# A realistic model of neutrino masses with a large neutrinoless double beta decay rate

###### Abstract

The minimal Standard Model extension with the Weinberg operator does accommodate the observed neutrino masses and mixing, but predicts a neutrinoless double beta () decay rate proportional to the effective electron neutrino mass, which can be then arbitrarily small within present experimental limits. However, in general decay can have an independent origin and be near its present experimental bound; whereas neutrino masses are generated radiatively, contributing negligibly to decay. We provide a realization of this scenario in a simple, well defined and testable model, with potential LHC effects and calculable neutrino masses, whose two-loop expression we derive exactly. We also discuss the connection of this model to others that have appeared in the literature, and remark on the significant differences that result from various choices of quantum number assignments and symmetry assumptions. In this type of models lepton flavor violating rates are also preferred to be relatively large, at the reach of foreseen experiments. Interestingly enough, in our model this stands for a large third mixing angle, , when is required to lie below its present experimental limit.

^{†}

^{†}preprint: CAFPE-165/11, UG-FT-295/11, FTUV-11-1128, IFIC/11-65, UCRHEP-T512

## I Introduction

Neutrino oscillations are the only new physics (NP)
beyond the minimal Standard Model (SM) observed
to date Nakamura:2010zzi () (for recent reviews see for instance
GonzalezGarcia:2007ib (); Mohapatra:2005wg (); Altarelli:1999gu ()).
They can be fully explained introducing rather small neutrino
masses and the corresponding (unitary) charged current
mixing matrix Pontecorvo:1967fh (); Maki:1962mu ().
The observed pattern of neutrino masses can be implemented,
however, in two quite different ways depending on the Dirac or
Majorana character of the neutrinos. In the first
case the SM is extended
by adding three SM singlets, , to provide Dirac masses
to the SM neutrinos, , through small Yukawa couplings.
Alternatively, we can consider extending the SM by adding
the only invariant dimension 5
(Weinberg) operator that can be written using the SM
field content Weinberg:1979sa () ^{1}^{1}1 and are the SM Higgs and
lepton doublets and ,
.,
.
In this case the SM
neutrinos acquire Majorana masses after electroweak symmetry breaking,
,
and are inversely proportional to , the NP
scale associated with . The small neutrino
masses then require that the coefficients of this operator be small, either due
to a very large NP scale or suppressed
dimensionless couplings.

Both alternatives (that light neutrinos are Dirac or
Majorana), are viable and indistinguishable if is very large,
except for the possible observation of lepton number violation (LNV)
in neutrinoless double beta () decay Furry:1939qr ()
^{2}^{2}2We shall not discuss the implications of requiring enough leptogenesis
to account for the observed baryogenesis Fukugita:1986hr () (for recent reviews
see Davidson:2008bu (); Branco:2011zb ()). (for a recent review see Vergados:2002pv ()).
Indeed, the SM extension with Dirac neutrino masses preserves lepton
number (LN), and hence decay is forbidden.
Whereas Majorana masses carry LN equal two, as does the
Weinberg operator, and this leads to a non-zero width.
Thus, the observation of decay would strongly favor Majorana
neutrino masses Schechter:1981bd (); hence the prime relevance of this type of experiments
(see Barabash:2011fg (); Avignone:2007fu () for recent reviews).

All this, however, relies on the assumption that both such minimal SM extensions describe the dominating NP effects up to very high scales. We will argue, however, that LNV and neutrino masses may be due to NP near the electroweak scale, in which case a much richer set of possibilities can be realized. The main point we wish to emphasize is that although both decay and Majorana neutrino masses do violate LN, they need not be both directly related to the Weinberg operator , unlike the above minimal SM extension consisting only of this dimension 5 operator. For instance, the leading LNV effects in decay could be mediated by an operator involving only right-handed (RH) electrons, such as, for example, which is the lowest-order LNV operator with two RH leptons and invariant under the SM gauge transformations. In this case (with the associated Majorana masses) is generated radiatively by at two loops, where the charged leptons suffer a chirality change through mass insertions and are transformed into neutrinos through the exchange of a boson. This results in a further suppression by two charged lepton masses divided by two powers of the NP mass scale, which we do assume to be close to the electroweak scale. These radiatively-generated Majorana masses will produce the usual contribution to decay, which, however, will be negligible compared to the one.

