TeV-scale seesaw mechanism catalyzed by the electron mass

We construct a model in which the neutrino Dirac mass terms are of order the electron mass and the seesaw mechanism proceeds via right-handed neutrinos with masses of order TeV. In our model the spectra of the three light and of the three heavy neutrinos are closely related. Since the mixing between light and heavy neutrinos is small, the model predicts no effects in pp and p \bar p colliders. Possible signatures of the model are the lepton-number-violating process e- e- -->H- H-, where H- is a charged scalar particle, lepton-flavour-violating decays like mu- -->e- e+ e-, or a sizable contribution to the anomalous magnetic dipole moment of the muon.


Introduction
The discovery of neutrino oscillations has established the existence of non-zero neutrino masses in the sub-eV range. 1 An appealing way of explaining such small neutrino masses is the seesaw mechanism [4], in which gauge-singlet right-handed neutrino fields ν R with Majorana mass terms are added to the Standard Model (SM). Those mass terms are not generated by the Higgs mechanism and may, therefore, be much larger than the electroweak scale. Let ν L be the neutral members of the left-handed leptonic SM gauge doublets and M D and M R the mass matrices of the fermionic bilinearsν R ν L andν R Cν T R , respectively. (C is the charge-conjugation matrix in Dirac space.) We denote the mass scales of M D and M R by m D and m R , respectively. If m R is much larger than m D , then an effective Majorana mass matrix is generated for the ν L . According to (1), the scale m ν of M ν is related to m D and m R through m ν ∼ m 2 D / m R . The mixing between the light and heavy neutrinos is of order m D /m R , hence very small.
We experimentally know that m ν is in the range of 0.1 eV to 1 eV, if m ν indicates the order of magnitude of the largest of the light-neutrino masses, but, in the framework of the seesaw mechanism, m D and m R remain a mystery. Since M D is the neutrino counterpart of the charged-fermion mass matrices, m D may vary in between 100 GeV (which is both the electroweak scale and the top-quark mass scale) and 1 MeV (the scale of the electron mass and of the up-and down-quark masses), and this spans a range of five orders of magnitude. As m R ∼ m 2 D / m ν , to these five orders of magnitude in m D correspond ten orders of magnitude in m R . If m D ∼ 100 GeV then m R ∼ 10 13 GeV; this is definitely below the typical GUT scale 10 16 GeV-identifying m R with the GUT scale would make the neutrino masses too small. If instead m D ∼ m τ , the mass of the tau lepton, then m R ∼ 10 9 GeV. If one wants to incorporate leptogenesis [5] into the seesaw mechanism, then the appropriate m R would rather be 10 11 GeV, which lies in between the two previous estimates. Finally, we may as well use m D ∼ m e , the electron mass, and then m R ∼ 1 TeV-this was noticed, for instance, in [6]. 2 Such a low m R has the advantages that it coincides with the expected onset of physics beyond the SM and that it might produce testable effects of the seesaw mechanism at either present or future colliders.
The possibility that m D ∼ m e and m R ∼ 1 TeV is the starting point of this Letter. We construct a seesaw model in which the vacuum expectation value (VEV) responsible for the mass matrix M D is of order m e . In our model, which is inspired by [9] and [10], each charged lepton α (α = e, µ, τ ) has its own Higgs doublet φ α , whose VEV generates the mass m α . On the other hand, there is only one Higgs doublet φ 0 which has Yukawa 1 Cosmological arguments [1] and the negative result of the direct search for neutrino mass in tritium β decay [2] are also crucial in eliminating the possibility that the neutrino masses may be higher than about one eV. For a review on neutrino masses see [3]. 2 Of course, the simplest way to obtain m D ∼ m e is to assume tiny Yukawa couplings, as was done for instance in [7]; this is the opposite of what we do in this Letter-we discuss a scenario with Yukawa couplings of order 0.01-1. Another avenue which has been pursued are technicolor models with a suppressed m D and m R < ∼ 10 3 TeV [8]; however, in those models m D is not claimed to be as low as m e .
couplings to the ν R and is, therefore, responsible for M D . We furthermore make use of a mechanism, first put forward in [11] and later extended in [12], for suppressing the VEV of φ 0 : this doublet has a positive mass-squared µ 0 (in the scalar-potential term µ 0 φ † 0 φ 0 ) and its VEV is triggered by a term φ † e φ 0 in the scalar potential. Since φ e is the Higgs doublet which gives mass to the electron, it must have a very small VEV, and this explains the smallness of the VEV of φ 0 .
We shall use indices k, l running over the range 0, e, µ, τ . The VEV of the neutral component of φ k is denoted v k .

