Interference bands in decays of doubly-charged Higgs bosons to dileptons in the minimal type-II seesaw model at the TeV scale

The dileptonic decays of doubly-charged Higgs bosons H^{\pm\pm} are investigated in the minimal type-II seesaw model with one Higgs triplet \Delta and one heavy Majorana neutrino N_1 at the TeV scale. We show that the branching ratios {\cal B}(H^{\pm\pm} \to l^\pm_\alpha l^\pm_\beta) depend not only on the mass and mixing parameters of three light neutrinos \nu_i (for i=1,2,3), but also on those of N_1. Assuming the mass of N_1 to lie in the range 200 GeV--1 TeV, we figure out the generous interference bands for the contributions of \nu_i and N_1 to {\cal B}(H^{\pm\pm} \to l^\pm_\alpha l^\pm_\beta): \sqrt{|\sin\theta_{i4} \sin\theta_{j4}|} \sim 10^{-8}--10^{-5}, where \theta_{i4} and \theta_{j4} measure the strength of charged-current interactions of N_1. We illustrate some salient features of the interference bands by considering three typical mass patterns of \nu_i, and stress that it is very difficult to distinguish the type-II seesaw model from the triplet seesaw model in such a parameter region at the Large Hadron Collider.


I. INTRODUCTION
The effort to build neutrino mass models at the TeV scale has recently revived [1], simply because this new energy frontier will soon be explored by the Large Hadron Collider (LHC). A naive but reasonable argument is that possible new physics, if it exists at the TeV scale and is responsible for the electroweak symmetry breaking, might also be responsible for the origin of neutrino masses. The latter is a kind of new physics which has been conceivably established by a number of neutrino oscillation experiments in the past decade [2].
Among many possibilities of generating tiny neutrino masses, a natural one is to extend the standard model by introducing a few heavy right-handed Majorana neutrinos [3] and (or) one Higgs triplet [4]. The gauge-invariant neutrino mass terms can then be written as where M R is the mass matrix of right-handed Majorana neutrinos, and denotes the Higgs triplet. After the spontaneous gauge symmetry breaking, one obtains the neutrino mass matrices M D = Y ν v/ √ 2 and M L = Y ∆ v ∆ , where H ≡ v/ √ 2 and ∆ ≡ v ∆ correspond to the vacuum expectation values of the neutral components of H and ∆. To minimize the degrees of freedom associated with M L , M D and M R , we may assume that there is only a single heavy Majorana neutrino (denoted as N 1 ) in the model. This assumption implies that M R and M D become 1 × 1 and 3 × 1, respectively, but M L remains to be 3 × 3. Such a simple seesaw scenario is phenomenologically viable and can be referred to as the minimal type-II seesaw model [5]. Its simplicity makes it interesting and instructive to reveal the salient features of the type-II seesaw mechanism. Therefore, we shall concentrate on this model in the present paper.
Our purpose is to investigate the dileptonic decays of doubly-charged Higgs bosons H ±± in the minimal type-II seesaw model. Such decays can naturally happen because ∆ is allowed to couple to the standard-model Higgs doublet H and thus the lepton number is violated by two units [4]. If the mass scale of ∆ is of O(1) TeV, then H ±± can be produced at the LHC via the Drell-Yan process qq → γ * , Z * → H ++ H −− or through the charged-current process qq ′ → W * → H ±± H ∓ . Note that the masses of H ±± and H ± are expected to be nearly degenerate in a class of seesaw models [4,6,7], so only H ±± → l ± α l ± β (for α, β = e, µ, τ ) and H ±± → W ± W ± modes are kinematically open. Note also that the dileptonic channels H ±± → l ± α l ± β become dominant when v ∆ < 1 MeV is taken [7]. Therefore, we focus our interest on the same-sign dilepton events of H ±± , which signify the lepton number violation and serve for a clean collider signature of new physics beyond the standard model [8]. The rates of H ±± → l ± α l ± β decays are given by from which one obtains the branching ratios [7] B where the Greek subscripts run over e, µ and τ . It becomes obvious that the magnitudes of B(H ±± → l ± α l ± β ) are only relevant to the matrix elements of M L . We find that the branching ratios B(H ±± → l ± α l ± β ) depend not only on the masses (m 1 , m 2 , m 3 ), flavor mixing angles (θ 12 , θ 13 , θ 23 ) and CP-violating phases (δ 12 , δ 13 , δ 23 ) of three light neutrinos ν 1 , ν 2 and ν 3 , but also on the mass (M 1 ) and mixing parameters (θ 14 , θ 24 , θ 34 and δ 14 , δ 24 , δ 34 ) of the heavy Majorana neutrino N 1 . When the former contribution is negligibly small, we can reproduce the case discussed in Ref. [6]; but when the contribution of N 1 is negligibly small, our results for B(H ±± → l ± α l ± β ) can simply reproduce those obtained in the triplet seesaw model [9,10]. The new and most interesting case, which has not been analyzed before, is the competition or interference between the contributions of light and heavy Majorana neutrinos. Typically assuming M 1 ∼ 200 GeV-1 TeV and taking three possible mass patterns of ν i as allowed by current neutrino oscillation data, we figure out the generous interference bands of ν i and N 1 contributions to B(H ±± → l ± α l ± β ): | sin θ i4 sin θ j4 | ∼ 10 −8 -10 −5 (for i, j = 1, 2, 3). We stress that both constructive and destructive interference effects are possible in this parameter region, in which it is very difficult to distinguish the type-II seesaw model from the triplet seesaw model at the LHC. We present some detailed numerical calculations of B(H ±± → l ± α l ± β ) in the interference bands. Although our numerical results are subject to the minimal type-II seesaw model, they can serve as a good example to illustrate the interplay between light and heavy Majorana neutrinos in a generic type-II seesaw scenario.

