Phenomenology of \(U(1)_{F}\) extension of inert-doublet model with exotic scalars and leptons

The self consistency of the standard-model (SM) has been established (tentatively at least) with LHC discovery of the scalar behaving like the Higgs scalar of the SM. And about years ago since the SM has been formulated, it has went through numerous experimental test and no major disagreement with its prediction has been discovered. But still our universe turn out to be at least little more complicated than the SM can anticipate. One major discovery that point towards incompleteness of SM is the presence of dark-matter (DM). Another was the discovery of neutrino oscillation which means that neutrinos has small masses where as in SM neutrinos are massless. Then also it turn out that CP violation in SM due to Kobayashi-Maskawa (KM) [1] phase turn out to be too small to explain the observed excess of matter over the anti-matter. So the most important questions that theoretical and experimental efforts in years to come will be related to nature of DM, mechanism behind the neutrino oscillation phenomena and search for new sources of CP violations.

In this work we will extend the SM by adding one more \(U(1)_{F}\) guage group to it with introducing only new exotic scalars and leptons. This simple extension turn out to explain the DM relic density, loop generated neutrino masses, Baryon genesis and muon (g-2) anomaly. One of the peculiar features of this particular realization of \(U(1)_{F}\) extension of SM is that, due to solutions to the axial anomaly free conditions, if the electric charge of \(F_{\tau }\) is taken as a free parameter, then the electric charge of \(F_{e}\) will have opposite sign to that of the \(F_{\mu }\), so if \(F_{\mu }\) to explain the observed deviation in muon (g-2) then \(F_{e}\) is require to be stable.

This paper is organized as follows. In the next section we introduce various particles in our model and write down Lagrangian invariant under all the symmetries imposed on the particles. Then in section III and IV we dwell into the consequences of the model related to muon (g-2), neutrino mass, DM, Baryon-Genesis and collider signature of the exotic charged leptons. We conclude in section V.

A non-trivial solution of Eqs. (1) and (2) for \(Y_{i}\)s and \(n_{i}\)s that is interesting is given when we set either \(n_{e}\) or \(n_{\mu }\) or \(n_{\tau }\) equal to zero. In this work we set \(n_{\tau } = 0\) which make the electric charge of \(F_{\tau }\) a free parameter. Then the four equations are solved for \(Y_{e}\), \(Y_{\mu }\), \(n_{e}\) and \(n_{\mu }\) if we set \(Y_{e} = -Y_{\mu }\) and \(n_{e} = -n_{\mu }\), but for \(F_{\mu }\) to explain the observed anomaly in \(\delta a_{\mu }\) [3], we require \(Y_{\mu } = Q_{F_{\mu }} = -1\) which implies that \(Y_{e} = Q_{F_{e}} = +1\) and so charge conservation and Lorenz invariance forbid \(F_{e}\) to contribute to \(\delta a_{e}\), which is consistent with the experimental findings as no deviation in \(\delta a_{e}\) from the SM prediction has been reported. If we require Yukawa terms such as \(y_{\mu }\bar{\mu }_{R}F_{\mu L}\phi \), to explain the observed anomaly in muon (g-2), then term such \(y_{e}\bar{e}_{R}F_{e L}\phi \) are forbidden by charge conservation and term such as \(y_{e}\bar{e}_{R}F_{e L}^{c}\phi \) is forbiden by Lorenz invariance, so \(F_{e}\) will be a stable heavy charged particle. Although \(Y_{\tau }\) is a free parameter, if we also set \(Y_{\tau } = Q_{F_{\tau }} = -1\), then we can expect a deviation in \(\delta a_{\tau }\) in future measurements coming from \(y_{\tau }\bar{\tau }_{R}\phi F_{\tau }\) term.

2.2 Fermionic sector

As mentioned before, the symmetries of the model allows the neutral fermions \(N_{iR}\) to have very large Majorana masses given as \(\bar{N}_{iR}M_{i}N_{iR}\) where \(M_{i}\) is a \(3\times 3\) Majorana mass matrix. We do not entertain in this work how such a very heavy Majorana masses can be generated which could come from unified theories such as supergravity etc., see [4, 5, 6, 7]. The symmetries of the model also allows Yukawa terms such as \(h_{ij}\bar{L}_{i}i\sigma _{2}\eta N_{iR}\), which along with the Majorana mass terms for the \(N_{iR}\) can generate neutrino masses at one loop [2]. We can also have Yukawa terms such as \(y_{i}\bar{l}_{iR}\phi F_{iL}\) where \(l_{iR}\) ’s are SM right handed charged leptons. But non trivial solutions to the \(U(1)_{F}\) anomaly free conditions restrict at maximum only two of the three leptons is allowed to have such Yukawa interactions. Since no deviation has been reported in the electron (g-2) other than the SM predicted value, in the following sections we will assume that \(F_{\mu L}\) and \(F_{\tau L}\) have electric charge given by \(Q_{F_{\mu , \tau }} = Y_{F_{\mu }, F_{\tau }} = -1\) and then the axial gauge anomaly free conditions sets \(Q_{F_{e}} = Y_{F_{e}} = +1\), so then only muon and tau (g-2) has contributions from respective exotic leptons in our model. The general fermionic sector Lagrangian invariant under the full symmetries of the model can be written as