In a companion paper delAguila:2011zz () we classify the different ways of generating decay and light neutrino masses by the addition of higher order effective operators. This has been studied in the literature Babu:2001ex (); Choi:2002bb (); Engel:2003yr (); deGouvea:2007xp (), but mostly for operators involving fermions and scalars; we will concentrate instead on operators involving gauge bosons but not quarks (thus, for example, excluding ab initio models with heavy leptoquarks from our analysis). Here we will instead provide a realistic testable model realizing the above scenario, where (i) LN is broken at the electroweak scale; (ii) decay into two RH electrons has a rate of the order of its experimental limit, through the tree-level exchange of new scalars; and (iii) it contains finite, and therefore calculable, neutrino masses. Despite a relatively small number of parameters this model can also accommodate the observed pattern of neutrino masses and mixings, which are generated at two-loop order (and whose contribution to decay is in this case negligible). This model is related to others that have been discussed in the literature Chen:2006vn (); Chen:2007dc (), the differences are crucial, and essential in maintaining the 3 features just mentioned; we discuss these points in detail below.

There are many SM extensions where decay
receive new contributions besides those from light Majorana neutrino masses
(see Mohapatra:1998rq () for a general overview).
The simplest scenario assumes the presence of heavy
Majorana neutrinos whose exchange generates
new contributions to decay, similar to
the ones generated by the light neutrinos.
(See for recent work Atre:2009rg (); Ibarra:2010xw (); Mitra:2011qr (); Blennow:2010th ().)
Other extensions with many more new particles
have also been studied; such as
left-right (LR) models Pati:1974yy (); Mohapatra:1974gc (); Senjanovic:1975rk ()
and supersymmetric extensions (see, for instance, for a review Nakamura:2010zzi (), and
references there in).
In such models there are several contributions
to the decay amplitude, any of which can dominate
depending on the region of parameter space being considered.
We are interested, however, in identifying the minimal SM
extensions leading to the largest possible contributions to decay
and containing no independent neutrino masses.
This means that, as argued above,
light neutrino masses only result from the unavoidable contributions
mediated by the LNV operators generating
decay; so that the
effective Lagrangian approach provides the proper
language for classifying such scenarios delAguila:2011zz ().
Here, we are only concerned with giving a specific example
of the case
where and neutrino masses are
generated by the same underlying physics but
at different orders in perturbation theory,
with the former appearing at tree level, while the latter
only at two loops
^{3}^{3}3LNV effective operators including quarks also generate neutrino masses
but in general at higher loop order, in fact too high in some cases to explain the observed
spectrum Duerr:2011zd (). Although such operators
are not considered in our analysis, we will comment on them further when
discussing our general set up
in detail delAguila:2011zz ()..
Simple models with finite neutrino masses at two loops have
been often discussed in the literature Zee:1985id (); Babu:1988ki (),
although not necessarily related to NP inducing decay
as it is the case here.

It is worth emphasizing again that the model we study is one particularly simple example of a class of models realizing the above scenario, and that all such models share many phenomenological implications. In our case the model contains a discrete symmetry, which is spontaneously broken. This model has the virtue of providing a direct analysis of the symmetries and scales, however, it also has a domain-wall problem Zeldovich:1974uw (); Vilenkin:1984ib (). One way of dealing with this problem is to allow the discrete symmetry to remain unbroken, but at the price of generating both neutrino masses and the amplitude at a higher loop order; a possibility we will not pursue. We instead discuss related but somewhat more involved models that avoid the domain wall problem while retaining the same low-energy phenomenology. We will restrict ourselves to the most relevant region of parameter space where the decay rate lies within the reach of the next round of experiments. With this assumption, together with the constraints from lepton flavor violation (LFV) processes such as , and with the requirement of perturbative unitarity (or if preferred, naturalness of perturbation theory), the model predicts that the neutrino masses obey a normal hierarchy, that the lightest neutrino mass lies in the range , and that the third mixing angle in the Pontecorvo-Maki-Nakagawa-Sakata mixing matrix Pontecorvo:1967fh (); Maki:1962mu () obeys . Value which lies well within the sensitivity of ongoing neutrino oscillation experiments. Besides, the new (charged) scalar masses can be within the LHC reach, and various LFV processes are predicted to have rates that will be probed when present precision is improved by the next generation of experiments.