II. INTERFERENCE BANDS
After the spontaneous electroweak symmetry breaking, we rewrite Eq. (1) as We assume the existence of only a single heavy Majorana neutrino N 1 in the type-II seesaw scenario. The 4 × 4 neutrino mass matrix in Eq. (5) is symmetric and can be diagonalized by the following unitary transformation: where M ν = Diag{m 1 , m 2 , m 3 } with m i being the masses of three light neutrinos ν i , and M 1 denotes the mass of N 1 . After this diagonalization, the flavor states of three light neutrinos ν α (for α = e, µ, τ ) can be expressed in terms of the masses states of both three light Majorana neutrinos ν i (for i = 1, 2, 3) and the heavy Majorana neutrino N 1 ; namely, ν α = V αi ν i + R α1 N 1 . Then it is straightforward to write out the standard charged-current interactions between ν α and α in the basis of mass states: We see that V describes the flavor mixing of three light neutrinos and three charged leptons, while R determines how strong the heavy Majorana neutrino interacts with three charged leptons. In other words, V and R are responsible for neutrino oscillations of ν i and collider signatures of N 1 , respectively. Note that V itself is not unitary, because V V † + RR † = 1 holds as a consequence of unitarity of the 4 × 4 transformation matrix in Eq. (6). The correlation between V and R can be parametrized as [11] where c ij ≡ θ ij , s ij ≡ sin θ ij andŝ ij ≡ e iδ ij s ij with θ ij and δ ij (for 1 ≤ i < j ≤ 4) being the rotation angles and phase angles, respectively. If the heavy Majorana neutrino N 1 is decoupled (i.e., θ 14 = θ 24 = θ 34 = 0), V will become a unitary matrix and take the standard form as advocated in Refs. [2,12]. Hence non-vanishing R measures the non-unitarity of V . Now we make use of Eqs. (6) and (8) to reconstruct M L , which determines the branching ratios of H ±± → l ± α l ± β decay modes. We obtain Then the explicit expressions of (M L ) αβ can be given in terms of the relevant neutrino masses, mixing angles and CP-violating phases. In view of current experimental constraints s 13 < 0.16 [13] and s i4 < ∼ 0.1 (for i = 1, 2, 3) [14], we may simplify the exact results of (M L ) αβ by taking c 13 ≈ c i4 ≈ 1. This good approximation allows us to arrive at As a consequence, By combining Eqs. (10) and (11) with Eq. (4), we are then able to calculate the branching ratios B(H ±± → l ± α l ± β ). There are two extreme cases. (1) If the heavy Majorana neutrino N 1 is essentially decoupled (i.e., θ i4 ≈ 0 for i = 1, 2, 3), the unitarity of V will be restored. In this case, the results of B(H ±± → l ± α l ± β ) are the same as those obtained in the triplet seesaw model [9,10].
Here let us explore the third interesting case, in which the contributions of ν i and N 1 to (M L ) αβ are comparable in magnitude and may give rise to significant interference effects on the branching ratios of H ±± → l ± α l ± β decays. To be explicit, we take ∆m 2 21 ∼ 8.0 × 10 −5 eV 2 and |∆m 2 32 | ∼ 2.5 × 10 −3 eV 2 [13] as the typical inputs and assume M 1 to lie in the range 200 GeV-1 TeV. There are three possible patterns of the light neutrino mass spectrum: (1) the normal hierarchy: m 3 ∼ 5.1 × 10 −2 eV, m 2 ∼ 8.9 × 10 −3 eV, and m 1 is much smaller than m 2 ; (2) the inverted hierarchy: m 2 ∼ 5.0 × 10 −2 eV, m 1 ∼ 4.9 × 10 −2 eV, and m 3 is much smaller than m 1 ; (3) the near degeneracy: m 1 ∼ m 2 ∼ m 3 ∼ 0.1 eV to 0.2 eV, which is consistent with the cosmological upper bound m 1 + m 2 + m 3 < 0.61 eV [13]. In each case, the contributions of ν i and N 1 to (M L ) αβ in Eq. (10) will be of the comparable magnitude if the mixing angles θ i4 satisfy the following condition 1 : where i, j = 1, 2, 3. In view of this rough estimate, which is essentially compatible with a more careful numerical analysis, we can generously set √ s i4 s j4 ∼ 10 −8 -10 −5 as the interference bands of B(H ±± → l ± α l ± β ) for M 1 ∼ 200 GeV-1 TeV. Because the CP-violating phases δ i4 are completely unrestricted, they may cause either constructive or destructive effects in the interference bands. We shall numerically calculate B(H ±± → l ± α l ± β ) in the subsequent section to illustrate the interference effects for different patterns of the light neutrino mass hierarchy.
