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Testing Higgs Triplet Model and Neutrino Mass Patterns

The observed neutrino oscillation data might be explained by new physics at a TeV scale, which is testable in the future experiments. Among various possibilities, the low-energy Higgs triplet model is a prime candidate of such new physics since it predicts clean signatures of lepton flavor violating processes directly related to the neutrino masses and mixing. It is discussed how various neutrino mass patterns can be discriminated by examining the lepton flavor violating decays of charged leptons as well as the collider signatures of a doubly charged Higgs boson in the model.


I. INTRODUCTION
The atmospheric, solar and reactor neutrino experiments [1,2,3,4] have firmly established the picture of three active neutrino oscillations, and provided us important information on two neutrino mass-squared differences and three mixing angles. Taking the most favorable parameter region of the solar neutrino oscillation (so-called LMA I), we have and the limit of sin 2 θ chooz < 0.038 coming from the non-observation of ν µ → ν e oscillation in the CHOOZ and atmospheric neutrino data [1,3].
Given such new experimental inputs, we could hope for uncovering new physics beyond the standard model, which must explain the observed neutrino data. In this regard, a "low-energy" model for neutrino masses and mixing is of particular interest since it may be tested in the future experiments observing lepton flavor violating processes in accelerators. A typical example of such a model would be the supersymmetric standard model with R-parity violation in which the flavor structure of neutrino mass matrix could be probed through the decay of the lightest supersymmetric particle [5]. Another example is the Zee model and its variations [6] which rely on radiative mechanism of neutrino mass generation.
In this paper, we consider the Higgs triplet model in which a triplet scalar field ∆ = (∆ ++ , ∆ + , ∆ 0 ) with the mass M is introduced to have the following renormalizable couplings; where L i = (ν i , l i ) L is the left-handed lepton doublet and Φ = (φ 0 , φ − ) is the standard model Higgs doublet. Due to the "µ" term in the above equation, the neutral component ∆ 0 of the triplet gets the vacuum expectation value (VEV), v ∆ = µv 2 Φ /2M 2 where v Φ = Φ 0 = 246 GeV. This leads to the neutrino mass matrix, We are interested in the possibility of the light triplet Higgs bosons, namely M ∼ v Φ , so that observations of various lepton flavor violating processes can provide a probe for the neutrino masses and mixing through the relation (3), and thus a direct test of the model. In this "low-energy triplet Higgs model", the small parameters f and ξ ≡ v ∆ /v Φ are required; f ij ξ ∼ 10 −12 (4) for M ν ij ∼ 0.3 eV. We will see later that such a smallness could be understood by a radiative mechanism. Here, let us note that we are interested in the case of very small ξ, say ξ < ∼ 10 −6 , so that the condition of ρ = m 2 Z /m 2 W c 2 W ≃ 1 is simply satisfied in our consideration. Phenomenological consequences of low-energy triplet Higgs bosons have been studied extensively in the past, in particular, centering around the exotic signatures of a doubly charged Higgs boson, ∆ ±± [7,8,9,10,11,12]. The main purpose of this work is to investigate how the observation of such phenomena can test the pattern of the neutrino masses and mixing. For this to happen, we will mostly assume that f > ∼ ξ to detect the lepton flavor violating processes induced by the coupling f . This paper is organized as follows. In section 2, we derive the flavor structure of the bileptonic couplings f ij depending on the acceptable neutrino mass patterns, based on which the observability of rare lepton decays such as µ → eγ, µ → 3e and τ → 3l will be discussed. In section 3, we will consider the production and decays of doubly charged Higgs bosons in colliders from which some information on the couplings f can be obtained. We will see when the collider effects of the coupling f can be observed in relation to the above discussion. Then, we examine how the neutrino mass patterns can be discriminated through the observation of ∆ ±± decays. In section 4, we present a model in which the smallness of the couplings f and µ is explained by a radiative generation at two-loop level. We conclude in section 5.

