New Class of Two-Loop Neutrino Mass Models with Distinguishable Phenomenology

We discuss a new class of neutrino mass models generated in two loops, and explore specifically three new physics scenarios: (A) doubly charged scalar, (B) dark matter, and (C) leptoquark and diquark, which are verifiable at the 14 TeV LHC Run-II. We point out how the different Higgs insertions will distinguish our two-loop topology with others if the new particles in the loop are in the simplest representations of the SM gauge group.

Introduction. The minimal particle content of the standard model (SM) of quarks and leptons does not allow a nonzero neutrino mass at the level of a renormalizable Lagrangian. However, it has long been known [1] that an effective dimension-five operator exists for obtaining a nonzero Majorana neutrino mass, i.e.
where (ν i , l i ), i = 1, 2, 3 are the three left-handed lepton doublets of the SM and (φ + , φ 0 ) is the one Higgs scalar doublet. As φ 0 acquires a nonzero vacuum expectation value φ 0 = v/ √ 2 = 174 GeV, the neutrino mass matrix is given by Tree-level [2][3][4][5][6] and one-loop realizations [7][8][9][10][11] of this operator have been discussed extensively in the literature, as well as some two-loop [12][13][14][15][16][17][18][19][20][21] and threeloop [22][23][24] examples. The two-loop case is particularly inviting because the smallness of the neutrino mass, i.e. of order 0.1 eV, agrees well with a mass scale of O(1) TeV for the heavy particles in the loops, without unduly small and large Yukawa and scalar couplings, as would be the case with one-loop and three-loop realizations. Generic structures of the two-loop diagrams involving fermions and scalars are shown in Fig. 1, where the   due to the different Higgs insertions. Such collider signatures may then be used to distinguish our two-loop topology ( Fig. 2(d)) with others ( Figs. 2(a, b, c)). The electric charge assignments of fermion ψ x , scalar doublets (η x+1 , η x ), (χ 2x+1 , χ 2x ), and singlet ρ 2x are as shown in Fig. 2(d). There are at least three natural realizations of this diagram: Table I. It is obvious that the doubly charged scalar is predicted in (A) and (B), while a dark matter candidate is embedded in (B). The leptoquark and diquark scalars are related to the neutrino mass generation in (C). It is easy to generate the neutrino mass around 0.1 eV when the new particles are O(1) TeV, without unduly small or large Yukawa and scalar couplings in these models. The effective Lagrangian related to our study is where i, j = 1, 2, 3 is the family index. Details of the neutrino mass generation are shown in the Appendix. Next we perform a collider simulation to explore the potential of the Large Hadron Collider (LHC) on discovering the three models. We focus on a 14 TeV LHC with an integrated luminosity (L) of 100 fb −1 and also the high-luminosity (HL) phase of 3000 fb −1 in this study, and for simplicity, we assume the neutrino mass matrix is diagonal for the collider phenomenology analysis.
Collider Phenomenology of Model (A). Discovery potential. The best way to probe our model is the pair production of the doubly charged scalar (χ ++ ) where ± represents the electron (e ± ) and muon (µ ± ). The event topology is characterized by four isolated charged leptons. The dominant backgrounds are γγ, Zγ, ZZ, four leptons (4 ), and four charged leptons plus one jet (4 1j). The null result in the search of doubly charged scalars via the pair production at the 8 TeV LHC imposes a bound, m χ ++ > 400 GeV, assuming χ ++ decays entirely into electron or muon pairs [25,26].