$$\begin{aligned} \mathcal {L}_{fermion}= & {} \sum ^{\tau }_{i = e}\bar{F_{i}}\gamma ^{\mu }(iD_{i\mu })F_{i} + \sum ^{\tau }_{i = e}\bar{F_{i}}\gamma ^{\mu }(i\partial _{\mu }P_{L})F_{i} \nonumber \\&+ \sum ^{\tau }_{i = e}\bar{N}_{Ri}M_{i}N_{Ri}^{c} \nonumber \\&+ \sum ^{\tau }_{i, j = e}h_{ij}\bar{L}_{i}i\sigma _{2}\eta ^{*} N_{jR} + \sum ^{\tau }_{i = \mu }y_{i}\bar{l}_{iR}\phi F_{iL} \nonumber \\&+ \sum ^{\mu }_{i = e}g_{F_{i}}\bar{F}_{iR}\phi _{i}F_{iL} + g_{F_{\tau }}\bar{F}_{\tau R}\phi _{1}F_{\tau L} + h.c \end{aligned}$$

(7)

where \(D_{i\mu } = \partial _{\mu }P_{R} - iY_{i}g^{'}B_{\mu } - in_{i}g_{F_{i}}Z_{F\mu }P_{R}\) with \(P_{R/L} = \frac{1}{2}(1 \pm \gamma _{5})\) and \(\phi ^{\dagger }_{e} = \phi _{\mu } = \phi _{2}\) as \(n_{e} = -n_{\mu }\) from the solutions of the anomaly free conditions. The \(B_{\mu }\) is the SM \(U(1)_{Y}\) gauge boson and \(Z_{F\mu }\) being the new \(U(1)_{F}\) gauge boson. We can express the \(B_{\mu }\) in terms of SM \(Z_{\mu }\) and \(A_{\mu }\) as

$$\begin{aligned} B_{\mu } = -\sin \theta _{W}Z_{\mu } + \cos \theta _{W}A_{\mu }, \end{aligned}$$

(8)

so then we can express the interaction of the new exotic leptons with the SM weak gauge boson \(Z_{\mu }\) and photon \(A_{\mu }\) as

$$\begin{aligned} \sum _{i = e}^{\tau }\bar{F}_{i}\gamma ^{\mu }Y_{i}g'B_{\mu }F_{i} = e\sum _{i = e}^{\tau }Y_{i}\bar{F}_{i}(-\tan \theta _{W}Z_{\mu } + A_{\mu })\gamma ^{\mu }F_{i} \end{aligned}$$

(9)

where \(g' = \frac{e}{\cos \theta _{W}}\). From the above Eq. (9), we can see that the production of these new leptons at LHC will be via the process \(pp \rightarrow Z^{*}/\gamma ^{*} \rightarrow F^{+}_{i}F^{-}_{i}\). Even though the production of the three exotic leptons are similar in nature, their signature mainly coming from their decays differs drastically. This is because, since \(y_{e}\bar{e}_{R}\phi F_{eL}\) term is forbidden by the axial gauge anomaly free conditions, in this model, the exotic lepton \(F_{e}\) will be a stable charged particle just like its SM counterpart, the electron. The other two exotic leptons, \(F_{\mu }\) and \(F_{\tau }\), are not stable as they can decay via \(F_{\mu /\tau } \rightarrow \mu /\tau + \phi \). Due to unbroken \(Z_{2}\) symmetry, \(\phi \) will be a stable particle if \(m_{F_{\mu }} > m_{\phi } + m_{\mu }\) is satisfied and so \(\phi \) can be a DM candidate. But we will see in Sect. 3, if \(y_{\mu }\bar{\mu }_{R}\phi F_{\mu L}\) to explain the muon (g-2) within 1 \(\sigma \) of the experimental value which require large \(y_{\mu }\), then contribution of \(\phi \) to the present relic density of DM will be negligibly tiny. In early universe, most of the \(F_{e}\) would have been depleted via \(F_{e}^{+}F_{e}^{-} \rightarrow Z_{F}/Z^{*}/\gamma ^{*} \rightarrow F_{\mu }^{+}F_{\mu }^{-}(F_{\tau }^{+}F_{\tau }^{-}) \rightarrow \mu ^{+}\mu ^{-}(\tau ^{+}\tau ^{-}) + missing\ energy\ (\phi \phi )\). The collider signature of \(F_{e}\) will be basically the collider signature of a very heavy stable charged particle, and given LHC reaching 13 TeV reported no such signatures of a stable heavy charged particle, we expect the mass of the \(F_{e}\) to be larger than few TeV. However for the other two exotic leptons, the main collider signatures will be