Symmetries:
The family symmetries of our model are the following: • Three 2 symmetries [9] z α : Notice that φ α does not change sign under z α . The symmetries z α may be interpreted as discrete lepton numbers.
Yukawa Lagrangian: The multiplets and symmetries of the theory lead to the Yukawa Lagrangian This has, remarkably, only two Yukawa coupling constants. The Lagrangian (4) enjoys the accidental symmetry Charged-lepton masses: The mass of the charged lepton α is m α = |y 1 v α |. Therefore m e : m µ : m τ = |v e | : |v µ | : |v τ |. In our model one cannot obtain a small m e by just tuning some Yukawa couplings-one really needs a small VEV v e . For instance, even if there were no further Higgs doublets in the theory and v τ by itself alone saturated the relation one would still need |y 1 | ∼ 0.01 because m τ ≈ 1.78 GeV. We would then have |v e | ∼ 50 MeV. On the other hand, we may assume that there are in the full theory some extra Higgs doublets other than the four φ k , which give mass to the quarks-in particular, for the top-quark mass a doublet with a large VEV is mandatory. Then |v τ | may be significantly smaller than 174 GeV and, accordingly, |v e | will be significantly smaller than 50 MeV too; in particular, |v e | ∼ m e is possible.
Soft symmetry breaking: We assume that the 3 symmetry and the three z α symmetries are softly broken in the dimension-three neutrino Majorana mass terms 4 We furthermore assume that the symmetries z h α are also softly broken, now by the dimension-two terms φ † k φ l (k = l) in the scalar potential. However, we shall assume that the combined symmetry is broken only spontaneously. Then the scalar potential is The quartic couplings in V , those with coefficients generically represented by the letter λ, obey all the family symmetries of the model. Because of the couplings λ ′′ kl φ † k φ l 2 , the only U(1) symmetry of V is the one associated with weak hypercharge; therefore, there are no Goldstone bosons in the model.
Suppression of v 0 : If µ 0 is positive, then v 0 will induced by v e and by the first term in the last line of (9): We envisage the possibility that |y 2 | and, possibly, also |y 1 | are of order 1, because that would enhance scalar effects and the experimental testability of our model, cf. sections 3 and 4 below. Still, it is possible, as we mentioned earlier, that |y 1 | ∼ 0.01 and |v e | ∼ 50 MeV. However, even in that case |v 0 | could easily be much smaller than |v e |, as we pondered in [12]. Indeed, |µ 0e | could naturally [14] be small, and there is no reason why µ 0 should not be rather large, maybe even of order TeV. Then |v 0 | might be much smaller than |v e |-this is the mechanism that we envisaged in the introduction. Thus: • If there are in the theory some Higgs doublets beyond the four φ k , the Yukawa coupling constant y 1 in (4) may be of order one and then |v e | ∼ m e . In that case, |µ 0e | and µ 0 may be allowed to be of the same order of magnitude and |v 0 | ∼ |v e | ∼ m e .
• If there are only the four φ k , 5 then y 1 is much smaller than one and |v e | ∼ 100 m e .
In that case, we may naturally assume |µ 0e | ≪ µ 0 because the theory acquires the extra symmetry z ν when µ 0e = 0. We might then still obtain |v 0 | ∼ m e .
Lepton mixing: The neutrino Dirac mass matrix is in our model proportional to the unit matrix: Therefore, the lepton mixing matrix U, which diagonalizes M ν , also diagonalizes M R : the m j and M j (j = 1, 2, 3) being, respectively, the light-and heavy-neutrino masses.
Consequently, there is a close relationship between the spectra of the light and heavy neutrinos: Following our rationale, we shall assume |y 2 v 0 | ∼ m e .