We generate both the signal and the background processes at the parton level using MadEvent [27] and impose basic cuts as follows: p ± ,j T > 5 GeV with η ± ,j < 5, where p T and η denote the transverse momentum and rapidity, respectively. We require the angular distance ∆R mn ≡ (η m − η n ) 2 + (φ m − φ n ) 2 between the objects m and n to be greater than 0.4 to obtain isolated objects. At the analysis level, all the signal and background events are required to pass a set of selection cuts [25,28]: where i with i = 1, 2, 3, 4 denotes the lepton ordered in accord with their p T 's. We demand only four leptons in the central region of the detector and veto extra jets if p j T > 10 GeV or |η j | > 3.5. In addition, we require the invariant mass of lepton pair to be away from m Z , with |m( ) − m Z | > 10 GeV, to suppress the dominant background containing Z-boson resonances. In order to suppress the 4 1j background, we require m( ± ∓ ) > 50 GeV. We end up with 26.05 background events in total at the LHC with L = 100 fb −1 , i.e. γγ (1.18), Zγ (2.67), ZZ (3.62), 4 (12.03) and 4 1j (6.56). The number inside the parenthesis denotes the number of events of each individual background.
We obtain a 5 standard deviations (σ) statistical significance using where n b and n s represent the numbers of the signal and background events, respectively. Figure 3 displays the discovery potential of χ ++ with BR(χ ++ → + + ) = 1 at the LHC. The χ ++ with m χ ++ < 566 GeV could be discovered with L = 100 fb −1 (red line). The HL-LHC extends the coverage to m χ ++ < 806 GeV (blue line).

Model Discrimination.
The doubly charged scalar appears in all the two-loop models depicted in Fig. 2 and yields exactly the same collider signature. Observing a doubly charged scalar alone cannot distinguish various models. However, different insertions of Higgs fields in Fig. 2 offer an opportunity to distinguish our two-loop topology with others if new particles inside the loops are in the simplest representations under the SM gauge group. Our model consists of two special ingredients: (i) two doublet scalars χ and η; (ii) the quartic coupling (χ η)(φ η).
The scalars in other two-loop models exhibit different weak quantum numbers. For example, the singly charged scalar (κ + ) and doubly charged scalar (χ ++ ) in Fig. 2(a) are both neutral under SU (2) L ; a good example is the socalled Babu-Zee model [14,15]. Fig. 2(b) consists of two singlet scalars and one doublet scalar with hypercharge 1/2 [29]. Fig. 2(c) is used in Ref. [18] to discuss neutrino mass generation which involves a triplet and two singlet scalars.
The simplest way to distinguish our model shown in Fig. 2

(d) from the models in Figs. 2(a) and (b) is the
This process is absent in the models shown in Fig. 2(a) and Fig. 2(b). The χ −− can be produced on-shell or offshell (labeled as ), depending on whether χ + is heavier than χ ++ or not.
Note that the χ ± scalar can also be probed in the χ + χ − pair production that yields a signature of multiple charged leptons and missing transverse momentum. However, in the parameter space of interest to us, m χ > 250 GeV, the cross section of pp → χ + χ − with subsequent decays χ ± → χ ±± W ∓ → ± ± ∓ + E T is around 10 −4 ∼ 10 −2 fb at the 8 TeV LHC while fixing m χ ++ = 400 GeV. Such a small cross section is consistent with current experimental bounds [30].
The typical cross section of the process pp → χ ±± χ ∓ → χ ±± χ ∓∓( ) W ± at the 14 TeV LHC is around 0.1 ∼ 1 fb. Hence the hadronic modes of the W boson, which exhibit large branching ratios, are considered in our simulation. For simplification, we assume χ ±± decays into same sign lepton pairs entirely, i.e. BR(χ ++ → + + ) = 1. To mimic the signal events, the SM backgrounds should consist of W/Z/γ in final state. We consider the following backgrounds: 4 2j, ttZ, Zγjj, γγjj and ttγ. The other backgrounds like W ZZ and h(→ ZZ )jj are negligible after imposing kinematic We plot the 5σ discovery potential of pp → χ ++ χ − at the LHC in Fig. 4(a), where the dashed vertical lines represent the discovery potential obtained from the doubly charged scalar pair production. Two benchmark integrated luminosities, L = 100 fb −1 (red) and HL-LHC (blue), are considered. It shows that χ ±± χ ∓ pair production could be discovered when m χ ++ < 550 GeV and m χ + ∼ 400 − 600 GeV with L = 100 fb −1 ; see the red contour. The parameter space extends to m χ ++ ,χ + 800 GeV at the HL-LHC. Hence, we are able to discriminate between our model shown in Fig. 2(d) and those models in Figs. 2(a, b).