If M 1 < ∼ O(1) TeV and the values of s i4 lie in the interference bands obtained above, it will be impossible to produce and observe N 1 at the LHC. The reason is simply that the interaction of N 1 with three charged leptons is too weak to be detected in this parameter space. Given the integrated luminosity to be 100 fb −1 , for example, the resonant signature of N 1 in the channel pp → µ ± N 1 with N 1 → µ ± W ∓ at the LHC has been analyzed and the sensitivity of the cross section σ(pp → µ ± µ ± W ∓ ) ≈ σ(pp → µ ± N 1 )B(N 1 → µ ± W ∓ ) to the effective mixing parameter S µµ ≈ s 4 24 /(s 2 14 + s 2 24 + s 2 34 ) has been examined in Ref. [15]. It is found that S µµ ≥ 7.2 × 10 −4 (or equivalently, s 2 24 ≥ 2.1 × 10 −3 for s 14 ∼ s 24 ∼ s 34 ) is required in order to get a signature at the 2σ level for M 1 ≥ 200 GeV. This result illustrates that there will be no chance to probe the existence of N 1 in the interference bands at the LHC.
Nevertheless, it is possible to produce H ±± at the LHC provided M H ±± < ∼ O(1) TeV, and it is also possible to observe the signatures of H ±± → l ± α l ± β decays [6,7,9,10]. In this case, however, the measurements of B(H ±± → l ± α l ± β ) themselves are very difficult to tell whether the existence of H ±± is due to a pure triplet seesaw model or due to a (minimal) type-II seesaw model.

IV. SUMMARY
We have studied the dileptonic decays of doubly-charged Higgs bosons H ±± in the minimal type-II seesaw model with only one heavy Majorana neutrino and one Higgs triplet. Their branching ratios B(H ±± → l ± α l ± β ) depend not only on the masses, flavor mixing angles and CP-violating phases of three light neutrinos ν i (for i = 1, 2, 3), but also on the mass (M 1 ) and mixing parameters (θ i4 and δ i4 ) of the heavy Majorana neutrino N 1 . We have focused our attention on the interference bands of B(H ±± → l ± α l ± β ), in which the contributions of ν i and N 1 are comparable in magnitude. Assuming M 1 ∼ 200 GeV-1 TeV and taking three possible mass patterns of ν i as allowed by current neutrino oscillation data, we have figured out the generous interference bands | sin θ i4 sin θ j4 | ∼ 10 −8 -10 −5 (for i, j = 1, 2, 3) and presented a detailed numerical analysis of B(H ±± → l ± α l ± β ). We stress that both constructive and destructive interference effects are possible in the interference bands of B(H ±± → l ± α l ± β ), and thus it is very difficult to distinguish the (minimal) type-II seesaw model from the triplet seesaw model in this parameter space. Although our numerical results are subject to a simplified type-II seesaw scenario, they can serve as a good example to illustrate the interplay between light and heavy Majorana neutrinos in a generic type-II seesaw framework. The latter involves more free parameters, so the corresponding interference bands of B(H ±± → l ± α l ± β ) will be in a mess. It is worth pointing out that the lepton-number-violating decays of singly-charged Higgs bosons H ± are also important for testing the gauge triplet nature of the Higgs field. For example, the observation of H + → l + ανα and H − → l − α ν α (for α = e, µ, τ ) decays will be particularly useful to determine the mass spectrum of three light Majorana neutrinos [10] because these processes are independent of the unknown Majorana phases in the triplet seesaw model. A similar study of the lepton-number-violating H ± decays can be done in the type-II seesaw model, where heavy Majorana neutrinos exist, although the interference bands of B(H + → l + ανα ) and B(H − → l − α ν α ) are expected to be different from those of B(H ±± → l ± α l ± β ). We shall carry out a systematic analysis of both H ±± decays and H ± decays in the minimal type-II seesaw scenario elsewhere [17].
It is certainly a big challenge to identify the unique or correct seesaw mechanism of neutrino mass generation, if such a mechanism really exists, at the upcoming LHC and the future International Linear Collider. In particular, the collider signatures of both the Higgs triplet and heavy Majorana neutrinos will have to be experimentally established before a claim of having verified the type-II seesaw mechanism can be made. While the running of the LHC itself might be very difficult to help us pin down the true flavor dynamics of leptons and quarks, we hope that it would at least shed light on what this dynamics looks like at the TeV energy scale.
One of us (Z.Z.X.) is grateful to W. Chao and S. Zhou for helpful discussions. This work was supported in part by the National Natural Science Foundation of China.