In next section we present our model; few details on the scalar spectrum and couplings are also worked out, and its relation to other models is briefly discussed. The reader mainly interested in the phenomenological implications of the model can go directly to Section III, where we evaluate the rate for decay. The requirement that this process can be observed in the next round of experiments, together with perturbativity, translates into upper bounds on the masses of the extra scalars. These, in turn, result in limits on their couplings if present bounds on LFV processes are to be fulfilled, as we show in Section IV. We calculate the neutrino masses in Section V, where we show that the model can accommodate the observed pattern of neutrino masses and mixings, though it predicts a somewhat small electron neutrino effective mass . Finally, the prospects for the discovery of the extra scalars at LHC are considered in Section VI. In connection with this it must be noted that models with extra scalars may allow for a would-be Higgs boson more elusive to LHC searches, for it can have further decay channels open (see for instance Shrock:1982kd (); Bertolini:1988ma (); Aparici:2009fh ()). Conclusions are drawn in last section. In three Appendices we collect some technical details.

## Ii A model with lepton number softly broken

This model only extends the SM Higgs sector, in order to allow for scalar couplings to lepton bilinears with non-zero LN. More precisely, as we look for separating the origin of neutrino masses from the mechanism mediating decay, we only introduce new scalar couplings to RH charged leptons, ensuring they are the only final state produced in tree-level decay. A simple way to achieve this is to enlarge the Higgs sector including, besides the SM scalar doublet of hypercharge 1/2, a complex scalar singlet of hypercharge 2, a real (neutral) scalar singlet , and an electroweak triplet of hypercharge 1. We also impose a symmetry, under which all SM particles and are even while both and are odd, which forbids their coupling to lepton bilinears. Thus, the Yukawa Lagrangian reads

(1) |

where we can assume to be a diagonal matrix with positive eigenvalues, and a complex symmetric matrix with three of its phases unphysical.

The most general Higgs potential consistent with the symmetries
is ^{4}^{4}4Terms like ,
or can easily be related to
the terms already included by using the fact that for two arbitrary
traceless matrices and , .
Other possible terms are forbidden by the discrete symmetry.

(2) | |||||

where by rephasing the fields we can always choose and real of either sign. For convenience, we will take positive and negative. In terms of charge eigenstates, the fields and are written

(3) |

The singlet is introduced to preserve the discrete symmetry that forbids the scalar triplet from coupling to leptons. This symmetry is broken spontaneously by the VEVs and .

For the phenomenological discussion it is important to note that in the limit of vanishing , lepton number can be defined to only act on ; this is enough to protect neutrinos from getting a Majorana mass. While the lepton number, which would forbid decay (into two ), is explicitly broken in the scalar potential due to the presence of the and terms. Therefore, the decay amplitude will be proportional to all three couplings: in Eq. (1) and and in Eq.(2); whereas neutrino masses will also depend on .

