UCRHEP-T317 September 2001
Veri?able Origin of Neutrino Mass at TeV Scale
arXiv:hep-ph/0109270v1 27 Sep 2001
Ernest Ma Department of Physics, University of California, Riverside, CA 92521, USA
Abstract The physics responsible for neutrino mass may reside at or below the TeV energy scale. The neutrino mass matrix in the (νe , ν? , ντ ) basis may then be deduced from future high-energy accelerator experiments. The newly observed excess in the muon anomalous magnetic moment may also be related.
————– Talk given at the 7th International Workshop on Topics in Astroparticle and Underground Physics, Assergi, Italy (September 8-12, 2001)
The conventional wisdom in neutrino physics is that the origin of neutrino mass is at a very high energy scale, say 1013 GeV or greater, in which case there is no hope of veri?cation experimentally. On the other hand, recent papers [1, 2, 3] have shown that it is just as natural to have the origin of neutrino mass at the TeV scale and be amenable to direct experimental veri?cation. In the minimal Standard Model with one Higgs doublet Φ = (φ+ , φ0 ) and 3 lepton doublets L = (ν, l)L and singlets lR only, neutrino mass must come from the e?ective dimension-5 operator [4, 5] 1 1 LLΦΦ = (νφ0 ? lφ+ )2 , Λ Λ (1)
which shows that the form of mν must necessarily be “seesaw”, i.e. v 2 /Λ where v = φ0 , whatever the underlying mechanism for neutrino mass is. The canonical seesaw mechanism  assumes 3 heavy right-handed singlet lepton ?elds NR with the Yukawa couplings f LNΦ and the Majorana mass mN , hence Eq.(1) is realized with the famous expression mν = f 2v2 . mN (2)
Note that lepton number is violated by mN in the denominator and it should be large for a small neutrino mass, i.e. mν ? f 1.0
1013 GeV mN
Higgs Triplet Model
An equally satisfactory realization of Eq.(1) is to use a Higgs triplet [7, 8] ξ = (ξ ++ , ξ + , ξ 0) with √ Lint = fij [ξ 0 νi νj + ξ + (νi lj + li νj )/ 2 + ξ ++ li lj ] √ ? + ?(ξ 0 φ0 φ0 ? 2ξ ? φ+ φ0 + ξ ?? φ+ φ+ ) + H.c.
This model violates lepton number explicitly, but if the parameter ? is set to zero, it becomes the Gelmini-Roncadelli model , which is now experimentally ruled out. On the other hand, with ? = 0 and m2 positive and large, i.e. mξ >> v, we have instead  ξ mν = 2fij ?v 2 = 2fij ξ 0 . 2 mξ (5)
Note that the e?ective operator of Eq.(1) is realized here with a simple rearrangement of the individual terms, i.e. Li Lj ΦΦ = νi νj φ0 φ0 ? (νi lj + li νj )φ+ φ0 + li lj φ+ φ+ . (6)
Note also that lepton number is violated in the numerator in this case. If fij ? 1, then ?/m2 < 10?13 GeV?1 . Hence mξ ? 1 TeV is possible, if ? < 100 eV. To obtain such a small ξ mass parameter, the “shining” mechanism of extra large dimensions  may be used. In
+ + that case, the doubly charged ξ ±± can be easily produced at colliders and ξ ++ → li lj is
a distinct and backgroundless decay which maps out |fij |, and thus determine directly the neutrino mass matrix of Eq. (5) up to an overall scale. This model also predicts observable ? ? e conversion in nuclei within the sensitivity of proposed future experiments.
Leptonic Higgs Doublet Model
Another simple and interesting way to have the origin of neutrino mass at the TeV scale has recently been proposed. As in the canonical seesaw model, we have again 3 NR ’s but they 3
are now assigned L = 0 instead of the customary L = 1. Hence the Majorana mass terms are allowed but the usual LNΦ terms are forbidden by lepton-number conservation. The LLΦΦ operator of Eq.(1) is not possible and mν = 0 at this point. We now add a new scalar doublet η = (η + , η 0 ) with L = ?1, then f LNη is allowed, and the operator LLηη will generate a nonzero neutrino mass if η 0 = 0. The trick now is to show how f η 0 < 1 MeV can be obtained naturally, so that mN ? 1 TeV becomes possible and amenable to experimental veri?cation, in contrast to the very heavy NR ’s of the canonical seesaw mechanism. Consider the following Higgs potential: V 1 1 = m2 Φ? Φ + m2 η ? η + λ1 (Φ? Φ)2 + λ2 (η ? η)2 1 2 2 2 + λ3 (Φ? Φ)(η ? η) + λ4 (Φ? η)(η ?Φ) + ?2 (Φ? η + η ? Φ), 12 (7)
where the ?2 term breaks lepton number softly and is the only possible such term. Let 12 φ0 = v, η 0 = u, then the equations of constraint for the minimum of V are given by v[m2 + λ1 v 2 + (λ3 + λ4 )u2 ] + ?2 u = 0, 1 12 u[m2 + λ2 u2 + (λ3 + λ4 )v 2 ] + ?2 v = 0. 2 12 Consider the case m2 < 0, m2 > 0, and |?2 | << m2 , then 1 2 12 2 v2 ? ? m2 1 , λ1 u?? ?2 v 12 . m2 + (λ3 + λ4 )v 2 2 (10) (8) (9)
Hence u may be very small compared to v(= 174 GeV). For example, if m2 ? 1 TeV, |?2 | ? 10 GeV2 , then u ? 1 MeV and 12 mν ? f 1.0
1 TeV mN
Since both mN and m2 are now of order 1 TeV, they may be produced at future colliders and be detected. (I) If m2 > mN , then the physical charged Higgs boson h+ , which is mostly 4
η + , will decay into N, which then decays into a charged lepton and a W boson via ν ? N miixing:
+ h+ → li Nj , ± Nj → lk W ? .
