Neutrino masses and gauged U(1)ℓ lepton number

We investigate the tree-level neutrino mass generation in the gauged U(1)ℓ lepton model recently proposed by us [1]. With the addition of one Standard Model(SM) singlet, ϕ1(Y = 0, ℓ = 1), and one SM triplet scalar, T (Y = −1, ℓ = 0), realistic lepton masses can be accommodated. The resulting magnitude of neutrino mass is given by ∼ vt3/vL2, where vt and vL are the vacuum expectation values of T and ϕ1, respectively, and it is automatically of the inverse see-saw type. Since vL is the lepton number violation scale we take it to be high, i.e. O ≳ (TeV). Moreover, the induced lepton flavor violating processes and the phenomenology of the peculiar triplet are studied. An interesting bound, 0.1 ≲ vt ≲ 24.1 GeV, is obtained when taking into account the neutrino mass generation, Br(μ → eγ), and the limits from oblique parameters, ΔS and ΔT. Collider phenomenology of the SM triplets is also discussed.


Introduction
It is now generally accepted that neutrino oscillation data indicate that at least two of the three active neutrinos have nonvanishing masses. This cannot be accommodated in the minimal Standard Model (SM)without adding new degrees of freedom such as two or more SM right-handed neutrinos. However, neutrino masses can be generated by the addition of the Weinberg operator [2], O 5 . This nonrenormalizable dimension five operator takes the form of O 5 = y Λ ℓ L ℓ L HH, where H is the SM Higgs field, ℓ L denotes a SM lefthanded lepton doublet, y is a free dimensionless parameter, and Λ is an unknown high scale. After H takes on a vacuum expectation value v ≃ 247 GeV, the electroweak symmetry is spontaneously broken, and we get a neutrino mass m ν ∼ yv 2 Λ . Since data indicate that m ν 1 eV, depending on the value of y, the scale Λ can range from 1 to 10 11 TeV. This elegant way of generating neutrino masses using only SM fields comes with the price of JHEP10(2018)015 nonrenomalizability. Furthermore, it reinforces the idea that the SM is an effective theory and the neutrino masses call for its extension.
Neutrino mass generated from the Weinberg operator is of the Majorana type, and it has lepton number ℓ = 2 provided the conventional lepton number assignments that all SM charged leptons e, µ, τ and their associated neutrinos ν e , ν µ , ν τ have ℓ = 1 and all other SM fields carry ℓ = 0 are assumed. Also the anti-leptons have ℓ = −1. This is a natural consequence if lepton number is a U(1) ℓ symmetry. Thus, the SM is largely invariant under this symmetry with a very small breaking by the Weinberg operator. However, the nature of this symmetry is unknown. Usually, the total lepton number is taken to be a global symmetry that is broken at a very high scale Λ 10 12 GeV by two or more SM singlet righthanded neutrinos N R with Majorana masses of O(Λ). Integrating them out gives rise to the Weinberg operator, and this is the celebrated type I seesaw mechanism [3][4][5][6][7]. Doing so raises the question of the origin of the Majorana mass bestowed to N R . One can add a Majorana mass for N R by hand. However, our current understanding is that masses of fermions are generated by the Higgs mechanism. It is interesting to also to apply this to U(1) ℓ . Doing so will lead to the existence of a Goldstone boson in the physical spectrum which can act as a candidate for dark radiation [8,9].
Moreover, it is phenomenologically and theoretically interesting to investigate the possibility of a gauged U(1) ℓ and study the spontaneously broken gauge theory. There are several possibilities. One can gauge the total lepton number as in [10]. 1 One can also gauge a combination of lepton generation number such as L µ − L τ [16,17]. In ref. [1], hereafter referred to as (I), we gauged each lepton family with the usual lepton number assignments for them. Of the just mentioned three examples only the second one is anomaly-free with only the SM fields. Gauging the total lepton will require extra leptons with very exotic lepton charges such as ℓ = 3 to cancel the anomalies from U(1) ℓ . In (I), the extra anomalies cancelations require two extra pairs of vector-like SU(2) doublet leptons with eigenvalues ℓ = 1, 0 for each family. We also did not include any singlet N R field, and the Weinberg operator is generated radiatively at 1-loop. The principal source of lepton number violation comes from a SM singlet scalar with ℓ = 2 which picks up a vacuum expectation value.
In this paper, we study a different mechanism of neutrino mass generation in the gauged lepton number scheme introduced in (I). The extra leptons presented before is sufficient to generate neutrino masses with the aid of a SM triplet scalar T and a SM singlet scalar φ 1 . T has ℓ = 0 whereas φ 1 is given ℓ = 1, with both fields being Higgssed. This naturally leads to an inverse seesaw mechanism (ISM) [18][19][20] for active neutrino mass. The novel feature here is that we do not add by hand any SM singlet leptons to implement ISM as is commonly done. The required leptons are dictated by anomaly cancelations. Details will be given in section 3. Since the physics involved with the gauge new gauge boson Z ℓ and the extra leptons are the same as in (I), we will not repeat their phenomenology here. Instead, we focus on neutrino physics and the phenomenology of T . We find that T has interesting different signatures at high energy colliders from previous studies of l = 2 Higgs triplets [21][22][23][24][25], which are commonly employed in the type-II see-saw model [26][27][28][29][30][31]. For a recent review see [32]. We organize the paper as follows. The next section we present our anomalies solution for completeness. Then we discuss lepton mass generation for one generation to illustrate the physics. This is followed by a realistic 3-generation study. Section 4 gives fits to the neutrino oscillation data. Constraints from charged lepton flavor changing neutral currents are given in section 5. Important electroweak precision constraints are studied in section 6. The productions of different new triplet scalars at the LHC and CLIC are examined in section 7. Our conclusions are given in section 8.