Note that the triplet scalar in the model of Fig. 2(c) can also generate the pp → χ ±± χ ∓ collider signature. To distinguish our model from it, we make use of the quartic coupling (χ η)(φ η); for example, it gives rise to unique decay modes of χ ++ and χ − as follows: which cannot occur in Figs. 2(a, b, c). The detailed analysis of the unique modes of χ ±± and χ ± mentioned above is beyond the scope of the current paper and will be presented elsewhere.

Comment on the T parameter.
It is well known that the triplet scalar violates the condition for the ρ parameter to be one if it develops a vacuum expectation value. That restricts the scalar potential of the model shown in Fig. 2(c). On the other hand, the scalar doublet η = (η 0 , η − ) in our model, where η 0 = (η 0 R + iη 0 I )/ √ 2, also contributes to the T parameter (which is proportional to ρ−1) and relaxes the constraint on the mass splitting between χ ++ and χ + .
The T parameter is not only sensitive to the mass splitting between η − and η 0 R,I , but also to the difference of m η 0 R and m η 0 I [29].
Relaxing the T parameter constraint imposes a strong correlation between (m χ ++ − m χ + ) and (m η 0 R − m η 0 I ). Figure 5(a) shows the allowed parameter space by the T parameter constraint at the 95% confidence level for the benchmark parameters of sin α = 0, m η 0 I = 200 GeV, m χ ++ = 400 GeV and m η + = (m η 0 R + m η 0 I )/2; see the green region. Here, α is the mixing angle between the two doubly charged scalars in our model. One can safely ignore it in the numerical evaluation of the T parameter although the neutrino mass demands a tiny nonzero sin α.
For completeness, we also investigate the potential of the LHC on discovering the neutral scalars η 0 R and η 0 I through the process of pp → Z → η 0 R (→ + − )η 0 I (→ + − ). This channel gives rise to exactly the same collider signature as the χ ++ χ −− pair production, and we show the 5σ discovery potential of η 0 R η 0 I pairs in Fig. 5(b). The meshed regions in Fig. 5(a) denote the 5σ discovery regions of the processes pp → χ ±± χ ∓ and pp → η 0 R η 0 I at the LHC. They show that the LHC with L = 100 fb −1 could probe the mass splitting between χ ++ and χ + up to 170 GeV maximally as well as the mass splitting of η 0 R and η 0 I up to 150 GeV (red region). The HL-LHC would extend the reach in both mass splittings to 224 GeV (blue region).

Collider Phenomenology of Models (B) and (C).
In (B), the vector-like charged fermion E + is odd under dark Z 2 and is connected to the dark matter candidate, i.e. η 0 R or η 0 I whichever is lighter. The best way to probe the dark fermion E + is through pp → E + E − → + − η 0 R/I η 0 R/I . It yields a collider signature of two charged leptons plus missing energy, which is shared by many neutrino mass models [31][32][33]. The dark matter relic abundance requires m η 0 R/I ∼ 63 GeV [29]. The mass of E + is severely constrained by the slepton search pp →˜ +˜ − → + −χ0χ0 [34,35], where˜ ± andχ 0 are the charged slepton and the neutralino, respectively. Suppose BR(E + → + η 0 R/I ) = 1 and m η 0 R/I = 63 GeV, we obtain m E + > 400 GeV.
We follow the slepton search at the 8 TeV LHC [34,35] to perform a simulation to estimate the potential of observing E + E − pairs at the 14 TeV LHC with L = 100 fb −1 . The dominant backgrounds include the diboson productions (W W , W Z and W γ), triple-boson productions (W W Z and W W W ), and tW production. We demand two isolated charged leptons with p That yields a discovery potential of E ± with BR(E + → + η 0 R/I ) = 1 depicted in Fig. 6(a), which shows E + could be discovered when m E + < 700 GeV with 100 fb −1 (red). The HL-LHC would extend the coverage to m E + < 1105 GeV (blue).
In (C), the neutrino mass is generated through SU (3) C colored particles, i.e. leptoquark and diquark scalars which also give rise to a very rich phenomenology. The current constraint of the first and second generation leptoquark is m η > 1 TeV [36][37][38]. The process pp → η 2/3 η −2/3 with subsequent decays of η ±2/3 → ± j can efficiently probe such a leptoquark and has been widely studied in the literature [39,40]. The diquark is severely constrained by the dijet data at the 13 TeV LHC [41]. However, the lower limit of the diquark mass depends critically on the Yukawa coupling h ij q c Ri q Rj χ. To generate the tiny neutrino mass, we choose h ij ∼ O(0.01) (see the appendix), which yields the cross section of pp → χ ±2/3 around 1 pb for m χ ±2/3 = 400 GeV at the 13 TeV LHC. That is much smaller than the current experimental constraint [41]. A much stronger constraint is obtained from the search of R-parity violating decays of the top squark in four-jet final states (pp →tt → 4j) at the 13 TeV LHC [42]. We find m χ ±2/3 is larger than 700 ∼ 800 GeV with the assumption of BR(χ ±2/3 → jj) = 1. This bound does not rely on the Yukawa coupling.
Summary. In this work, we have constructed a new class of two-loop models of neutrino mass where the two external Higgs lines are attached differently from previous examples. There are three natural realizations containing (A) doubly charged scalar, (B) dark matter candidate, and (C) leptoquark and diquark scalars. We show that these new particles have promising discovery potentials at the √ s = 14 TeV LHC with an integrated luminosity L = 100 fb −1 and 3000 fb −1 . We demonstrate as well how the different Higgs insertions offer an opportunity of distinguishing these models from others under the assumption that the new particles in the loop are in the simplest representations of the SM gauge group. As an example, we propose that the pp → χ ±± χ ∓ production process in (A) could be used to distinguish our model topology (see Fig. 2(d)) from Fig. 2(a) and Fig. 2(b). It is also possible to distinguish it from Fig. 2(c) by using the quartic coupling (χ η)(φ η). In addition, two doublet scalars inside the loop of our model could be useful to relax the constraint from the T parameter. Models (B) and (C) also share the same topology with (A) and the same promise to be probed at the LHC, with possible signatures to distinguish their different Higgs insertions. m E in addition to that of Eq. 6. The mass matrix, similar to that in the Babu-Zee model, is given by [43,44] M ab = c,d .
Similar to I acbd , the loop function I acbd is also symmetric from exchanging t 1 and t 2 . We also note that there is a strong correlation between I acbd and I acbd due to the similar loop integration. However, I acbd is proportional to the integration momentum p 2 , while I acbd is depending on the fermion mass m c m d . That leads to the different power dependence on t 1 and t 2 . In the limit of m c,d → 0, I acbd (m c , m d , m η , m Hi ) is given by Note however that m c , m d ∼ O(m Hi , m η ) in (B) which does not apply in the case of Ref. [43]. To estimate this new contribution, we set t 1 = t 2 = t, and obtain Our numerical result shows that the two terms M ab and M ab are of the same order, and the neutrino mass matrix is sensitive to the mass splitting between H 2 and H 1 . For illustration we choose r 1 = 1, r 2 = 0.8, sin α = 0.1, f ac = f bd = h cd = λ = 0.05. That yields M ν ∼ 0.1 eV.