Possible collider effects
In our model we assume m D ∼ m e ∼ 1 MeV while m R ∼ 1 TeV. Otherwise our model is a normal seesaw model, therefore in it the mixing between the light and the heavy neutrinos is of order m D /m R ∼ 10 −6 . This small mixing suppresses most possible signatures of heavy right-handed neutrinos that have been considered in the literature [15]. For instance, the Drell-Yan production of a virtual W boson and its subsequent decay W ± * → ℓ ± N j , where ℓ ± is a charged lepton and N j a heavy Majorana neutrino, is negligible because it is suppressed by the mixing between the light and heavy neutrinos. Suppose that the Higgs doublet φ q (which may be one of our four doublets or an additional one) couples to the quarks and, in particular, generates the top-quark mass.
Then we may envisage the Drell-Yan production of a virtual φ ± q at the LHC, followed by the transition φ ± q → φ ± 0 and the decay φ ± 0 → ℓ ± N j , finally leading to heavy-neutrino production. However, since the VEV of φ 0 q is necessarily large and the VEV of φ 0 0 is very small, the mixing φ ± q -φ ± 0 will in general be small, unless we invoke finetuning in the scalar potential.
In contrast to what happens at hadron colliders, in an electron-electron collider the interesting lepton-number-violating process e − e − → H − H − might occur [6]. This process is due to the Majorana nature of the heavy neutrinos and is one of the processes, other than neutrinoless ββ decay, via which it might be possible to probe lepton-number violation [16]. Let us suppose for simplification that H − ≡ φ − 0 , the charged component of the scalar doublet φ 0 . Then, the relevant Yukawa Lagrangian for e − e − → H − H − is given by where P L is the negative-chirality projection matrix. This leads to the total cross section (see [6] for a special case) where s is the square of the energy of the e − e − system in its centre-of-momentum reference frame, β = (1 − 4m 2 Notice that the cross section (16) depends on the Majorana phases of the products (U ej U * ek ) 2 (j = k). In figure 1 we have plotted σ (e − e − → H − H − ) as a function of the light-neutrino mass m 1 in a number of cases. Following our rationale, we have taken |y 2 v 0 | = m e in equation (14). On the other hand, in equation (16) we have taken |y 2 | = 1, bearing in mind that σ (e − e − → H − H − ) depends very strongly on this Yukawa coupling. As for the U ej matrix elements, we have used the values |U e1 | 2 = 0.7, |U e2 | 2 = 0.3 and U e3 = 0, which almost coincide with the present best fit [17]; this has the advantage that then the third-neutrino mass m 3 and the type of neutrino mass spectrum become irrelevant-we only have to take into account the experimental value of m 2 2 −m 2 1 , which we fixed at 8 ×10 −5 eV 2 . We have moreover assumed that the phase of (U e1 U * e2 ) 2 is zero-this choice maximizes σ (e − e − → H − H − ). In figure 1 this cross section is given in units of σ QED = 4πα 2 / (3s), with α = 1/128.
Therefore, in that limit where m ββ = j m j (U ej ) 2 is the effective mass measured in neutrinoless ββ decay. The approximation (19) indicates a close relationship and, indeed, an approximate proportionality between σ (e − e − → H − H − ) and m 2 ββ . Equation (19) overestimates the cross section: σ (e − e − → H − H − ) is much smaller than indicated by that approximation whenever m 1 > ∼ 0.5 eV. On the other hand, for m 1 < ∼ 0.1 eV the approximation (19) becomes quite good, but at the same time the cross section becomes small. With the values of U used in figure 1, for m 1 = 0.1 eV, √ s = 10 3 GeV and m H = 120 GeV, equation (19) gives a cross section about 14% too large; for smaller s or larger m H the discrepancy is smaller. Note that a cross section of about 10 −2 × σ QED is still considered reasonable for detection of e − e − → H − H − at an e − e − collider [6].

Possible non-collider effects
We next investigate whether large Yukawa couplings y 1 and y 2 in equation (4) might induce measurable effects in non-collider physics.

The magnetic dipole moment of the muon
A promising observable is a µ , the anomalous magnetic moment of the muon. There is a puzzling 3σ discrepancy [18] a exp µ − a SM µ = 255(63)(49) × 10 −11 (20) between experiment and the SM prediction for that observable. In our model there are contributions to a µ from one-loop diagrams involving either charged or neutral scalars. We consider the latter firstly. The fields φ 0 k are written in terms of the physical neutral The scalar S 0 1 corresponds to the Goldstone boson eaten by the gauge boson Z 0 and is, therefore, unphysical. The complex 4 × 8 matrix V is such that The contribution of the physical neutral scalars to a µ is given by where A b = y 2 1 V 2 µb . One sees that there is a term δ b ln δ b /(16π 2 ) which is ∼ 10 −7 when m b ∼ 100 GeV. Thus, if one assumes |A b | ∼ 1 then |a µ (φ 0 )| might well be very large.
We proceed to analyze the diagrams involving charged scalars in the loop. The three left-handed neutrinos ν Lα and the three right-handed neutrinos ν Rα are written in terms of the six physical Majorana neutrinos χ i (i = 1, . . . , 6) as where P L,R are the chirality projectors in Dirac space and R, S are 3 × 6 matrices. The matrix R S (26) is 6 × 6 unitary [20]. The fields φ + k are written in terms of the physical charged scalars S + a (a = 2, 3, 4) as The scalar S + 1 corresponds to the Goldstone boson eaten by the gauge boson W + and is, therefore, unphysical. The complex 4 × 4 matrix U is unitary [19].
Let m i be the mass of the physical neutrino χ i and, for a ≥ 2, let m a be the mass of S + a . The contribution of the diagrams with charged scalars to a µ is given by The three light neutrinos χ 1,2,3 have masses much smaller than m µ , hence these masses are negligible. For those neutrinos the matrix elements S µi ∼ 10 −6 are also negligible. One then has where δ a = m 2 µ /m 2 a . For m a ∼ 100 GeV one thus has a µ (φ + ) light neutrinos ∼ −10 −9 |y 1 | 2 , which is not very significant. The three heavy neutrinos χ 4,5,6 have masses comparable to those of the charged scalars. For those neutrinos the R µi ∼ 10 −6 . One then has where For Yukawa couplings of order 1, and for heavy neutrinos with masses of order 1 TeV, one obtains a µ (φ + ) heavy neutrinos ∼ 10 −10 , which is smaller than the experimental error in equation (20) and hence irrelevant. So one concludes that the largest non-standard contribution to a µ in our model is in principle a µ (φ 0 ), which is proportional to |y 1 | 2 and may be as large as 10 −7 if |y 1 | 2 ∼ 1. One must invoke either a small Yukawa coupling y 1 or cancellations between the contributions of the various neutral scalars for our model not to give too large a contribution to a µ . One may also view (24) as a possible way to explain the discrepancy (20).