This is one of a class of models with the same low energy physics.
In order to understand the common LNV features it is convenient
to consider the above theory, but with complex,
and the modifications needed to insure a
real Lagrangian.
In this case LN is exactly conserved; if leptons are assigned LN ,
carries LN equal to , while
and both carry LN equal to ,
so that type II see-saw Yukawa couplings
are forbidden.
The vertex ,
however, is allowed and generated at two loops,
providing light neutrinos a finite mass
(which we calculate exactly)
after spontaneous symmetry breaking proportional to
.
Obviously, this model has a Majoron Chikashige:1980ui () because LN, which is a global abelian
symmetry, is spontaneously broken. One can
then consider the region of parameter space where
such interesting models are phenomenologically viable ^{5}^{5}5
In this model the Majoron will be mainly a singlet Chikashige:1980ui ().
In which case its couplings to ordinary matter are small and then little constrained, while its
coupling to the Higgs boson
is essentially free. Thus, singlet Majorons can result in invisible Higgs decays
(see Ghosh:2011qc () for a recent example). Moreover, they can have interesting implications
in astrophysics and cosmology because they can substantially affect the cooling of supernovas Choi:1989hi () and the
neutrino relic abundance, due to the possibility of neutrino decaying Chikashige:1980qk () or annihilating Choi:1991aa ()
into Majorons. They can be even used as a dark matter candidate (see for instance Gu:2010ys ()) if massive (pseudo-Majorons).,
or promote the symmetry to
a local one by gauging, for instance, baryon minus lepton number (BL).
Although, this is an interesting possibility, too, with renewed experimental
interest (see, for instance, for a review Basso:2011hn ()),
it requires adding RH neutrinos
that provide
a new potential source of
light neutrino masses; this lies outside the goal
of the present investigation that is centered on theories where
neutrino masses are generated by the LNV effective operator accounting
for decay.

For this reason we have assumed to be real,
breaking LN explicitly but leaving as a remnant an exact
symmetry.
The general features of the theory
remain, though the Majoron does not appear and LN is reduced
to the above symmetry ^{6}^{6}6One can also assign an odd parity to the leptons,
in which case this discrete symmetry equals ..
The spontaneous breaking of
this discrete symmetry will generate domain walls. This poses no
problem provided is sufficiently large (with
correspondingly small while keeping
their product fixed)
guaranteeing the formation of such defects before the
inflationary epoch.
We have assumed
instead that is not too large because we prefer
avoid requiring unnaturally small;
we may then avoid a domain-wall problem by adding soft-breaking terms,
such as to the potential (such a modification would not require the introduction
of tree-level Yukawa interactions violating the discrete
symmetry). We do not pursue this discussion because within this class
of models there is an even simpler one, with the same
neutrino physics at low energy, and none of these
potential drawbacks.
It is the same model presented above, but with
replaced by its vacuum expectation value, ; up to coupling constant redefinition this yields
the same potential as Eq. (2) without the
terms containing , except for the last term
that becomes ,
with

(4) |

of the order of a TeV.
Obviously, the resulting renormalizable Lagrangian has the same quantum
behaviour than ours; in particular,
neutrino masses are finite and generated at two loop order
and can be obtained from our results by eliminating
using (4).
We will
prefer to discuss the model including the real scalar field
because the perturbative and symmetry analyses appear to be
more transparent to us.
In the minimal model with the SM addition of only and
, and the scalar potential terms relating their LN charges
and
the LN of the field is not
well defined since the first term requires it to be 1 while the second 0 (though never 2);
as a consequence the effective vertex
,
is finite and generated radiatively ^{7}^{7}7Note that in the theory
containing an extra
real scalar , one could also reason
differently to justify this result. Indeed, we could assign
LN equal to 0; then the
quartic term fixes the
LN also equal to 0, while the trilinear term
breaks LN softly.
In any case the discrete symmetry guarantees that
the neutrino masses stay finite..
Despite the parallels of this discussion with models implementing a
type II see-saw mechanism (which contain a tree-level coupling) there is an important
difference, namely, that in such theories both the expression
for the neutrino masses and the decay amplitude
are linear in ; in contrast the corresponding results in
our model are proportional to (see Eqs. (25)
and (33) below).