(II) If mN > m2 , then
± Ni → lj h? ,
h+ → t? b,
the latter coming from Φ ? η mixing. In either case, m2 and mN can be determined kinematically, and |fij | measured up to an overall scale. In summary, the particle spectrum of the leptonic Higgs doublet model consists of the usual Standard-Model particles, including the one physical Higgs boson h0 , 3 heavy NR ’s 1 at the TeV scale, and a heavy scalar doublet (h± , h0 , A) of individual masses ? m2 . The 2 charged Higgs boson h± can be pair-produced at hadron colliders, whereas NR (h± ) can be produced at lepton colliders via the exchange of h± (NR ).
The Size of Lepton Number Violation
It has been shown in the above that whereas Majorana neutrino masses have to be tiny, the actual magnitude of lepton number violation may come in all sizes. (1) Large: mN ? 1013 GeV in the canonical seesaw mechanism. (2) Medium: |?2 | ? 10 GeV2 in the leptonic Higgs doublet model with mN ? 1 TeV. 12 (3) Small: ? ? 10 eV in the Higgs triplet model (mξ ? 1 TeV) with a singlet bulk scalar in extra large dimensions. In (2) and (3), direct experimental determination of the relative magnitudes of the elements of Mν is possible at future colliders.
Muon Anomalous Magnetic Moment
The recent measurement  of the muon anomalous magnetic moment appears to disagree with the Standard-Model prediction  by 2.6σ, i.e. ?a? = aexp ? aSM > 215 × 10?11 ? ? (14)
at 90% con?dence level. The origin of this discrepancy may be directly related to the TeV physics responsible for neutrino mass. If the leptonic Higgs doublet model  is combined with a similar model of quark masses  to become a supersymmetric model  with 4 ? ? Higgs doublets, then the loop contribution of N and h+ will in general cause the transition li → lj γ. Hence there are predictions  for the muon anomalous magnetic moment as well as lepton ?avor violating processes such as ? → eγ, τ → ?γ, etc. If the neutrino mass matrix is hierarchical, then Eq. (14) implies B(τ → ?γ) > 8.0×10?6 , which contradicts the experimental upper bound of 1.1 × 10?6 . To avoid this restriction, the neutrino mass matrix has to be nearly degenerate, in which case we have the interesting prediction of Γ(τ → eγ) Γ(τ → ?γ) Γ(? → eγ) : : 5 5 m? mτ m5 τ 1 1 = (?m2 )2 : (?m2 )2 : (?m2 )2 , sol sol atm 2 2 where bimaximal mixing has been assumed. In Fig. (1), the branching fractions of τ → ?γ and ? → eγ, and the ? ? e conversion ratio in
Al are plotted using the lower bound of Eq. (14), as a function of the common neutrino
mass mν . The values (?m2 )atm = 3 × 10?3 eV2 , (?m2 )sol = 3 × 10?5 eV2 , (16)
have been chosen according to present data from neutrino-oscillation experiments. At mν ? 0.2 eV, which is in the range of present upper limits on mν from neutrinoless double beta 6
decay, B(? → eγ) and R?e are both at their present experimental upper limits. Hence Eq. (15) will be tested in new experiments planned for the near future which will lower these upper limits.
Physics at the TeV scale may reveal the true origin of neutrino mass, so that accelerator experiments will become complementary to nonaccelerator experiments in determining the neutrino mass matrix without ambiguity.
This work was supported in part by the U. S. Department of Energy under Grant No. DEFG03-94ER40837.
This talk was being given in Assergi, Italy on September 11, 2001, at the moment the infernal attack on the World Trade Center in New York began. All of humanity is the victim today and for years to come.
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Doublet model BR
Figure 1: Lower bounds on B(τ → ?γ), B(? → eγ), and R?e .