_2+{\lambda }_3)v_{\Delta }^2-\frac{({\lambda }_1+{\lambda }_4)}{2}{v}_{\Phi }^2+\frac{\mu }{\sqrt{2}}\frac{v_{\Phi }^2}{v_{\Delta }}.$$

(A3)

Physical states are obtained by a rotation of the triplet and doublet fields. Here, we are presenting that dependence in a simplified way:

$$\begin{array}{ccccccc}h\hfill & =& \hfill cos\alpha & h_{\varphi }\hfill & +& \hfill sin\alpha & h_{\Delta },\hfill \\ H\hfill & =& \hfill -sin\alpha & h_{\varphi }\hfill & +& \hfill cos\alpha & h_{\Delta },\hfill \\ H^{\pm }\hfill & =& \hfill -sin\beta & w_{\Phi }^{\pm }\hfill & +& \hfill cos\beta & w_{\Delta }^{\pm },\hfill \\ H^{\pm \pm }\hfill & =& \multicolumn{5}{c}{{\delta }^{\pm \pm },}\end{array}$$

(A4)

where $\alpha$ and $\beta$ are functions of the scalar potential parameters and multiplets VEV’s. The exact formulas can be found in [49] and in the Appendix of [37]. In this paper, only the approximation for $v_{\Delta }\ll v_{\Phi }$: $sin\alpha \sim 2\frac{v_{\Delta }}{v_{\Phi }}$ [53], $tan\beta =\frac{\sqrt{2}{v}_{\Delta }}{v_{\Phi }}$ is presented. The physical states masses are as follows:

$$\begin{array}{ccc}{M}_h^2\hfill & =\hfill & \lambda v_{\Phi }^2{cos}^2\alpha +\left(\frac{\mu v_{\Phi }^2}{\sqrt{2}{v}_{\Delta }}+2v_{\Delta }^2({\lambda }_2+{\lambda }_3)\right){sin}^2\alpha +2\left(v_{\Phi }{v}_{\Delta }({\lambda }_1+{\lambda }_4)-\sqrt{2}\mu v_{\Phi }\right)cos\alpha sin\alpha ,\hfill \\ M_H^2\hfill & =\hfill & \lambda v_{\Phi }^2{sin}^2\alpha +\left(\frac{\mu v_{\Phi }^2}{\sqrt{2}{v}_{\Delta }}+2v_{\Delta }^2({\lambda }_2+{\lambda }_3)\right){cos}^2\alpha -2\left(v_{\Phi }{v}_{\Delta }({\lambda }_1+{\lambda }_4)-\sqrt{2}\mu v_{\Phi }\right)cos\alpha sin\alpha ,\hfill \end{array}$$

$$\begin{array}{ccc}{M}_{H^{\pm }}^2\hfill & =\hfill & \frac{(2\sqrt{2}\mu -{\lambda }_4{v}_{\Delta })}{4v_{\Delta }}(v_{\Phi }^2+2v_{\Delta }^2),\hfill \\ M_{H^{\pm \pm }}^2\hfill & =\hfill & \frac{\mu v_{\Phi }^2}{\sqrt{2}{v}_{\Delta }}-\frac{{\lambda }_4}{2}{v}_{\Phi }^2-{\lambda }_3{v}_{\Delta }^2.\hfill \end{array}$$

(A5)

Appendix A.2. Scalar-Leptons Couplings

Here we present the exact formulas for the $H^{\pm \pm }-l_i-l_j$ and $H^{\pm }-l_i-{\nu }_j$ vertices, derived from the Yukawa Lagrangiang for the scalar triplet $\Delta$ (see Equation (2)):

$${\mathcal{V}}^{\pm \pm }=\left\{\begin{array}{ccc}{l}_i^{+}-l_j^{+}-H^{--}\hfill & =\hfill & i\left({\mathcal{Y}}_{ij}+{\mathcal{Y}}_{ji}\right),\hfill \\ l_i^{-}-l_j^{-}-H^{++}\hfill & =\hfill & i\left({\mathcal{Y}}_{ij}^{*}+{\mathcal{Y}}_{ji}^{*}\right),\hfill \end{array}\right.$$

(A6)

$${\mathcal{V}}_{\Delta }^{\pm }=\left\{\begin{array}{c}{\tilde{\nu }}_i-l_j^{+}-H^{-}=\frac{i}{\sqrt{2}}cos\beta \left({\mathcal{Y}}_{ij}+{\mathcal{Y}}_{ji}\right),\\ {\nu }_i-l_j^{-}-H^{+}=\frac{i}{\sqrt{2}}cos\beta \left({\mathcal{Y}}_{ij}^{*}+{\mathcal{Y}}_{ji}^{*}\right),\end{array}\right.$$

(A7)

where ${\mathcal{Y}}_{ij}$ is the Yukawa coupling, and from the neutrino mass diagonalisation we obtain [37] the following formula. $m_{{\nu }_i}$ means neutrino mass and $V_{PMNS}$ is the Pontecorvo– Maki–Nakagawa–Sakata matrix, parametrised as in the Equation (A9) ($s_{ij}$, $c_{ij}$ denote $sin\left({\theta }_{ij}\right)$ / $cos\left({\theta }_{ij}\right)$).

$${\mathcal{Y}}_{ij}=\frac{1}{\sqrt{2}{v}_{\Delta }}{V}_{PMNS}^{*}{D}_{\nu }{V}_{PMNS}^{†},D_{\nu }=\mathrm{diag}\{m_{{\nu }_1},m_{{\nu }_2}.m_{{\nu }_3}\},$$

(A8)

$$V_{\mathtt{PMNS}}=\left[\begin{array}{ccc}{c}_{12}{c}_{13}{e}^{i{\alpha }_1}& s_{12}{c}_{13}{e}^{i{\alpha }_2}& s_{13}{e}^{-i{\delta }_{CP}}\\ (-s_{12}{c}_{23}-c_{12}{s}_{23}{s}_{13}{e}^{i{\delta }_{CP}})e^{i{\alpha }_1}& (c_{12}{c}_{23}-s_{12}{s}_{23}{s}_{13}{e}^{i{\delta }_{CP}})e^{i{\alpha }_2}& s_{23}{c}_{13}\\ (s_{12}{s}_{23}-c_{12}{c}_{23}{s}_{13}{e}^{i{\delta }_{CP}})e^{i{\alpha }_1}& (-c_{12}{s}_{23}-s_{12}{c}_{23}{s}_{13}{e}^{i{\delta }_{CP}})e^{i{\alpha }_2}& c_{23}{c}_{13}\end{array}\right].$$

(A9)

Appendix A.3. Feynman Diagrams

Below, we list the most relevant diagrams which contribute to the processes considered in the work, in which one doubly charged Higgs boson is present (so those marked with blue in Table 1). We keep the same numeration as in the mentioned table, so we had chosen the processes $\left(II\right)$, $\left(IV\right)$, $\left(VII\right)$, $\left(VIII\right)$ and $\left(IX\right)$.

Figure A1. Feynman diagrams for singly and doubly charged scalars production in $pp$ collision, associated with other spin 0 particles (gauge and scalar bosons).