4.2
The decay µ − → e − e + e − In our model, there are also lepton-flavour-changing processes due to the soft breaking of the family lepton numbers [20]. The process most likely to be observable is in general [20] µ − → e − e + e − , 7 which is mediated by µ − → e − S 0 b * if the Yukawa couplings are large. Using the formulas in [20] we obtain the approximate upper bound where µ being an arbitrary mass parameter on which F finally does not depend. Clearly, if y 1 , y 2 and F are all of order one, then BR (µ − → e − e + e − ) will be much too large in our model, since experimentally that branching ratio cannot be larger than 1.0 ×10 −12 at 90% confidence level [18]. One possibility to avoid this problem is by making |y 1 | ∼ 0.01, which is possible as we have seen in section 2, and would have the further advantage of also suppressing a µ (φ 0 ), while keeping y 2 at order one in order not to suppress σ (e − e − → H − H − ).
Another possibility is to assume that the mass spectrum of the light neutrinos is almost degenerate, which in turn renders the mass spectrum of the heavy neutrinos almost degenerate too. With this condition and assuming U e3 to be exactly zero, we estimate Taking, for instance, m 1 = 0.3 eV, one obtains |F | 2 ∼ 10 −7 , which is sufficient to make equation (33) compatible with the experimental bound. Therefore, our model may be compatible with the present experimental bound on BR (µ − → e − e + e − ), if the mass spectrum of the light neutrinos is almost degenerate or the Yukawa coupling y 1 is of order 0.01. On the other hand, the Yukawa coupling y 2 may perfectly well be of order one.

Conclusions
We have started in this Letter with the trivial observations that, in the seesaw mechanism, a mass scale of 1 MeV in the neutrino Dirac mass matrix corresponds to right-handed neutrinos in the TeV range, and that the scale 1 MeV might be provided by the electron mass. We have then constructed a simple model, which extends the SM with three righthanded neutrino singlets and four Higgs doublets, in which that observation is realized. Our model has the following properties: 1. Each charged lepton α has a corresponding Higgs doublet φ α which, through its VEV v α , generates the charged-lepton mass m α = |y 1 v α |.
2. The Dirac mass matrix M D of the neutrinos is generated by another Higgs doublet, φ 0 , whose VEV v 0 is induced by the VEV v e such that v 0 ∼ v e or smaller, hence v 0 ∝ m e .
3. In the appropriate weak basis, the mass matrix of the charged leptons is diagonal while M D is proportional to the unit matrix; this yields the simple relation M j ∝ m 2 e / m j between the masses M j of the heavy Majorana neutrinos and the masses m j of the light Majorana neutrinos. 4. Moreover, the diagonalization matrix of the light-neutrino mass matrix-which is just the lepton mixing matrix-and the diagonalization matrix of the heavy-neutrino mass matrix are complex conjugate of each other.
5. The mixing between light and heavy neutrinos is small, of order 10 −6 .
The last property prevents heavy-neutrino production in pp and pp colliders. However, at an e − e − collider one might test the mechanism of the model by searching for the process e − e − → H − H − , where H − is a charged scalar; this process is somehow a high-energy analogue of neutrinoless ββ decay. The charged scalar would mainly decay into a light neutrino and the electron, if the heavy neutrinos are heavier than the scalar. Thus the signal would amount to e − e − plus missing momentum. At an e + e − collider one should, of course, look for e + e − → H + H − . If the Yukawa coupling y 1 of our model is of order one, then the anomalous magnetic moment of the muon may be too large, and the branching ratio of the flavour-changing process µ − → e − e + e − as well; in this case the suppression mechanisms discussed in section 4 have to be invoked. The simplest way to make our model compatible with experimental constraints is by choosing |y 1 | ∼ 0.01. However, this does not impede the process e − e − → H − H − , which proceeds through a different Yukawa coupling y 2 .