$$\begin{aligned}&e^{+}e^{-}/pp \rightarrow Z^{*}/\gamma ^{*} \rightarrow F_{\mu }^{+}F^{-}_{\mu }(F_{\tau }^{+}F^{-}_{\tau }) \rightarrow \mu ^{+}\mu ^{-}(\tau ^{+}\tau ^{-}) \nonumber \\&\quad +\, missing\ energy\ (\phi \phi ), \end{aligned}$$

(10)

and other key collider signatures will be

$$\begin{aligned}&e^{+}e^{-}/pp \rightarrow Z^{*}/\gamma ^{*} \rightarrow \gamma + \phi _{1}/\phi _{2} (via\ triangle\ loop) \nonumber \\&\quad \rightarrow \gamma + missing\ energy\ (\phi _{1}/\phi _{2} \rightarrow \phi \phi ) \end{aligned}$$

(11)

and

$$\begin{aligned}&e^{+}e^{-}/pp \rightarrow Z^{*}/\gamma ^{*} \rightarrow \gamma + \phi _{1}/\phi _{2} (via\ triangle\ loop)\nonumber \\&\quad \rightarrow \gamma + (\gamma + \gamma (or\ Z))(via\ triangle\ loop). \end{aligned}$$

(12)

In this sense we think \(e^{+}e^{-}\) colliders such as ILC (being a precision machine) will be more sensitive in detecting the missing energies in the final states of above reactions than Hadron colliders such as LHC.

Open image in new window

In this work we have proposed a simple extension of the SM by only adding three \(Z_{2}\) odd exotic charged leptons (\(F_{e},\ F_{\mu },\ F_{\tau }\)) whose right handed component are charged under a new \(U(1)_{F}\) gauge symmetry, three \(Z_{2}\) odd neutral fermions (\(N_{1},\ N_{2},\ N_{3}\)) singlet under the entire gauge symmetry, one \(SU(2)_{L}\) doublet (\(\eta \)) odd under a discrete \(Z_{2}\) symmetry whose VEV is zero (the inert-doublet model), one \(Z_{2}\) odd scalar (\(\phi \)) singlet under the entire gauge group and whose VEV is zero, one \(Z_{2}\) even scalar (\(\phi _{1}\)) also singlet under the entire gauge group which develops a non zero VEV \(v_{1}\), one more \(Z_{2}\) even scalar (\(\phi _{2}\)) charged under the \(U(1)_{F}\) whose non-zero VEV breaks the \(U(1)_{F}\) gauge symmetry spontaneously and give mass to the gauge boson \(Z_{F}\). With addition of the above new particles, we have been able to build a model which can give two DM candidate in terms of lightest neutral component of the inert-doublet (\(\eta _{R}\) in our case) and \(\phi \) but if \(\phi \) to explain the muon (g-2) anomaly than \(\phi \) will contribute only a tiny fraction to the present DM relic density, so most of the present relic density of DM will consist of \(\eta _{R}\) whose mass can be in the range 500 GeV to 130 TeV. In this model neutrino masses are generated at one loop level as well as baryon-genesis via lepto-genesis is also possible. We have also given key signatures of these new exotic leptons at LHC and future \(e^{+}e^{-}\) colliders. We find the the key signature will be in the form of \(e^{+}e^{-}/pp \rightarrow Z^{*}\gamma ^{*} \rightarrow F_{\mu }^{+}F^{-}_{\mu }(F_{\tau }^{+}F^{-}_{\tau }) \rightarrow \mu ^{+}\mu ^{-}(\tau ^{+}\tau ^{-}) + missing\ energy\ (\phi \phi )\) and \(e^{+}e^{-}/pp \rightarrow Z^{*}\gamma ^{*} \rightarrow \gamma + \phi _{2} (via\ triangle\ loop) \rightarrow \gamma + missing\ energy\ (\phi _{2} \rightarrow \phi \phi )\). One of the most peculiar signature of this model is the existence of a stable very heavy (about \(m_{F_{e}} \ge \) few TeV) charged lepton partner to the electron.