UCRHEP-T334 April 2002
arXiv:hep-ph/0204013v1 1 Apr 2002
Neutrino Mass from Triplet and Doublet Scalars at the TeV Scale
Ernest Ma Physics Department, University of California, Riverside, California 92521
Abstract If the minimal standard model of particle interactions is extended to include a scalar triplet with lepton number L = ?2 and a scalar doublet with L = ?1, neutrino 2 5 ?2 eV is possible, where v ? 102 GeV is the electroweak masses mν ? ?4 12 v /M ? 10 symmetry breaking scale, M ? 1 TeV is the typical mass of the new scalars, and ?12 ? 1 GeV is a soft lepton-number-violating parameter.
In the minimal standard model of particle interactions, neutrinos are massless, but if new interactions exist at a higher scale, then they may become massive through the unique e?ective dimension-?ve operator  Lef f = fij Li Lj ΦΦ, Λ (1)
where Li = (νi , li )L is the usual left-handed lepton doublet, Φ = (φ+ , φ0 ) is the usual scalar Higgs doublet, and Λ is an e?ective large mass. There are three tree-level realizations  of this operator: (I) the canonical seesaw mechanism  using one heavy right-handed neutrino NR for each νi ; (II) the addition of a heavy Higgs triplet (ξ ++ , ξ + , ξ 0) which couples to Li Lj directly [4, 5]; and (III) the replacement of NR with (Σ+ , Σ0 , Σ? )R [6, 7]. If the mass of N or ξ or Σ is very large, then each realization is the same as any other at low energies, because the only observable e?ect would be the appearance of small Majorana neutrino masses. It has recently been pointed out that simple extensions of the above minimal scenarios for neutrino mass are possible for which the scale of new physics may be only a few TeV and thus be observable at future accelerators. There are two speci?c proposals: (A) the Higgs triplet ξ may be only a few TeV, whose decay into two leptons would map out all elements of the neutrino mass matrix ; and (B) the fermion singlets NR as well as a second Higgs doublet η = (η + , η 0 ) may be only a few TeV . In (A), the notion of lepton-number violation as a distance e?ect from the separation of our brane from another in the context of large extra dimensions is invoked to explain the smallness of the trilinear scalar coupling of ξ to ΦΦ. In (B), there is no need to consider large extra dimensions. Instead, the e?ective operator of Eq. (1) is suppressed because Φ is replaced with η , which has a naturally small vacuum expectation value. In this note, the two mechanisms are synthesized so that there is a Higgs triplet ξ from (A), and a Higgs doublet η from (B), but no NR . Neutrino masses come from ξ (which is assigned lepton number L = ?2), and its interaction with η (which has L = ?1). The smallness of mν comes from 2
the soft breaking of L in the scalar sector, with the result mν ?
2 ?4 12 v ? 10?2 eV, M5
where v ? 102 GeV is the electroweak symmetry breaking scale, M ? 1 TeV is the typical mass of the new scalars, and ?12 ? 1 GeV is a soft lepton-number-violating parameter. This model requires neither large extra dimensions nor NR to obtain naturally small Majorana neutrino masses, and have veri?able experimental consequences at the TeV scale. In the Higgs triplet model, ξ couples to leptons according to √ LY = fij [ξ 0 νi νj + ξ + (νi lj + li νj )/ 2 + ξ ++ li lj ] + H.c., (3)
resulting in (mν )ij = 2fij ξ 0 . Therefore, in order to explain why neutrino masses are so small, a natural mechanism for ξ 0 to be small is needed. Consider now the Higgs sector consisting of the usual standard-model doublet Φ (with L = 0), a second doublet η (with L = ?1), and a triplet ξ (with L = ?2). Let ? ? √ ++ + 2 ξ ξ / √ ?, ?≡? ξ0 ?ξ + / 2 then the most general L-conserving Higgs potential is given by V
? 2 ? 2 ? = m2 1 Φ Φ + m2 η η + m3 T r ? ?
1 1 1 λ1 (Φ? Φ)2 + λ2 (η ? η )2 + λ3 T r ?? ? 2 2 2
1 + λ4 T r ?? ?? (T r ??) 2
+ λ5 (Φ? Φ)(η ? η ) + λ6 (Φ? Φ) T r ?? ? + λ7 (η ? η ) T r ?? ? + λ8 (Φ? η )(η ? Φ) + λ9 Φ? ?? ?Φ + λ10 η ? ?? ?η + ? η ? ?? η + H.c., (5)
where η ? = (? η 0 , ?η ? ) and the parameter ? has the dimension of mass. Lepton number is then assumed to be broken by explicit sof t terms, i.e.
? ? ′ ? + H.c. η + ?′′ Φ? ?Φ Vsof t = ?2 12 Φ η + ? Φ ??
Let φ0 = v1 , η 0 = v2 , and ξ 0 = v3 , then the minimum of V is given by
2 2 2 2 2 2 Vmin = m2 1 v1 + m2 v2 + m3 v3 + 2?12 v1 v2
1 1 1 4 4 4 2 2 2 2 2 2 λ1 v1 + λ2 v2 + λ3 v3 + (λ5 + λ8 )v1 v2 + λ6 v1 v3 + λ7 v2 v3 2 2 2 (7)
2 2 + 2?v2 v3 + 2?′ v1 v2 v3 + 2?′′ v1 v3 ,
′ ′′ where ?2 12 , ?, ? , and ? have been assumed real for simplicity. The equations of constraint
are obtained from ?Vmin /?vi = 0, i.e.
2 2 2 ′′ 2 ′ v1 [m2 1 + λ1 v1 + (λ5 + λ8 )v2 + λ6 v3 + 2? v3 ] + ?12 v2 + ? v2 v3 = 0, 2 2 2 2 ′ v2 [m2 2 + λ2 v2 + (λ5 + λ8 )v1 + λ7 v3 + 2?v3 ] + ?12 v1 + ? v1 v3 = 0, 2 2 2 2 ′ ′′ 2 v3 [m2 3 + λ3 v3 + λ6 v1 + λ7 v2 ] + ?v2 + ? v1 v2 + ? v1 = 0.
(8) (9) (10)
2 2 2 ′ ′′ Consider now the case m2 1 < 0, but m2 > 0 and m3 > 0 with small ?12 , ? and ? . The
solutions to the above equations are then
2 v1 ? ?
m2 1 , λ1
(11) (12) (13)
v2 ? ? v3
?2 12 v1 , 2 2 m2 + (λ5 + λ8 )v1 2 ?v 2 + ?′ v1 v2 + ?′′ v1 ? ? 2 2 . 2 m3 + λ6 v1
Since ?′ violates L by 1 unit and ?′′ violates L by 2 units, it is reasonable to assume that ?′′ /?′ ? ?′ /? ? v2 /v1 . Thus for m2 , m3 , and ? all approximated by M ? 1 TeV, v2 ? This shows that v3 << v2 << v1 , and v3 ?
2 ?4 12 v1 , M5
?2 12 v1 , M2
2 v2 . M
i.e. the analog of Eq. (2). For v1 ? 102 GeV and ?12 ? 1 GeV, the solutions are v2 ? 0.1 MeV and v3 ? 10?2 eV as desired. 4
In conclusion, a scenario has been presented where a Higgs triplet at the TeV scale is
+ + responsible for neutrino masses. The decay of ξ ++ into li lj would then map out the neutrino
mass matrix, as proposed previously . In addition, a second Higgs doublet at the TeV scale is predicted  so that ξ ++ → η + η + is also possible if kinematically allowed.
This work was supported in part by the U. S. Department of Energy under Grant No. DEFG03-94ER40837.
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