U(1) ℓ anomalies cancelations and new fields
We extend the SM gauged group by adding a U(1) ℓ and is explicitly given as G = SU(2) × U(1) Y × U(1) ℓ . All SM leptons have the conventional value of ℓ = 1. We will concentrate on one family. This can be trivially extended for all 3 SM families.
The new anomaly coefficients are where A 5 stands for the lepton-graviton anomaly. While new chiral leptons are introduced to cancel eq. (2.1), one also needs to make sure that the SM anomalies of  The minimal set of scalar fields, by utilizing the triplet scalar for neutrino mass generation, can be obtained by examining the gauge invariant set of lepton bilinears that can be formed from the above fields. They are given in table 2. where all H, φ 1 , and T develop non-zero VEVs.

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The Yukawa interactions are where all the generation indices are suppressed. The full gauge invariant and renormalizable scalar potential reads, where we have used the bi-doublet form for T as below 2 The following conditions must hold (λ 4t = λ 4 +λ t ) so as to ensure that the potential is bounded from below. After SSB, And the minimal condition for the scalar potential become Note that λ 5,6 do not come into play here. From the above equations, the tree-level mass squared for T − and ℜT 0 are From phenomenology we expect that v t ≪ v (see section 8) and before scalar mixings considerations we have which are above a TeV if κ takes a phenomenologically interesting value around the electroweak scale. However, in general, κ is a free parameter. The scalar potential after SSB gives a small mass splitting between T − and T −− . The mass squared difference can be worked out to be which is ∼ O(v 2 ) provided λ 6 is not much smaller than λ 4,5 . Therefore, it is a good approximation to assume that T − and T −− are degenerate. However, we should keep in mind the mass splitting could be about the Fermi scale. Similarly, ignoring the contribution from v t , we have Since we expect that v L ≫ v, i.e. lepton symmetry breaking to be above the Fermi scale, we obtain (2. 16) JHEP10(2018)015 Table 3. Couplings of gauge bosons to triplet fields.
Thus, it is also required to have |λ 2 | ≪ λ L . As expected, there will be mixing among the three neutral scalars H = (ℜH 0 , ℜT 0 , ℜΦ). They are related to the physical states h = (h SM , t 0 , φ 0 ) via the usual unitary rotation given by Details of this transformation are not important for this study and we will not present them.
For completeness, we discuss the imaginary parts of the scalar fields. ℑΦ is the wouldbe Goldstone for the gauge boson Z ℓ . Moreover, the would-be Goldstone bosons eaten by W ± , Z, the physical singly charged scalars, h ± , and the pseudoscalar, A 0 , can be identified as: Since v t ≪ v from the electroweak precision studies (see section 8), it is a good approximation to treat T ± and ℑT 0 as the physical states. Being the only degree of freedom with two units of electric charge, T ±± are the physical scalars.
Since the symmetry G forbids T from coupling to two SM fermions simultaneously, its gauge interactions become the most relevant for phenomenology. From the G-covariant derivative we obtain the Feynman rules for its triple couplings to gauge bosons, displayed in table 3, where P stands for the photon, and all the momenta are incoming.

