A Model of Neutrino and Higgs Physics at the Electroweak Scale

We present and explore the Higgs physics of a model that in addition to the Standard Model fields includes a lepton number violating singlet scalar field. Based on the fact that the only experimental data we have so far for physics beyond the Standard Model is that of neutrino physics, we impose a constraint for any addition not to introduce new higher scales. As such, we introduce right-handed neutrinos with an Electroweak Scale mass. We study the Higgs decay $H \to \nu \nu$ and show that it leads to different signatures compared to those in the Standard Model, making it possible to detect them and to probe the nature of their couplings.


I. INTRODUCTION
Neutrino physics has received a tremendous amount of experimental input in the last decade [1,2,3,4,5,6,7]. Neutrino oscillations are now completely determined and thus neutrinos are massive. On the theoretical side, the origin of neutrino masses and their observed patterns (for the neutrino mass squared differences) as well as the mixing angles still represent a mystery [8]. There are some ideas that have been widely used in order to explore the situation, like the Zee model [9] or the see-saw mechanism [10,11] in its several incarnations [12], but we are far from a profound understanding. Most of the actual realizations of these mechanisms postpone much of the desired knowledge to very high, experimentally unaccessible, energy scales. Concretely, since the introduction of Right-handed (RH) neutrinos seem to be the obvious addition needed in order to write a Dirac mass for the neutrinos, and the seesaw can be used to explain the smallness of the neutrino mass scale, most models assume their existence with a mass scale typically of size ∼ 10 13−16 GeV [11,12].
In this paper we adhere to the idea that our current (experimental) knowledge of particle physics should be explored by a "truly minimal" extension of the Standard Model (SM).
In this tenor we consider the possibility of having only one scale associated with all the high energy physics (HEP) phenomena. Since the SM is consistent with all data so far (modulo neutrino masses), we propose a minimal extension of the SM where new phenomena associated to neutrino physics can also be explained by physics at the Electroweak (EW) scale which we take to be in the range from 10 GeV to 1 TeV (similar approaches can be found in [13,14,15,16]). Thus, we assume • SM particle content and gauge interactions.
• Existence of three RH neutrinos with a mass scale of EW size.
• Global U(1) L spontaneously (and/or explicitly) broken at the EW scale by a single complex scalar field.
• All mass scales come from spontaneous symmetry breaking (SSB). This leads to a Higgs sector that includes a Higgs SU(2) L doublet field Φ with hypercharge 1 (i.e. the usual SM Higgs doublet) and a SM singlet complex scalar field η with lepton number sector of the SM, where the hierarchy (naturalness) problem resides. By enlarging the SM to explain the neutrino experimental results, we can get a richer spectrum of signals for Higgs physics and it is expected that once the LHC starts, it will allow us to test some of the theoretical frameworks created thus far. In any case, in order to fully probe whether the Higgs bosons have "Dirac" and/or "Majorana" couplings, we might have to wait until we reach a "precision Higgs era" at a linear collider [17].

II. THE MODEL
Taking into account the previous assumptions it is straightforward to write the Lagrangian. The relevant terms for Higgs and neutrino physics are with where N R represents the RH neutrinos, ψ c = Cγ 0 ψ * and ψ c R ≡ (ψ R ) c = P L ψ c has left-handed chirality. The potential is given by Note that the fifth term in the potential breaks explicitly the U(1) associated to lepton number. This is going to be relevant when we consider the Majoron later in the paper.
Assuming that the scalar fields acquire vacuum expectation values (vevs) in such a way that Φ and η are responsible for EW and global U(1) L symmetry breaking respectively, and using the notation where v/ √ 2 and u/ √ 2 are the vevs of Φ and η respectively, we obtain the following mini-mization conditions: We can also obtain the mass matrix for the scalar fields and it is given by where r ≡ −κ/u. The mass for the σ (Majoron) field is Note that, as expected, M 2 σ is proportional to the parameter κ associated to the explicit breaking of the U(1) L symmetry.
We are working under the assumption that the explicit breaking is very small, i.e. κ << EW scale. This is why we are minimizing the potential with respect to η thus assuming it does break the symmetry spontaneously. Furthermore we expect the SSB generated by the vev of η = u to be of EW scale size and so we work under the assumption r ≡ −κ/u << 1.
For example, taking −κ ∼ KeV one obtains r ∼ 10 −7−9 which then leads to a Majoron mass of hundreds of KeV.
From Eq.(7) we see that it is useful to define the mass eigenstates Using these definitions to rewrite Eq.(2) we obtain We now make some comments regarding neutrino mass scales. Since we are interested in RH neutrinos at the EW scale, we take their masses to be in that scale, i.e. anywhere from a few to hundreds of GeV. The Dirac part on the other hand will be constrained from the seesaw. Writing the neutrino mass matrix as where (m D ) αi = y αi v/ The mass eigenstates are denoted by ν 1 and ν 2 and are such that where θ = m D /m 2 ≈ 10 −(5−6) .
The relevant terms in the Lagrangian become where y * ν = y ν and Z ≡ Z 11 . As discussed in the introduction we are interested in exploring the Higgs decays to neutrinos and their signatures in this model. The possible decay modes involving the Majoron and its relation to Dark Matter will be considered elsewhere. Then, using Eq. (13) we compute the following decay widths 1 :