The extension of the SM model we consider is related to the one presented in Refs. Chen:2006vn (); Chen:2007dc () as far as particle content is concerned (we differ by adding the singlet ). However, the symmetries and quantum number assignments are different, which proves a crucial difference. Had we included the hard term as was done in the above references, should have been assigned LN ; this would necessitate also including the tree-level coupling that would lead to the usual type II see-saw scenario. In particular it would have been inconsistent to assign LN zero to or to arbitrarily exclude this Yukawa coupling from the Lagrangian, for its coefficient would receive divergent radiative corrections; in consequence the neutrino masses are not calculable. If one requires to have LN equal to , as was done in these publications, the quartic coupling must be absent; but then LN remains unbroken and the light neutrino masses must vanish to all orders which is again inconsistent with the results presented there.

### ii.1 The scalar spectrum

The requirement on the scalar potential of being bounded from below
is fulfilled restricting the quartic couplings in Eq. (2)
adequately; these conditions include ^{8}^{8}8Notice that the term
is always positive.

(5) |

We have also checked that there is a non-trivial minimum on which the scalar neutral components acquire non-zero expectation values: , , , with

(6) |

In Appendix A we comment on the experimental limit on the scalar triplet VEV , which we will find to be of the order of few GeV; to be conservative we will assume , this satisfies where (derived from Eq. (45) and the experimental limit on the parameter, Nakamura:2010zzi ()). In this approximation the minimization conditions can be easily solved:

(7) |

Notice that the phase choice is consistent with being real and positive. We set the and mass squared terms in the Higgs potential negative to favour the development of such a minimum. Though we choose the mass squared term positive, this field also acquires a small VEV induced by the doublet and singlet VEVs, similar to the case observed in see-saw of type II models Konetschny:1977bn (); Cheng:1980qt (); Schechter:1980gr ().

The scalar masses can be obtained by substituting Eq. (6) in the potential. Using the exact minimization conditions to eliminate and in favour of the VEVs, the mass terms for the charged scalars can be written

(8) | |||||

(9) | |||||

(10) |

Analogously for the neutral sector

(11) | |||||

(12) | |||||

(13) |

All eigenvalues of these mass matrices must be positive (except for the would-be Goldstone bosons providing the longitudinal vector boson degrees of freedom) in order to guarantee that the solution to the minimization conditions corresponds to a local minimum. This sets further constraints on the model parameters, which can be satisfied rather easily, especially in the limit .

Thus, we are left with two massive doubly-charged scalars ,

(14) |

with

(15) |

and only one massive, mainly triplet, singly-charged scalar ,

(16) |

Similarly, there is a neutral scalar with imaginary components,

(17) |

There are also three neutral scalars along the real components . We will denote these mass eigenfields by (mainly doublet), (mainly triplet) and (mainly singlet). They are obtained rotating the current fields, what introduces other three mixing angles. Notice that in the limit , we have and , with all mixings small.

### ii.2 Some scalar couplings of phenomenological interest

Once the quadratic terms of the Lagrangian are diagonalized we can read the interactions for the mass eigenfields. In the following we will need the scalar coupling to RH electrons

(18) |

and the corresponding doubly-charged scalar couplings to gauge bosons

(19) |

as well as their trilinear couplings

(20) |

which can be also expressed in terms of the mass eigenfields using Eqs. (14–16), and the corresponding VEVs in Eq. (6). Finally, the Yukawa coupling changing charge and chirality writes

(21) |

where is the would-be Goldstone boson providing the third component to the .

## Iii Neutrinoless double beta decay

Both doubly-charged scalars have components along the singlet and the triplet . Therefore, they (respectively) contain couplings to RH electrons and to ’s, generating an effective vertex that mediates decay. In this section we calculate this contribution to decay and obtain the explicit constraints on the model parameters derived from the assumption that decay will be observed in the next round of experiments.

Assuming that and integrating out the heavy and modes we find, after a straightforward calculation, that the effective Lagrangian contains the term

(22) |

as announced in the Introduction, and discussed in the companion paper
delAguila:2011zz (). One can better understand the origin
of this LNV interaction by considering the contribution of the dominant diagram
in Fig. 1,
where the different couplings and VEVs involved are displayed explicitly.
The corresponding vertex at low energy ()
can be written as ^{9}^{9}9
It must be emphasized that the Lagrangian
violating LN by 2 units for decay
is proportional to .
This must be compared with the linear dependence in obtained when
the LN assignment is 2.