Lepton masses for 1 generation
The physics of how the new leptons affect the SM charged leptons is best seen in the one family scenario. In the basis {e, E 1 , E 2 }, the Dirac mass matrix is In general the electron will mix with E 1,2 and the mixing depends on f 1 and f 2 . In that case, the charged-current interaction of the SM leptons could deviate JHEP10(2018)015 from the canonical SM (V − A) form due to their mixings with L 2R and E 2L . Moreover, the SM gauge couplings are flavor non-diagonal. Physically, this mixing must be very small and we can take the limiting case of f 1 = f 2 = 0 and eliminate the mixing of the electron with the new charged leptons. 3 In general, we can write the physical mass eigenstates where V L/R is the left-handed/right-handed unitary matrix that diagonalizes the charged lepton mass matrix so that For the limiting case of f 1 = f 2 = 0 and f = f 3 = f 4 (1 + δ) with |δ| ≪ 1, the mass eigenvalues can be worked out to be One can see that the leading mass splitting between E + and E − , apart from the phase convention, comes from the SM Higgs Yukawa interaction, h 2,3 , and to a very good approximation, In the basis {ν L , N 1L , N c 2R }, the neutrino mass matrix is and ǫ t = vt v L < ǫ v . Again, we consider the case that f 1 ≪ 1 and y 1 ∼ y 2 = y. The eigenvalues can be worked out to be around (yǫ t /f ) 3 , −1 + (yǫ t /2f ), and 1 + ( It is natural to identify the first term as the mass of the active neutrino. For yv t ∼ 0.1GeV and f v L ∼ 3TeV, the resulting active neutrino mass is about (yv t ) 3 /(f v L ) 2 ∼ 0.1 eV. From electroweak precision measurements we expect v t O(1) GeV. We see that the desired neutrino mass can be obtained without much tuning of the Yukawa couplings.
Notice that the neutrino mass matrix given in eq. (3.5) is of the inverse seesaw type [18,19], and a review can be found in [20]. The novel feature here is that we do not require ad hoc addition of the SM singlet leptons. The additional leptons are dictated by anomaly cancelation and are SM doublets.

JHEP10(2018)015 4 3-generation lepton masses
One can extend the above to the realistic 3-generation case. Without losing any generality, we can start with the basis that the Yukawa couplings for N 2R N 1L are diagonal. And we can go to the basis where the SM charged leptons are in their mass eigenstates by bi-unitary transformation among the e R and e L . Similarly, we have the freedom to start

Charged lepton mass matrix
For simplicity, let's consider that f 1,2 = 0, f 3,4 ∼ f , and the heavy charged lepton are roughly degenerate. Then, in the basis (e, E 1 , E 2 ) where each entry is a 3-vector in family space, the most general (9 × 9) mass matrix for charged leptons looks like where h 1 and δ 1,2 are 3 × 3 diagonal matrices and 1 is the unit matrix. For convenience, δ 1,2 which encodes the small splitting of the heavy charged leptons are separated out from the leading term. One can first perform a rotation among the heavy charged leptons by U = V B , which is a (9 × 9) generalization of eq. (3.4). Then the small perturbation can be separated from the leading order mass eigenvalues, Or by using so that

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It is clear that the 6 heavy charged leptons will form 3 nearly degenerate pairs. And as in the 1-generation case, the mass splitting for each pair is mainly controlled by h 2,3 . Moreover, they decouple from the SM charged leptons.