III. NUMERICAL RESULTS
We have computed the branching ratios for the Higgs decays and the results are presented in Figure 1. In each plot we have included the results for three values of cos α (0.1, 0.5 and 0.9). The three graphs correspond to the values of m 2 = 10, 60 and 100 GeV respectively.
Only the dominant contributions are shown for clarity, i.e. h → ν 2ν2 , ZZ, W W, bb and ττ .
It is interesting to note that for the whole range where it is possible, the decay h → ν 2ν2 dominates in all three cases for small cos α and it is still relevant for large cos α. This is a clear distinctive signature of our model.
In order to study the specific signatures that would be observed in this scenario, we consider the ν 2 decays. In Table I we present the possible signatures of these decays.
h → ν 1 ν 1 --- Since we are interested in a Higgs mass in the natural window of 100 − 200 GeV, and in neutrino masses such that they can appear in Higgs decays, we will consider neutrino masses of order 10 − 100 GeV, therefore we need to consider the 2 and 3-body decays ν 2 → V + l and ν 2 → ν 1 + V * (→ ff ′ ), where V * = W * , Z * One can also evaluate the branching ratios for the neutrino radiative decay, but since this is a loop-process, it is quite suppressed unless the mass differences among the right and left-handed neutrinos are very small. We have not included such process in this work.
Consider the process in Figure 2. Its decay width is given by where For the 2-body decays the result is We have evaluated the branching ratios for these processes and they are presented in table II. We show the results for m 2 = 100 GeV as the results are similar in all the m 2 range considered in this paper. We find that the dominant contributions are the ones associated to the W * decay process.

IV. DISCUSSION AND CONCLUSIONS
A longstanding question in neutrino physics has been to determine whether neutrino masses are of Dirac or Majorana type, which in turn motivates the terminology used in m 2 (GeV ) Z ν W l ν l + l − ν ν ν ν q uqu ν q dqd l ± l ± ν l ± qq ′  calling the neutrinos either Dirac or Majorana. As the Higgs mechanism employed in this paper is at the root of the origin of both types of masses, we find it reasonable to ask the same question for the Higgs couplings, namely, we would like to determine wether the Higgs couplings are dominated by its Dirac or Majorana components.
In fact, the interaction eigenstates that appear in our model, Φ 0 and η 0 , do have well defined couplings to neutrinos: of Dirac type the former and Majorana type the latter.
Although one may think that such a question is academic, we argue that it is not the case, and that it will be possible to study the experimental signatures that would distinguish among both types of couplings at coming colliders.
At the base of our discussion is the fact that the Dirac couplings φ 0ν l ν R , involve both types of chiralities (L and R), whereas the Majorana one η 0 ν c R ν R , involves only one chirality. Therefore, in a Higgs decay of Dirac type, one would have a fermion of a given chirality and an anti-fermion with the opposite chirality, while in the Majorana case, the Higgs decay would involve a fermion pair of like chiralities. In our model, as the decays h → ν 1 ν 1 would escape detection, while the decay h → ν 1 ν 2 will only have one detectable neutrino, which does not allow the possibility to correlate chiralities, we are left only with the decay: h → ν 2 ν 2 . Let us follow the decay chain produced after the neutrino decays into a lepton and a pair of jets, namely ν 2 → l + qq ′ . It is then possible to a have a pair of same-sign charged leptons, plus jets, which should help in order to discriminate against backgrounds. Furthermore, the charged leptons will inherit the chiralities of the neutrinos, and its measurement will allow to test the Higgs couplings. A detailed simulation study is needed, but this is beyond the scope of the present paper.
We close this discussion with a few comments on the possibility to measure the Dirac or Majorana coupling of the Higgs bosons in models with a richer Higgs spectrum. For instance we could have models with Higgs triplets that will include double-charged Higgs states δ ++ , which then couple to a lepton pair, therefore violating lepton number. As such state would decay into e + e + or µ + µ + , it would then be possible to measure the chiralities of those light leptons appearing in the final state, and therefore to test the Dirac or Majorana nature of their couplings.
It is quite interesting that the Higgs sector presented in this paper can lead to substantial modification of the signatures of the Higgs bosons. Although these signatures seem quite different from those expected in the SM, they represent the kind of variations that one would have when new physics beyond the SM is included, which in our case is well motivated by the plethora of recent neutrino physics experiments. As the LHC is expected to start operation very soon, it is very important to have an open mind regarding possible variations of the signals expected from the SM Higgs.