(23) |

where we have summed up all possible mass insertions in the internal propagator, and used Eq. (15). This expression coincides with Eq. (22) when the scalar doublets develop a VEV in the limit of large (), i.e., and , .

Let us particularize to the case , relevant for , and further integrate the two ’s to obtain the appropriate 6-fermion contact interaction

(24) |

where denotes the proton mass and

(25) |

This type of interactions has
been already considered in the literature Pas:2000vn (), where
limits from the most sensitive experiments at that moment were derived.
Since they have not been substantially improved, we will directly use
the results in Pas:2000vn () from the Heidelberg-Moscow
experiment corresponding to which yields
^{10}^{10}10There is a misprint in Ref. Pas:2000vn (). We thank the authors of this reference
for providing us with the correct limit on .
at 90% C.L.
On the other hand, experiments in the near future will be sensitive
to lifetimes of the order of years Barabash:2011fg (), i.e.
a reduction factor on the coupling of roughly . Then, in order to
decay be observable in the next
round of experiments but still satisfy the present limits, we must require

(26) |

where is the proton mass and the inequality with the superscript "" corresponds to the requirement that decay will be observed in the next generation of experiments Barabash:2011fg (). While the inequality without the superscript stands for the present experimental limit at the 90% C.L.. The conditions in Eq. (26) will prove rather restrictive because its range of variation is relatively narrow. In fact, reducing the lower limit will appreciable enlarge the allowed parameter region as discussed below.

Thus, the above lower limit together with the requirement of perturbative unitarity (indicated by the "Pert" superscript in the inequalities below) or naturality, which bounds from above the product of couplings and VEVs , translate into an upper limit on the product of the scalar masses . These, however, are not precisely established because the perturbative bounds are in fact estimates that vary with the approach. In Fig. 2 we show the allowed region, where and in the limit of small mixing angle , for (blue, darker) and 5 (orange, lighter) GeV (see Appendix A), respectively, assuming perturbative unitarity (left) and a maximum LN breaking scale (right).

In the first case (see Eqs. (47) and (48) in Appendix B)

(27) |

whereas in the second one

(28) |

It must be noticed that all (pseudo-)observables violating LN are proportional to . Hence, an increase in can be traded by the corresponding increase in , and vice-versa. So, the orange areas in Fig. 2 can be also interpreted as the allowed regions for GeV and (left) and TeV (right). One may wonder at this point why we choose the bound of 20 TeV for in Eq. (28); or equivalently, what is the effect of varying such a value. The answer is simple. The blue, darker region in the right panel of Fig. 2 disappears for TeV, which only reflects the narrowness of the range allowed by Eq. (26), as required by our main working assumption that decay will be observed in the next round of experiments. The allowed regions in Fig. 2 are appreciably enlarged by reducing the lower limit in this equation. This can also be achieved by further increasing .

These areas are also bounded from below due to the non-observation of doubly-charged scalars; we can then assume GeV, as discussed in Section VI. However, the enclosed areas in Fig. 2 are further reduced by a more stringent and subtle constraint. As we shall discuss below, bounds on LFV processes (see section IV) like banish to large values if the corresponding coupling product is sizeable, which is required because neutrino masses are proportional to (see section V), and and must be large in order to accommodate the observed neutrino spectrum. Moreover, both and enter in the two-loop integrals generating neutrino masses, but these tend to zero in the limit . As a result, both scalar masses are constrained, but differently, by the bound. The regions in Fig. 2 satisfy all experimental restrictions, including the upper bound in Eq. (26). The LHC will further reduce the allowed regions, mainly in the case of large LN breaking scale . We also provide a “benchmark point”, denoted by a cross in the figures, where all constraints are satisfied, and which we will use as reference throughout the paper. As can be deduced from this Figure the parameter space is rather constrained in this simple model when we require that the values of couplings and scalar masses stay natural, but one can think of other models within this class of theories where these constraints are significantly relaxed (at the price of complicating the spectrum through the introduction of additional scalars).