II. NEUTRINO MASS PATTERNS AND LOW-ENERGY LEPTON FLAVOR VI-OLATION
Current neutrino data (1) give us the following neutrino mixing matrix; In the below, we will show the ratios;
Given the information on ∆m 2 (1), one has a variety of possibilities for the neutrino mass eigenvalues. Assuming CP conservation, the following different patterns can be allowed: (i) Hierarchy with m 1 < m 2 < m 3 which gives where R ≡ ∆m 2 atm /m 2 1 . Since the recent WMAP results put a limit of m 1 < 0.23 eV [13], the ratio R has to be larger than about 0.02.
The schematic form of the bilepton couplings (2) can be written explicitly as where we used the matrix form of the triplet field; The above Lagrangian induces the tri-leptonic and radiative decays of a charged lepton at tree and one-loop level, respectively [12]. Let us now discuss the observational possibilities of such lepton flavor violating decays of muon or tau in the triplet Higgs model. Table I shows the current limits on the products of couplings for various decay modes, and their future experimental sensitivities. For the discovery of some lepton flavor violating decay modes, one needs where i, j, k = 1, 2 as indicated in Table I.
In the cases of (IN2), (DG3) and (DG4), neither µ → eγ nor τ → 3l can be observed as the strong constraint from the µ → 3e pushes them outside the future experimental sensitivity. To see this, let us note that f 11 f 12 ∝ sin 2θ 3 cos 2θ 3 / √ 2 from Eqs. (10), (14) and (15), and cos 2θ 3 > 0.1 from Eq. (1), which shows that where R has to be included in the (DG) case. The situation can be different in other cases where one has the following relations for the ratio f 11 f 12 : (f f † ) 12 : f ii f 23 ; Mode Current limit [14,15] Future sensitivity [15,16] Bound on the couplings where f ii = f 22 for (HI) and f 11 otherwise. From this, one can see that the decay modes other than µ → 3e can be seen only if the coupling f 12 is made small and thus the following relation is fulfilled; In this case, one predicts An ideal case is to observe both τ → 3l and µ → eγ decays which will enable us to discriminate the different mass patterns.

III. COLLIDER TEST: PRODUCTION AND DECAYS OF HIGGS TRIPLET
Some of striking collider signals in the triplet Higgs model comes from the decays of a doubly charged Higgs boson, such as ∆ −− → l i l j , W − W − , which have been studied extensively in the past years [7,8,9,10,11,12]. We are interested in the situation that the decays ∆ −− → l i l j are sizable so that the neutrino mass structure can be tested in colliders. Depending on the masses of the triplet components, the fast decay process like Note that the triplet VEV is given by v ∆ = µv 2 Φ /2M 2 ∆ 0 . In this theory, the mass eigenstates consist of ∆ ++ , H + , H 0 , A 0 and h 0 . Under the condition that |ξ| ≪ 1, the first five states are mainly from the triplet sector and the last from the doublet sector. The approximate mass diagonalizations are given as follows. For the neutral pseudoscalar and charged scalar parts, where G 0 and G + are the Goldstone modes, and for the neutral scalar part, . The masses of the Higgs bosons are The mass of h 0 is given by m 2 h 0 = 4λ 1 v 2 Φ as usual. When λ 5 > 0, we have M ∆ ±± < M H ± < M H 0 ,A 0 , so that the doubly charged Higgs boson ∆ −− can only decay to l i l j or W − W − through the following interactions; The corresponding decay rates are giving rise to the decay rate where y ≡ 2|λ 5 |/g 2 . This can be rewritten as Γ( Here, let us remark that, after the diagonalization in Eqs. (21) and (22), we also get couplings for the interactions, H + → ud, h 0 W + , ZW + and H 0 , A 0 → ff , W + W − , ZZ, h 0 h 0 , Zh 0 , all proportional to ξ, and thus they should be considered as Before going to our main discussion, let us note that the triplet Higgs decay is short enough to occur inside colliders. Assuming Eq. (25) as the main decay rates and recalling one obtains the following form of the total decay rate: When M ∆ ±± > 2M W , one finds the minimum value of the total decay rate given by • Single production of ∆ ±± : e + e − → e ± l ± ∆ ∓∓ In the e + e − colliders, an energetic virtual photon emitted from e ± leads to the enhanced e ∓ γ scattering producing l ± i ∆ ∓∓ when a coupling f 1i is sizable. Adopting the result of Ref. [9] with the p T cut (p T = 10 GeV) and neglecting the final state lepton masses, we obtain the following pairs of M ∆ ±± and f 2 1i : Let us first consider the cases of (IN2), (DG3) and (DG4) where the couplings f 2 1i are strongly constrained as seen in Eq. (17). In each case, we get (f 2 11 , f 2 12 , f 2 13 ) ≈ (cot 2θ 3 , neglecting a small deviation due to the contribution of s 2 . Thus, if µ → 3e decay is found near the current experimental limit and θ 3 is close to 45 o , the final states µ ± ∆ ∓∓ and τ ± ∆ ∓∓ could be observed with for smaller values of the triplet mass, say M ∆ ±± < 700 GeV. In the cases of (IN1), (DG1) and (DG2), one has f 2 12 ≪ f 11 f 12 and f 2 13 ≪ f 2 11 and thus the characteristic signature is a copious production of the final state, e ± ∆ ∓∓ . If the low energy decay τ → 3l or µ → eγ is observed, the value of f 2 11 is determined by the following comparison with f 11 f 23 and (f f † ) 12 triggering the decays τ →μee and µ → eγ, respectively: for the cases of (IN1), (DG1) and (DG2), respectively. Here, x cannot be specified as 12 can be vanishingly small in the case (DG1). This shows that f 2 11 ≫ 10 −6 and thus the production of e ± ∆ ∓∓ can be detected even for M ∆ ±± ∼ 1 TeV. Even in the case that only µ → 3e decay is observed, there is some allowed parameter space for the production of e ± ∆ ∓∓ as we have ] f 11 f 12 (34) For the case of (HI), we have when f 12 is made small to suppress the decay µ → 3e. This shows that the decay τ → 3µ and µ → eγ could be observed together with the collider signals of producing the events e∆ and τ ∆ satisfying the relation Let us note that no signal of l∆ production can be observed if only the decay µ → 3e is observable in the case (HI).
When the couplings f ij are much smaller than the electroweak gauge couplings, which is always the case except for (DG1), pairs of doubly charged Higgs bosons can be produced through the gauge interactions exchanging γ or Z, if allowed kinematically. Then, the produced ∆ ±± may decay mainly to a pair of same-sign charged leptons through the couplings f . In this case, we can measure the relative sizes of the branching ratios B(∆ −− → l i l j ) and thus • (HI) 2r sin 4 θ 3 : In the above expressions, we assumed that s 2 is negligible.
In the linear collider with √ s = 1 TeV, the pair production cross section is σ ≈ (100 −10) fb for M ∆ ±± = (100 − 450) GeV [9]. Thus, taking L = 1000/fb, the number of the produced ∆ ±± will be N = (10 5 − 10 4 ). In LHC with L = 1000/fb, the number of the reconstructed pair production events is expected to be N = (10 5 −10 3 ) for M ∆ ±± = (100−450) GeV and it becomes down to N = 10 for M ∆ ±± = 1000 GeV [10]. Thus, both the linear collider and LHC can produce enough numbers of ∆ ±± to probe the neutrino mass pattern if M ∆ ±± < ∼ 450 GeV. In this case, the precise measurement of the branching ratios can also determine the neutrino oscillation parameters such as r, R or θ 3 . It is amusing to note that LHC has a good potential to confirm the triplet Higgs model as the source of neutrino mass matrix up to the triplet mass around 1 TeV. For this, the observation of the leading decay modes will be enough to discriminate the neutrino mass patterns as follows: Here we assumed that cot 2θ 3 and tan θ 3 sit at their lowest allowed values and thus give a sub-leading effect.

