Probing the Majorana Nature in Radiative Neutrino mass models with the same-sign dilepton final states at future colliders

TL;DR

This paper explores how future colliders can detect the Majorana nature of neutrinos in models where neutrino masses are generated radiatively at three loops, using same-sign dilepton signals.

Contribution

It demonstrates the potential of future lepton colliders to probe the Majorana nature of neutrinos in three-loop radiative mass models through specific collider signatures.

Findings

Same-sign dilepton production can indicate Majorana neutrinos.

Future colliders like FCC-ee and ILC can test these models.

Collider signals can distinguish between Majorana and Dirac neutrinos.

Abstract

Neutrinos in the Standard Model (SM) are considered to be massless which is in contradiction with the evidence from the neutrino oscillation data. These experiments established that at least two SM neutrinos have non-zero masses and that individual lepton numbers are violated. This is strong evidence of new physics beyond the SM that should be responsible for generating non zero mass for the neutrinos. In this work, we study the collider phenomenology of an extension of the SM where neutrinos are generated radiatively at three-loop. We show that the production of same-sign dilepton at lepton colliders (such as FCC-ee and ILC) can be used to probe the Majorana nature of neutrinos in this class of models.

Figures

Tables

Table 1. Table 1: The cut sets for the process e − ​ e − → e − ​ e − + E m ​ i ​ s ​ s → superscript 𝑒 superscript 𝑒 superscript 𝑒 superscript 𝑒 subscript 𝐸 𝑚 𝑖 𝑠 𝑠 e^{-}e^{-}\rightarrow e^{-}e^{-}+E_{miss} and e − ​ e − → e − ​ μ − + E m ​ i ​ s ​ s → superscript 𝑒 superscript 𝑒 superscript 𝑒 superscript 𝜇 subscript 𝐸 𝑚 𝑖 𝑠 𝑠 e^{-}e^{-}\rightarrow e^{-}\mu^{-}+E_{miss} at different CM energies. E e subscript 𝐸 𝑒 E_{e} ( E μ subscript 𝐸 𝜇 E_{\mu} ) represents the energy of the electron(muon), M m ​ i ​ s ​ s subscript 𝑀 𝑚 𝑖 𝑠 𝑠 M_{miss} indicates the invariant mass of the possible missing energy contributions. θ e subscript 𝜃 𝑒 \theta_{e} ( θ μ subscript 𝜃 𝜇 \theta_{\mu} ) refers to the emission angle of the electron (muon) in the final states.

Channel	Selection Cuts
$e^{-}{e}^{-}\to e^{-}{e}^{-}+E_{miss}$	, $M_{miss}>580$,
$e^{-}{e}^{-}\to e^{-}{\mu }^{-}+E_{miss}$	, $E_{\mu }>300$, $M_{miss}>350$, ,

Equations

$$\mathcal{L}={\mathcal{L}}_{SM}+\{f_{\alpha \beta }{L}_{\alpha }^{T}Ci{\tau }_2{L}_{\beta }{S}_1^{+}+g_{i\alpha }{N}_i^C{\ell }_{\alpha R}{S}_2^{+}+\frac{1}{2}{M}_{N_i}{N}_i^C{N}_i+h.c\}-V(\Phi ,S_1,S_2)$$

$$V(\Phi ,S_{1,2})$$

$$+\frac{{\eta }_1}{2}{\left(S_1^{\ast }{S}_1\right)}^2+\frac{{\eta }_2}{2}{\left(S_2^{\ast }{S}_2\right)}^2+{\eta }_{12}{S}_1^{\ast }{S}_1{S}_2^{\ast }{S}_2+\left\{{\lambda }_s{S}_1{S}_1{S}_2^{\ast }{S}_2^{\ast }+h.c\right\}$$

$$\mathcal{B}(\mu\to e\gamma)=\frac{\alpha v^{4}}{384\pi}\Bigg{[}\frac{|f_{\mu\tau}f_{\tau e}^{*}|^{2}}{M_{S_{1}}^{4}}+\frac{36|g_{1e}g_{1\mu}^{*}|^{2}}{M_{S_{2}}^{4}}\,F^{2}_{2}(M_{N_{1}}^{2}/M_{S_{2}}^{2})+\frac{36|g_{2e}g_{2\mu}^{*}|^{2}}{M_{S_{2}}^{4}}\,F^{2}_{2}(M_{N_{2}}^{2}/M_{S_{2}}^{2})\\ +\frac{36|g_{3e}g_{3\mu}^{*}|^{2}}{M_{S_{2}}^{4}}\,F^{2}_{2}(M_{N_{3}}^{2}/M_{S_{2}}^{2})\Bigg{]}$$

$$\Delta a_{\mu}=\frac{m_{\mu}^{2}}{96\pi^{2}}\Bigg{[}\frac{|f_{\mu\tau}|^{2}+|f_{\mu e}|^{2}}{M_{S_{1}}^{2}}+\frac{6|g_{1\mu}|^{2}}{M_{S_{2}}^{2}}F_{2}(M_{N_{1}}^{2}/M_{S_{2}}^{2})+\frac{6|g_{2\mu}|^{2}}{M_{S_{2}}^{2}}F_{2}(M_{N_{2}}^{2}/M_{S_{2}}^{2})\\ +\frac{6|g_{3\mu}|^{2}}{M_{S_{2}}^{2}}F_{2}(M_{N_{3}}^{2}/M_{S_{2}}^{2})\Bigg{]}$$

$${\Omega }_{N_1}{h}^2$$

$$f_{e\mu }$$

$$g_{i\alpha }$$

$$m_{N_i}$$

Full text

1 Introduction

The Standard Model (SM) of the electroweak and strong interactions is extremely successful in explaining most of the measurements in particle physics at high energies that are accessible today. Nonetheless, despite its enormous success in describing the elementary particle interactions, there appears to be a need for physics beyond the SM at about the weak scale, with additional features that could explain the presence of Dark Matter (DM) and its nature in the universe  [1] in addition to several other questions which remain unanswered like the neutrino masses and mixing  [2], the origin of the Baryon Asymmetry of the Universe (BAU). So far, none of these issues are successfully explained within the SM framework. In the SM, the neutrinos are considered to be massless which is in contradiction with the evidence from neutrino oscillation experiments. In addition to this, the absolute neutrino mass scale remains unknown, although it is possible to measure the mass difference between neutrino flavors through neutrino oscillations. Thus, these limitations necessitate the introduction of additional components to the SM and going beyond its actual framework.

To tackle this open question, many generic models, which are available in the market, are trying to investigate this issue, for instance: Seesaw mechanism (Type I,II or III) [3], Radiative Models: one loop (A. Zee [4], Ma [5]), two loops (Zee-Babu), or three loops (Krauss, Nasri and Trodden (KNT)). The focus in this work will be on the latter model (KNT) in [6] which proposed the extension of the SM with two electrically charged $SU(2)_{L}$ singlet scalars and one right handed (RH) neutrino field $N_{1}$, where a $\mathbb{Z}_{2}$ symmetry was imposed at the Lagrangian level in order to forbid the Dirac neutrino mass terms. After the breaking of the electroweak symmetry, neutrino masses are generated at three-loop, which makes their masses naturally small due to the high loop suppression. Moreover, the field $N_{1}$ is odd under $\mathbb{Z}_{2}$ symmetry, and thus it is guaranteed to be stable, which makes it a good candidate for DM. It was shown also that in order to fit the neutrino oscillation data and be consistent with different recent experimental constraints such as lepton flavor violation (LFV), one needs to have three RH neutrinos.

The aim of this paper is to elaborate an overview as well as systematic assessment of one of the possible signatures for heavy neutrino searches that can be explored mainly at future lepton colliders. Within the proposed class of neutrino mass models, the neutrino mass is generated radiatively at three-loop, the lightest right-handed neutrino ($N_{1}$) is a good dark matter candidate, and the scalar sector can give a strongly first order electroweak phase transition as required for baryogenesis. In our previous works, we considered similar studies at the CERN Large Hadron Collider (LHC) [7] and the International Linear Collider (ILC) [8]. In this work we focus on new detailed physics studies that will be anticipated on the final states that can be produced due to the new interaction terms and extra matter to the SM Lagrangian introduced by the RH degrees of freedom, which can hint at the existence of the RH Majorana neutrinos.

This paper is organised as follows, In Sec. I, we introduce the model and its phenomenology. Then in Sec II, we study the physics analysis and provide the simulation details of the same-sign dilepton signatures at lepton colliders. Finally, our conclusions and outlook are presented in the last Sec IV.

2 The model and its phenomenology

In our work we consider the model  [6] which is an extension of the SM. It has basically two charged scalar singlets $S_{1,2}$ under $SU(2)_{L}$ gauge group and three right handed neutrinos. A $\mathbb{Z}_{2}$ symmetry was imposed at the Lagrangian level to forbid Dirac neutrino mass terms. The model Lagrangian is written as:

[TABLE]

where $f_{\alpha\beta}$ denotes the Yukawa couplings, $\alpha$ and $\beta$ represents the lepton family indices, and $L_{\alpha}$ is the LH lepton doublet. The Majorana RH neutrino masses are $M_{N_{i}}$. The tree-level scalar potential $V(\Phi,S_{1,2})$ is given by:

[TABLE]

Here $\Phi$ represents the SM Higgs field doublet. One of the features of this model is that by imposing $\mathbb{Z}_{2}$ symmetry, Dirac neutrino mass term at all levels of perturbation theory, is forbidden. After the electroweak symmetry breaking, the neutrino masses are produced at three-loop, where $N_{1}$ is considered to be the lightest RH neutrino in the model and hence the most stable and designated candidate for DM. The three-loop diagram that generates the small non-zero neutrino mass matrix elements is shown in Fig. 1. The model should obey the following constraints:

The generated small neutrino masses at three-loop fit the neutrino oscillation data. 2. 2.

Furthermore, no conflict with the bounds on Lepton Flavor Violating (LFV) processes, the muon anomalous magnetic moment and $0\nu\beta\beta$ decay. 3. 3.

$N_{1}$ is the considered DM candidate with a relic density that is in agreement with the observation for masses around the EW scale. 4. 4.

Yields a strong first order electroweak phase transition that ensures any baryogenesis scenario and its testability with the Higgs mass measurement currently provided by the CERN-LHC experiments.

The most stringent bound on lepton flavor violation (LFV) comes from from the processes $\l_{\alpha}\rightarrow\l_{\beta}\gamma$, with branching fraction given by

[TABLE]

where $v$ is the vaccum expectation value of $H$, $\alpha$ denotes the the electromagnetic fine structure constant and $F_{2}$ is the loop function given by $F_{2}(x)=(1/6(x-1)^{4})(1-6x+3x^{2}+2x^{3}-6x^{2}-6x^{2}\log x)$. In this case, the considered value shall be $\mathcal{B}(\mu\to e\gamma)<5.7$ x $10^{-13}$ in accordance with the MEG collaboration upper bound  [9].

On the other hand if $\l_{\alpha}=\l_{\beta}=\mu$, new contribution is added to the muon anomalous magnetic moment $\Delta a_{\mu}$ described as:

[TABLE]

with the upper limit $\Delta a_{\mu}<2.89$ x $10^{-9}$ [10].

The DM emerges as the lightest RH neutrino in the model ($N_{1}$). The fermion masses and couplings values in the radiative neutrino mass model should fall within the relic density range in [11] which represents the abundance of the DM candidate. This reads:

[TABLE]

3 Analysis and discussion

To validate the model parameters, we check the feasibility of the benchmark that should satisfy all the experimental constraints mentioned above (relic density and lepton flavor violation) and that will be considered at a later stage. In this case, the benchmark selection is explored to determine the range of the model parameters.

Considering the benchmark with the model parameter values:

[TABLE]

To define the feasible benchmarks, we probe the parameter space of neutrino masses and mixing (after satisfying the LFV, muon anomalous magnetic moment and relic density constraints). Figure 1 illustrates the range of RH neutrino ($N_{1}$) versus the charged scalars ($S_{1}$ and $S_{2}$) masses that can yield possible DM relic density in the considered parameter phase space (left panel). Clearly the charged scalar masses tend to choose $M_{S_{1}}$ to be larger than $M_{S_{2}}$. Furthermore, we investigate the Yukawa couplings $g_{i\alpha}$ and $f_{\alpha\beta}$ and their magnitude (right panel). The results are shown on the right side of Figure 1. One can notice that the $f_{\alpha\beta}$ couplings are generally smaller than the couplings $g_{i\alpha}$. Consequently, it becomes mandatory to require the large coupling in the $g_{i\alpha}$ set that satisfy LFV and muon anomalous magnetic moment.

We study the same-sign dilepton production via heavy Majorana neutrinos at lepton colliders. The natural roadmap would be to initially provide the cross section sensitivity of the same-sign dilepton processes at high energy. In our analysis we take into account the simple setting for neutrino mass spectrum, i.e. $M_{N_{1}}<M_{N_{2}}<M_{N_{3}}$. In this analysis, the considered physics processes are $e^{-}e^{-}\rightarrow e^{-}e^{-}+E_{miss}$ and $e^{-}e^{-}\rightarrow e^{-}\mu^{-}+E_{miss}$. It is worth mentioning that we do not consider the same-sign dimoun final state since it is background free. The missing energy contribution at the final state is $E_{miss}\subset\{\nu_{e}\bar{\nu}_{e},\nu_{\mu}\bar{\nu}_{\mu},\nu_{\mu}\bar{\nu}_{\tau},\nu_{\tau}\bar{\nu}_{\mu},\nu_{\tau}\bar{\nu}_{\tau},N_{i}N_{k}\}$ where $i,k=1,2,3$. Figure 2 shows the cross section for each subprocess in $e^{-}e^{-}\rightarrow e^{-}e^{-}+E_{miss}$ (left) and $e^{-}e^{-}\rightarrow e^{-}e^{-}+E_{miss}$ (right) versus the $E_{CM}$ energies. Both possible signature and expected background contributions that can mimic the signal are plotted together.

We extensively study the signal and background kinematic distributions in order to define the cut sets where an excess in the number of events in the signal over the background can be observed distinctly. Table 1 summarises the useful kinematic variables and their corresponding cut values in both considered signatures, where an excess in the number of events of the signal over background is prominent. We extend our analysis to explore the dependence of the same-sign dilepton production cross section in terms of the mass if the DM candidate ($M_{N_{1}}$), as well as the mass of the charged scalar ($M_{S_{2}}$). The results are shown in Figures 4 and 5 for $M_{N_{1}}$ and $M_{S_{2}}$ respectively in both channels at $E_{CM}=1$ TeV. One can see that all the missing energy contributions can be produced at higher energy whereas at lower energy few of them can be probed due to small cross section values.

4 Conclusions

Although this work is still ongoing, the preliminary results show promising sensitivity of the same-sign dilepton production via Majorana neutrinos in Radiative Neutrino mass models at lepton colliders [12]. The study will be extended to investigate the detectability of a larger class of radiative neutrino mass models at future lepton colliders using polarized beams where the signal is enhanced and can be observed with smaller integrated luminosity as compared to the case of unpolarized beams.