## Iv Lepton flavor violation constraints

We will show in the next section that in order to obtain neutrino masses in agreement with experiment, the doubly-charged scalar Yukawa couplings and the ratio cannot be too small. In such a case some of the predicted LFV rates can become large enough to be at the verge of their present experimental bounds, especially for very rare processes like or . Thus, we can use LFV processes to further constrain the model, and perhaps to confirm or exclude it in the near future.

In this section we will briefly discuss the most restrictive process , whose tree-level amplitude is obtained by the single exchange of the doubly-charged scalar . The corresponding branching ratio equals

(29) |

where takes into account the fact that there may be two identical particles in the final state, as in our case, and

(30) |

(If , the effective mass since we, in practice, assume that are never very different.) Then, the current experimental limit on Nakamura:2010zzi () translates into

(31) |

which is mainly a constraint on because must be relatively large if decay has to be observable at the next generation of experiments.

Related processes provide weaker constraints. Thus, proceeds at one loop and is suppressed by the corresponding loop factor, and similarly for conversion in nuclei. The bounds from (muonium-antimuonium conversion) or muon-positron conversion, although tree-level processes, are also less restrictive (for a discussion of LFV processes mediated by doubly-charged scalar singlets in a similar model see Nebot:2007bc (); Raidal:1997hq ()). All these processes and the analogous ones involving leptons, as well as the corresponding (anomalous) magnetic moments will be discussed in detail with more generality elsewhere.

Here we shall be mainly interested in the interplay of a large decay rate and a realistic pattern of Majorana masses, and for this purpose it is sufficient to show the restrictions on these (pseudo-)observables in simple SM extensions as the one at hand, and indicate which further processes may be within the reach of new experiments. In our case, the most restrictive process besides is , which will be discussed below taking into account the neutrino mass requirements.

## V Neutrino mass generation and expectation

In the model under consideration LN is not conserved when the couplings , , and are non-vanishing. In this case there is no protection against the neutrinos acquiring Majorana masses , which will then be proportional to all four couplings, are finite, and appear at the two-loop level, as explained in Section II and shown by explicit calculation in Appendix C. (If any of these couplings vanishes a conserved lepton number remains after spontaneous symmetry breaking and the neutrino masses will vanish.) These masses are generated by the non-renormalizable interaction generated at two loops, when and develop VEVs; the corresponding coupling being . In contrast, the see-saw type II coupling violates the symmetry and is forbidden to all orders.

In Fig. 3 we draw one of the diagrams.

Defining the neutrino mass matrix as usual,

(32) |

we can write, taking (see Eq. (46) and Appendix A),

(33) |

where is the sum of the (rescaled) loop
integrals from the different graphs ^{11}^{11}11
Again we note that neutrino masses, which violate LN by 2 units, are
proportional to in our case.
In contrast the dependence is linear in the estimate for the model in
Refs. Chen:2006vn (); Chen:2007dc (), showing that must be assigned LN 2,
as in see-saw models of type II.
Besides, the neutrino masses are in fact infinite in that particular case, a point obscured because divergent diagrams were omitted.
. can be estimated in the mass
insertion approximation with . For instance,
in this limit one of the contributions, , corresponding to the diagram in
Fig. 3 gives (neglecting the lepton masses in the denominator
and assuming equal masses for the doubly and singly-charged triplet components)

(34) |

In this approximation the full is a dimensionless function of of order one, except for , in which case it tends to zero as . In contrast with which tends faster to zero, as , for vanishing . is also bounded from above, going to a constant of order for . A complete calculation taking into account the -mass, as well as the corrections and the new scalar mass scales, is presented in Appendix C. A reasonable approximation is to neglect higher effects and take all triplet masses equal ; in fact, in the physical limit , is mainly the doubly-charged triplet component, the singly-charged one, and the imaginary part of the neutral triplet component.