IV. A MODEL: TWO-LOOP GENERATION OF LL∆ AND ΦΦ∆
An unnatural feature of the Higgs triplet model generating the neutrino mass is that the model requires another hierarchy of couplings; the smallness of f or µ. This would have the same origin as the hierarchies of the usual quark and lepton Yukawa couplings, which is one of the difficult problems in particle physics. In this section, we separate the neutrino sector from the other and try to explain the smallness of f or µ through a radiative mechanism.
In the case of f ≫ µ, a way to get the small µ has been explored in Ref. [17] in which the operator ΦΦ∆ has been obtained at two loop. A variant of such a scheme can be found to explain the smallness of both f and µ. For this, let us introduce the following new scalar fields and a Z 3 discrete symmetry; where the SU(3) c × SU(2) L × U(1) Y × Z 3 charge of each field is specified in the second line and α = e 2π/3 . We assign the Z 3 charge α to L and α 2 to e c and ∆. All the other fields are neutral under Z 3 . The allowed couplings are QQX T , Ld c X Q , d c d c X u , X Q X Q X T S * , ∆X T X u S.
Then the operators LL∆S 2 and ΦΦ∆S arise from the two-loop diagrams as in Figure 1 and thus the small values of f and µ can obtained when S gets a VEV of the order v Φ .

V. CONCLUSION
We have investigated the testability of the low-energy Higgs triplet model and the resulting neutrino masses and mixing in the future collider experiments. The bileptonic couplings f ij can be large enough to yield observable lepton flavor violating decays of a charged lepton such as µ → 3e, µ → eγ or τ → 3l depending on the neutrino mass patterns. For this to happen, the coupling f 12 needs to be vanishingly small in order to satisfy the current bound on the µ → 3e decay. Another effect of the bileptonic couplings is the production of a doubly charged Higgs boson accompanied by a charged lepton l i in the e + e − collider. In this case, we have identified the characteristic flavor structure of the final state, l ∓ i ∆ ±± , for each neutrino mass pattern. We have shown that copious production of the doubly charged Higgs boson pairs through the gauge interactions in the linear collider and LHC provides a promising way to test not only the triplet Higgs model but also the resulting neutrino mass matrix even when f is very small. In LHC, in particular, we expect sufficient production of the doubly charged Higgs bosons up to the mass ∼ 1 TeV which will enables us to determine the neutrino mass pattern only by observing the leading decay channels. A problem in the low-energy triplet Higgs model is how to understand the smallness of the couplings f and µ. We have also worked out a radiative mechanism as one of possible solutions.
Acknowledgment: EJC was supported by the Korea Research Foundation Grant, KRF-