Neutral lepton mass matrix
Using the notation of the charged leptons and factor out the common mass, we write the general (9 × 9) neutrino mass matrix as to accommodate the small non-degeneracy among the three heavy N s. First, the leading mass diagonalization can be made by the same rotation V B , similar to eq. (3.4), as in the charged lepton case. This results in a symmetric 3 × 3 matrix, δ 3 ≡ −δ + ǫ 2 /2, in the diagonal blocks as the perturbation. Assume there exists , the re-scaled neutrino mass matrix can be brought into the following form One can see that after this rotation, the leading mass eigenstates are nothing but the Cartesian basis. By the standard perturbation techniques, it is easy to see that the SM neutrinos will acquire nonzero masses at the second order perturbation. For example, at this order, and it is indeed of the order of O(ǫ 2 1 ǫ 2 ) as in the 1-generation case. The active neutrino masses can also be understood diagrammatically. By integrating out the heavy N , the corresponding Feynman diagram in the weak basis, displayed in figure 1, and can be seen to give the same conclusion. It also reveals that the low energy effective operator for active neutrino mass is not given by the Weinberg operator. If we assume a hierarchy that v L ≫ v ≫ v t , and T is the only beyond SM degree of freedom left below v L , the active neutrino masses are generated by a dimension six operator where c is a constant and Λ L is the lepton number breaking scale JHEP10(2018)015 Figure 1. Diagrammatic representation of the ǫ 3 -suppression for the active neutrino masses. Superscripts denote family indices. Upper(green) arrows denote flow of lepton charge.
related to v L . After T picks up a VEV, v t , the neutrino mass is given by It is also clear that O 6 has a higher dimension than the Weinberg operator. Together with the fact that v t ≪ v., they allow the lepton breaking scale to be much lower than the usual type I seesaw mechanism. Now, the upper-left (3 × 3) sub-matrix, denoted as N ν , of U for active neutrinos is in general non-unitary, N ν N † ν = 1. This non-unitarity will result in various observable effects. However, one expects that the off-diagonal elements of |N ν N † ν | are of the order of which is roughly below the current experimental limits, 10 −5 [36,37]. Therefore, we will leave the comprehensive study of these precision tests to future work.

Neutrino oscillations and data fitting
First, we provide a simple, realistic solution which can accommodate the neutrino data. Then we move on to the more general numerical survey where the solutions will be fed into the later study of lepton flavor changing processes.
To simplify the discussion, we assume that the heavy N 's are degenerate(δ = 0), y 2 ∝ 1, and all the Yukawa couplings in y 1 are of the same order and there is no hierarchy among them. The (9 × 9) mass matrix looks like is an unknown overall constant which controls the amplitude of perturbation and the elements of y 1 are of ∼ O(1). As discussed previously, in the leading approximation, the (3 × 3) active neutrino mass matrix reads If y 1 is highly democratic, namely, 3)

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the resulting active neutrino mass matrix also has the pattern M ν ∝ I c which is of rank one and it has two zero eigenvalues. It naturally leads to the normal hierarchical neutrino masses. Taking into account the data, the realistic mass matrix for normal hierarchy(NH) instead takes the form if m 1 ≃ 0, and, to simplify the discussion we set δ CP = 0. A simple solution to arrive such pattern is which has the apparent µ − τ symmetry. This can be realized in the extra-dimensional models by arranging the amount of overlap in higher dimensional fermion wavefunctions, see for example [33][34][35].
On the other hand, a more subtle construction of y 1 is required to accommodate the inverted hierarchy( IH ) case. For example, if m 3 ≃ 0, δ CP = 0, the following realistic neutrino masses matrix can be generated by For both NH and IH cases, Taking v t = 1GeV and v L = 1(5)TeV, we have y ∼ 0.03(0.09). This simple solution with y 2 ∼ y · 1 gives us a rough idea of the Yukawa coupling strengths.
For the realistic data fitting, we perform a comprehensive numerical scan with the working assumption that |(y 2 ) ij | ≃ y 2 and that the heavy N 's are nearly degenerate. These assumption can be relaxed giving rise to more free parameters to fit the data. Moreover, the Yukawa couplings are taken to be complex in the numerical study to accommodate the nonzero CP phase, δ CP which current data give a hint of. However, it is clear that the resulting neutrino mass is about m ν ∼ (y 2 1 y 2 v 3 t /M 2 N ). We adopt the following 3σ ranges from [38] for the neutrino oscillation parameters. For NH, As for the IH case, the corresponding 3σ ranges are: The lightest neutrino masses, m lightest , for both NH and IH are allowed to vary in the range between 10 −4 eV and 0.2eV so that the cosmological bound, j m j < 0.57eV at 95% C.L.(from CMB spectrum and gravitational lensing data only) [39], 4 can be met. Once m lightest is fixed, m 1,2,3 can be determined from the measured mass squared differences. Then the effective active neutrino mass matrix can be obtained by (5.11) where the shorthand s 12 ≡ sin θ 12 and the like are used. Each element of the y 2 Yukawa matrix is a random number in between 0.7 and 1.3 times an overall unknown factor y 2 with either sign. And we require that the ratio of the largest to the smallest absolute values in y 1 to be smaller than 10. About 10 5 such solutions are prepared for both NH and IH cases. The realistic Yukawa coupling configurations can be used for predicting the lepton flavor violating processes. The results will be displayed in the next section. the V B rotation, the triplet coupling can be approximated as where i, j = 1, 2, 3 are the generation indices, and ± denote the different mass eigenstates within each generation. The 1-loop contributions can be calculated to be , and See figure 3 for the plot of this function. When x ≪ 1 and x ∼ 1, the loop function can be expanded as The first term in the square bracket of eq. (6.2) is the contribution where the photon attaches to the heavy charged lepton. The second and third terms are the contributions where the photon attaches to the T −− and T − , respectively. Because of the electric charge, the T −− contribution has an extra factor 2. Note also the one half factor associated with the T − contribution which is due to the extra 1/ √ 2 factor in the singly charged tripletfermion vertex coupling. Moreover, assuming that M E ∼ M N ∼ M ( so that all I 1 ∼ 1/12) JHEP10(2018)015 the (g − 2) µ can be related to the neutrino mass and eliminating the y 1 dependence, Similar calculation can be carried out for the µ → eγ dipole transition amplitude.
where q µ ≡ (p 2 −p 1 ) µ is the photon 4-momentum, andR = (1+γ 5 )/2. Then, the branching ratio is [41] Br Or, assuming that M E ∼ M N ∼ M , the LFV process can be related to the neutrino mass, Note that the mass squared of heavy leptons in numerator and denominator cancel out, and the branching ratios is not very sensitive to the masses of heavy degree of freedom.
Comparing to the most recent bound, Br(µ → eγ) < 4.2 × 10 −13 at 90% C.L. [42], our numerical results are shown in figure 4. As can be seen from the plot, it is easier to find the solutions for larger m lightest in the IH case. For y 2 v 3 t = 1(GeV ) 3 , the µ → eγ branching ratio is right below the current experimental limit for m lightest 10 −2 eV. Note that the branching ratios have lower bounds, around ∼ 10 −16 /10 −15 for NH/IH case with y 2 v 3 t = 1(GeV ) 3 . Therefore, for this model to admit a realistic solution which accommodates simultaneously the neutrino oscillation data and the current µ → eγ bound, the predicted lower bounds must stay below the experimental limit. It is required that  Since in our model the triplet does not carry lepton number, there is no tree-level contribution to µ → 3e and the similar τ decays. The dipole induced Br(µ → 3e) will be small comparing to µ → eγ. The ratio [41] is given by which makes Br(µ → 3e) < 3 × 10 −15 in this model. Similarly, the branching ratios of τ → lγ (l = e, µ) are and we adopt the measured Br(τ → eν e ν τ ) = 17.82% [43]. The predicted branching ratios of τ → lγ in our model are displayed in figure 5 which are much smaller than the current experimental bound; Br(τ → eγ) < 3.3 × 10 −8 and Br(τ → µγ) < 4.4 × 10 −8 at 90%C.L. [43]. Note that in the IH cases the two have same statistics which is due to the complex conjugated pair solutions to the y 1 Yukawa for a given m lightest and U P M N S . As pointed out in [44], the double ratios, for example, Br(µ → eγ)/Br(τ → eγ), are independent of the unknown parameters y 2 , v t and the masses of the heavy degrees of freedom. They are complementary handles to the long baseline experiments for determining the type of neutrino mass hierarchy. Unfortunately, we have not found any notable statistical difference between the double ratios of NH and IH in this model.

Triplets at colliders
The phenomenology of the Z ℓ and the charged heavy leptons are the same as in (I), and we shall not repeat them here. The triplets are the new players and we will discuss their signatures at the LHC below. We start with a list of their dominant decay modes.

Decays of the triplet
Due to the gauge couplings and SSB, the triplet scalar can decay into (a) two SM gauge bosons collectively called V (b) a lighter triplet partner plus a V, e.g. T −− → T − W + , and (c) two light triplets, e.g. T −− → 2T − . The later two require huge mass splitting or the rates are suppressed by v t , thus can be ignored here. 6 Therefore, T → V 1 V 2 (V 1,2 = W ± , Z) are the dominant decays since T does not couple to two SM fermions simultaneously in the weak basis. This is very different from the cases of triplet with l = 2 as discussed, for example, in [45]. Parameterizing the vertex T V µ 1 V ν 2 Feynman rule as iκ V 1 ,V 2 g µν , it's JHEP10(2018)015 Table 4. Feynman rules for T V V vertices. The g µν factors are omitted.
straightforward to calculate the following decay widthes: The couplings are listed in table 4. The typical decay widthes for charged triplets are narrow, around O(10 −2 ) MeV, for v t ∼ 1GeV and M T ∼ 1TeV. However, the charged triplet still decays promptly once produced. Moreover, the signal of triplet will be 4 fermion final state from the decay of two gauge bosons or 2 fermion plus a high energy photon.
On the other hand, if there is mixing between ℜT 0 and the Higgs boson, t 0 can decay into fermion pairs. The two body decay width of t 0 is given by where U h is given in eq. (2.17). This will be dominated by the tt final state if M T ≫ M t = 174 GeV. LHC-1 gave a bound on the SM signal strength that µ = 1.09 ± 0.11 [46], which implies that |U 12 h | 2 < 0.13 at 2σ level. For M T = 0.5(1.0)TeV, the 2-body decay width has an upper bound Γ(t 0 → tt) < 8(36) MeV, and Γ(t 0 → bb) < 0.57(1.1) MeV. The mixing with the SM Higgs will also provide additional 2 gauge bosons decay widthes, Finally, we discuss the t 0 → 2h SM decay. Since |U 12 h | ≪ 1, the relevant Lagrangian is roughly and the κ term dominates. We have

Triplet production at hadron colliders
As seen in the previous section, the production and decay of t 0 is very sensitive to its mixing with the SM Higgs. We will start with the case that the mixing between h, ℜT 0 is negligible and focus on the production of the charged triplet at the collider. The pair production at the LHC is mainly by the Drell-Yan processes through the T T V vertices. The gauge boson associated production cross section, σ(pp → V T ), is proportional to v 2 t and negligible. If ignoring the mixing and mass differences, σ(pp → T + T −− ) = σ(pp → T * 0 T − ) and σ(pp → T − T ++ ) = σ(pp → T 0 T + ) for they have the same couplings and mediated by the s-channel W -exchange diagrams. The cross sections at LHC14 for some typical triplet masses, listed in table 5, are evaluated by the program CalcHep [47] with the CTEQ6l1 [48] PDF.
Note that pp → ttW will be dominant SM background for T −− T ++ . After applying proper cuts, a doubly charged of mass up to about 0.7 TeV, and it decays mainly into diboson, can be probed at LHC14 with an integrated luminosity of 300 f b −1 [45]. However, we defer a full study of the signal and proper treatment of the background to a future study.
In contrast, the real part of the neutral triplet 7 can be singly produced via gluon fusion through the mixing (U 12 h ). Our estimates of the production cross sections at the LHC and future hadron colliders are given in table 6. The SM backgrounds are estimated by evaluating the production cross section with the di-boson invariant mass in the M T ±50GeV range. Derived from the numbers listed in table 6, the 5σ limits on the 2-dimensional |U 12 h | 2 and effective luminosity plane is shown in figure 6. The limit is determined by where ξ V V is the efficiency of detection of V V final states, and L 0 is the integrated luminosity. It can be seen that t 0 with a mass of 1TeV and |U 12 h | 2 = 0.05 could be directly studied at the LHC14 with ∼ 1ab −1 effective luminosity.  Table 6. Gluon fusion neutral Triplet boson production cross sections(in f b) at the LHC and beyond. Here we assume that |U 12 h | 2 = 0.1.  calculated to be:

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The cross sections are displayed in figure 7. Note that the interference between photon and Z contributions is destructive/constructive for T + T − /T −− T ++ production cross section. Because the electric charge squared, T ±± has the largest production cross section. We use CalcHEP to estimate the SM backgrounds and find that they are about three orders of magnitude smaller than the signals, and thus negligible.

Tree-level ρ-parameter
Since T gets a VEV, v t , the tree-level ρ−parameter is less than unit: Therefore, the loop-induced ∆T loop (> 0) can be compensated by ∆T tree (< 0). For ∆T = 0.08 ± 0.12 [43], the 2σ range is The above only uses tree level contributions from the SM triplet implies that v t < 5.94 GeV. Combining with neutrino mass generation and the µ → eγ limit, we obtain the following interesting limit 0.107 < v t < 5.94 GeV .

Loop corrections
Since anomaly cancelation mandates the addition of extra leptons, it is important to know how quantum corrections to ∆T and ∆S from these new states will alter the above bound on v T . For each generation, the contributions from exotic leptons are [1] ∆T be ignored. The di-photon decay width is thus where τ i ≡ (m H /2m i ) 2 , and all the loop functions can be found in [51]. For the exotic leptons, the Yukawa couplings are parameterized as L ⊃ −y E iĒ i E i h SM in the mass basis. Assuming that T − and T −− are degenerate, the width reads for M E i = 1000(100)GeV, and M T − = 1.0(0.5)TeV. The first two numbers are the dominate SM contributions from W ± and top quark, respectively. The dominant SM Higgs production channel at the LHC is through gluon fusion which is intact in this model. Therefore, the signal strength of pp → h → γγ is It is expected that |y E i | ∼ m l /v h ≪ 1 [1], and the charged leptons contribution can be ignored. Comparing to the data µ γγ = 1.18(+0.17 − 0.14) [52], it is safe even |λ 1,6 | ∼ O(1). This agrees with the general analysis given in [53].

Conclusions
We have studied a novel neutrino mass generation mechanism in the recently proposed gauged lepton number model by us [1]. The model is free of anomalies by the addition of two sets of exotic chiral leptons for each generation. The U(1) l gauge symmetry is spontaneously broken when a l = 1 SM singlet, φ 1 , gets a VEV, v L . In addition, one l = 0 SM triplet, T , is introduced for neutrino mass generation. The triplet in this model differs from the well-studied l = 2 triplet in the type-II see-saw model. Since it carries no lepton number, the triplet does not couple to the SM leptons. An immediate consequence is that there is no doubly charged triplet contribution to the neutrino-less double decays of nuclei which in our model is given mainly by the exchange of light neutrinos. The VEV of the charge-neutral parts of T , v t , and the SM Higgs H, v ≃ 246GeV, breaks the SM electroweak gauge symmetry and the custodial symmetry. With only two exotic scalars, φ 1 and T , and no RH SM singlet neutrino, the resulting neutrino mass is of the inverse see-saw type. Since the phenomenology of the obligatory new gauge boson Z ℓ and the JHEP10(2018)015 exotic leptons have been studied in [1], we have focused on the physics of neutrino mass and the new l = 0 triplet in this work. We begin the discussion of the one-generation case since the physics is clear in this simple setting. Since the exotic leptons required for anomaly cancelations will in general mix with the SM leptons we require that the Yukawa couplings f 1,2 to be very small. This discussion is later extended to the realistic three-generation case, and we have carefully investigated the physics of active neutrino masses in this model. The active neutrino masses are of the order of v 3 t /v 2 L given by the dimension-six operator O 6 . Since the electroweak precision requires a relatively small v t , no further parameter fine-tuning is required other than taking f 1 ≃ 0 mentioned before. Both realistic NH and IH neutrino masses can be accommodated in this model. If assuming a democratic structure of the Yukawa couplings, it is more natural to get an NH pattern. For IH, it requires a more subtle Yukawa pattern and prefers to have the lightest neutrino mass 10 −2 eV, which is promising for the neutrinoless double beta decays searches.
It is worth noting that O 6 produces elements of the active neutrino mass matrix that is Majorana-like, i.e of the form ν ic L ν j L where i, j are family indices. This is the same as O 5 would. Thus, low energy neutrino measurements such as neutrinoless double beta decays of nuclei, tritium β decays spectrum endpoint, and cosmological neutrino mass bounds cannot distinguish between O 6 or the Weinberg operator as the origin of neutrino masses. In order to do that one needs to explore the TeV scale to discover whether there are new degrees of freedom. O 5 assumes that there are none whereas O 6 requires new leptons below 10 TeV. 8 In addition, a detailed program searching for CLFV decays of muon and τ will also be useful since O 5 and O 6 have very different UV completions and thus will yield different results for these processes.
We have calculated the 1-loop triplet contributions to a µ and the LFV processes l 1 → l 2 γ(l 1,2 = e, µ, τ ). ∆a µ is negative but negligible in this model. Thus, it cannot resolve the discrepancy between the data and the SM expectation [54]. On the other hand, we have found an interesting connection between the neutrino masses and the LFV branching ratios. Taking into account the current limit on Br(µ → eγ) < 4.2 × 10 −13 , we have obtained an interesting lower bound on v t 0.1 GeV. Since T does not couple to SM leptons, the LFV process µ → 3e and the τ counterparts are mediated by the photon dipole transition and thus predicted to be very small, Br(µ → 3e) 10 −15 .
The triplet gets a VEV so that the constraint from ∆T can be relaxed. We have carefully analyzed the limits from both ∆S and ∆T and arrived an upper bound for v t 24.1 GeV if assuming the mass squared differences among the isospin components of the triplet and the heavy leptons to be at most electroweak, v 2 . Combing with the neutrino masses and LFV bounds, we have 0.1 v t 24.1GeV in this model. The lower bound of v t also implies that M T 8TeV provided that κ ≃ v.
We have studied the decays of the triplet. For T ± and T ±± , the dominant decay channel is into di-boson. Depending on the scalar potential, the T 0 component of the triplet in 8 We have seen previously that the mass splitting |M 2 E − M 2 N | v 2 . Leptons with the mass around 10 TeV will give a splitting of < 1 GeV. This is much smaller than what we have encountered and will require very delicate tuning of parameters.

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general mixes with the SM Higgs doublet, although the mixing squared is limited to be smaller than 0.13 at 2σ level [46]. However, even allowing for this mixing the dominant decay channel of T 0 is still the SM di-boson modes. Due to their SM gauge interactions, the charged triplets can be pair produced via Drell-Yan processes at the LHC. In addition to the SM gauge couplings, due to its mixing with the SM Higgs, the neutral triplet can be singly produced via the gluon fusion. At LHC14, it is possible to probe t 0 of mass up to 1TeV and |U 12 h | 2 ∼ 0.1 with an integrated luminosity of 300 f b −1 . At the linear colliders, the signal of triplet pair production will be very clean once the center-of-mass energy is higher than the mass threshold. For the mass range of triplet we are interested in, we have found that the bound from the current h SM → 2γ measurement is weak.