In our model the Yukawa couplings appear in the neutrino mass matrix, the decay amplitude and the amplitudes for the LFV processes, and this translates into rather stringent constraints on the allowed neutrino mass matrices (once one insists in dealing with a perturbative theory up to several tens of TeV). We now consider these constraints.

Assuming and , and taking , Eq. (33) gives . How large can it be in general ? Using perturbativity limits (27) and , we get

(35) |

where the upper limit is obtained by taking (see Fig. 2, left). Alternatively, we can translate the limits on decay in Eq. (26) into bounds on , but for large scalar masses this limit is less stringent than (35). In either case, is typically less than .

There is in addition a quite strong bound on from . Substituting Eq. (31) in the generic neutrino mass expression in Eq. (33) we find

(36) |

The constraint that a signal is seen in near-future decay experiments (left inequality in Eq. (26)) can then be used to eliminate ; in this way we obtain

(37) |

where in the second inequality we used , , the naturality limit on (Eq. (27)), and . If we had used the LN breaking scale TeV and the production limit on TeV, then .

The final result of this phenomenological discussion is that both, and , must be below , and this follows from requiring that (i) decay is at the reach of the next round of experiments, and (ii) that the theory is perturbative and free of unnatural fine tuning up to several tens of TeV. These limits could be somewhat relaxed: in the case by making doubly-charged scalar masses larger, and in the case by allowing for a smaller . However, this is at the price of generating some tension with the naturality constraint in the former case, and spoiling the possibility of observing decay induced by scalars in the near future in the latter one. There are additional but less severe bounds on the remaining from other LFV processes; we discuss them below, when presenting the plots for the relevant neutrino mass (pseudo-)observables satisfying present experimental restrictions.

### v.1 Prediction for the third neutrino mixing angle

The question now becomes whether it is possible to accommodate the observed spectrum of neutrino masses and mixing angles in this type of models once the above experimental constraints are imposed. In the following we will use the standard parameterization of the neutrino mass matrix Bilenky:1980cx (); Schechter:1980gr (); Bilenky:1987ty (); Nakamura:2010zzi () in terms of 3 mass parameters, 3 mixing angles and 3 phases:

(38) |

and

(39) |

where and . A global fit to neutrino oscillation data gives (see, for instance, Schwetz:2011zk ()) , , , , . Neutrino oscillations are not sensitive to the phases and , nor to a common mass scale which is conventionally chosen to be the lightest neutrino mass. , which appears multiplied by , is beyond present experimental sensitivity. The sign of is not presently known, and could be negative (known as inverted hierarchy), however, in this case and cannot be accommodated within our model; we will therefore consider only the normal hierarchy case . Finally, recent data on electron neutrino appearance at T2K Abe:2011sj () and Double Chooz dc-3393-v3:2011 () experiments point out to a mixing angle different from zero.

A possible way of identifying the allowed region in parameter space would be to first generate random values for masses, angles and phases within the 1 regions experimentally allowed in Eqs. (38) and (39), and obtain scatter plots for . Then, using Eq. (33) we can solve for , up to an overall factor , and then find the values of , which respect the constraints discussed in the previous sections. The potential problem we face is due to the specific form of the neutrino mass matrix, which contains (see Eq. (33)), and is therefore suppressed for the first generations due to the light charged-lepton mass factors. To compensate this may require to be too large to meet the bounds required by decay (Section III), LFV processes (Section IV) and perturbative unitarity (Appendix B). An alternative way to proceed is noticing that in practice (see (35) and (37)) we are asking if is consistent with neutrino oscillation data (a question also of general interest not only within the model under consideration). These additional constraints will hold only within a limited region of the allowed neutrino masses and mixing parameters, which then implies that the type of models under consideration gives rather clear predictions about some of these parameters.

In order to see how this comes about it is useful to go through a straightforward parameter counting exercise: