# Lower Mass Bound on theW′ Mass via Neutrinoless Double Beta Decay in a 3-3-1 Model

Authors: A. C. O. Santos; P. Vasconcelos
Journal: Advances in High Energy Physics
 (2018)

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## Abstract

The discovery of neutrino masses has raised the importance of studies in the context of neutrinoless double beta decay (0νββ), which constitutes a landmark for lepton number violation (LNV). The standard interpretation is that the light massive neutrinos that we observed oscillating in terrestrial experiments mediate double beta decay. In the minimal 3-3-1 model (3-3-1M), object of our study, there is an additional contribution that stems from the mixing between a new charged vector boson, W′, and the Standard Model W boson. Even after setting this mixing to be very small, we show that tight constraints arise from the nonobservation of 0νββ. Indeed, we derive bounds on the mass of the W′ gauge boson that might exceed those from collider probes and most importantly push the scale of symmetry breaking beyond its validity, leading to an exclusion bound for the minimal 3-3-1 model.


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## Body

## 1. Introduction

The Standard Model (SM) has thrived after a multitude of precision tests at low and high energy scales [1]. However, the existence of neutrino masses is one of the main motivations for physics beyond the SM [2]. If neutrinos are majorana particles, neutrinoless double beta decay (0νββ) should occur. Double beta decay is the transition of a nucleus with mass and atomic number A and Z to a nucleus with A and Z+2, accompanied by the emission of two electrons only. Its possible discovery will represent an irrefutable proof of lepton number violation, LNV. The standard diagram that leads to such lepton number violation process is exhibited in Figure 1.Figure 1
Canonical0νββ due to light massive neutrinos.At present, a lot of experiments using different isotopes and techniques are operating or under development in the search for neutrinoless double beta decay (see [3]). In this work we investigate the implications of the nonobservation of neutrinoless double beta decay in the context of 3-3-1 models [4–9]. 3-3-1 models are plausible extensions of the SM where fermions are placed in the fundamental or adjoint representation of SU(3). Generally these models while being consistent with colliders data [10, 11] nicely explain why we have three fermion generations [5, 12, 13], accommodate neutrino masses [14–16], and may feature several dark matter candidates [17, 18] to the dark matter problem. Moreover, they could also address low energy anomalies [19, 20] and might feature interesting astrophysical [21, 22] and collider phenomenology via the presence of exotic gauge bosons which are popularly used in many different contexts [23, 24]. See [25–31] for many other exciting phenomenological studies.In this work, we focus our attention on the minimal 3-3-1 model (3-3-1M), one of the possible versions of 3-3-1 models, in which the electric operator takes the formQ=T3+βT8+X considering Ti as λa/2 (Gell-Mann matrices with a=1,…,8), β=-3 for SU(3)L (see explanation below), and X as U(1)X charge, where no new leptons are evoked (right-handed neutrinos as well), thus featuring a minimal fermion content. Since we are dealing with an extended gauge group, there will be additional gauge bosons, one of them being singly charged, W′, due to the chosen charge operator parameter. This new gauge boson might induce the neutrinoless double beta decay if it mixes with the W boson. In order to successfully explain neutrino masses in this model without the use of nonrenormalizable operators [14], a scalar sextet should be introduced [32, 33]. This scalar sextet embeds the scalar triplet model, often used in the context of neutrino masses [34–37]. The first component of the scalar sextet is a neutral field which is often assumed to yield a null vacuum expectation value (vev). In this work we make assumption otherwise. If such scalar acquires a nonvanishing vev then W′ mixes with the SM W boson, giving rise to new contributions to the neutrinoless double beta decay. This scenario has been investigated before in the context of majoron in [38–41]. Here we revisit the implications in a more general setting in perspective with other existing colliders constraints. We will ignore contributions arising from the scalar fields because they yield less restrictive bounds [39] and because a bound on the W′ mass is already implied into a lower mass bound on the entire particle spectrum of model, since the W′ mass is directly connected to the scale of symmetry breaking of the model. It is worth mentioning that in this work our parameter space provides a very small Z-Z′ mixing and does not bring relevant constraints coming from STU parameters [41, 42].We will show that ifW′ does feature a mixing with the SM W boson then limits stemming from neutrinoless double beta decay supersede LHC probes, highlighting the importance of multiple new physics searches probes.The paper is organized as follows: In Section2 we describe the model and explain how the W′-W mass mixing can be generated. In Section 3 we derive the contributions to neutrinoless double beta decay. In Section 4 we draw our conclusions.


## 2. The Model

In the minimal 3-3-1 model leptons are arranged as(1)Ll=νlllcL~3,0,where ~ represents the way the triplet transforms under SU(3)L⊗U(1)N, with l=e,μ,τ, representing the three known generations. Notice that the third component of the lepton triplet is the right-handed lepton. The quarks are placed as follows:(2)Q1L=u1d1J1L~3,23,QiL=di-uiJiL~3¯,-13,i=2,3.Here uaL and daL (a=1,2,3) correspond, respectively, to flavors u, c, t and d, s, b of the Standard Model (SM) quarks. The first generation transforms as a triplet, while the second and third generations transform as antitriplets under SU(3)L. In addition to the left-handed field we have right-handed quarks as singlets under SU(3)L: uaR~(1,2/3), daR~(1,-1/3), J1R~(1,5/3), JiR~(1,-4/3), with a=1,2,3 and i=2,3. J1 and Ji are exotic quarks predicted by the model with electric charges of 5/3 and -4/3, respectively. These exotic quarks are also known as leptoquarks in the literature [43, 44].The mass generation mechanism relies firstly on the presence of three scalar triplets; namely,(3)η=η0η1-η2+,ρ=ρ+ρ0ρ++,χ=χ-χ--χ0,with them transforming as (3,0), (3,1), and (3,-1) under SU(3)L⊗U(1)N, respectively. These scalar triplets couple with the fermions fields through the following Yukawa Lagrangian:(4)-LY=12∑ll′Gll′ϵlmnLllc¯Lml′ηn+G1auQ¯1LηuaR+G1adQ¯1LρdaR+G11J1Q¯1LχJ1R+G2asQ¯2Lη∗daR+G2acQ¯2Lρ∗uaR+G2iJ2Q¯2Lχ∗JiR+G3abQ¯3Lη∗daR+G3atQ¯3Lρ∗uaR+G3iJ3Q¯3Lχ∗JiR+h.c.,with l,l′=e,μ,τ. Equation (4) is sufficient to generate masses to charged leptons and quarks but neutrinos remain massless. In order to obtain nonzero neutrino masses, a scalar sextet should be introduced:(5)S=σ10h2-2h1+2h2-2H1--σ202h1+2σ202H2++~6,0.The presence of such scalar sextet gives rise to the Yukawa term,(6)LYS=-12∑ll′GLlc¯S∗Ll′,with Lc=CL¯T, C being the charge conjugate matrix.Expanding (6) we have explicitly(7)LYS⊃-12∑ll′Gll′νlLc¯νl′Lσ10+lLc¯lL′H1+++lR¯l′cLH2--+lLc¯νl′L+νlLc¯lL′h2+2+lR¯νl′L+νlLc¯l′cLh1-2+lR¯lL′+lLc¯lR′cσ202+h.c.Notice that a new contribution to the charged lepton masses arises ifvσ2≠0. Moreover, neutrinos remain massless unless 〈σ10〉≠0. Typically, this scalar sextet is absent in 3-3-1 studies, and consequently neutrino masses are not addressed. In this work, we discuss a more general setting, where the scalar sextet is present and vσ1,2≠0. In this setup, the spontaneous symmetry breaking occurs as follows: firstly, χ0 develops vev (vχ), breaking SU(3)L⊗U(1)N into SU(2)L⊗U(1)Y. Later ρ0 and η0 acquire a nonzero vev with vρ~vη, breaking SU(2)⊗U(1)Y into U(1)QED, that is, the electromagnetism gauge group.This symmetry breaking pattern generates mass to all SM fermions and gauge bosons. We highlight that, due to the enlarged gauge group, 3-3-1 models feature five new gauge bosons. In the minimal 3-3-1 model they are identified asW′±, U±±, and Z′. An important result of the spontaneous symmetry breaking is the rising of a mass mixing between the W and W′ bosons that yields the mass mixing matrix:(8)Wμ+Wμ′+MW2MWW′2MWW′2MW′2Wμ-W′μ-,where(9)MWW′2=g222vσ1vσ2,g=e/sW, and(10)MW2=g22vη2+vρ2+vσ22+vσ12,MW′2=g22vη2+vχ2+vσ22+vσ12.If we takevσ1,2=0, then MWW′2=0, and consequently vη2+vρ2=v2, where v=175 GeV. In this way, the W mass is correctly obtained. After diagonalization we find two mass eigenstates W1+ and W2+ with(11)MW1,2=12MW2+MW′2±MW2-MW′22+4MWW′41/2,which are related to the mass eigenvectors via(12)W+W′+=cθ-sθsθcθW1+W2+,with(13)tan⁡2θ=-2MWW′2MW2-MW′2.Considering the limitMWW′2→0, we get MW1=MW and MW2=MW′. The charged currents associated with these gauge bosons are(14)Lcc⊃g2u¯LγμVCKMdL-ν¯LγμUνVllLWμ++g2lc¯LγμVlUν†νLWμ′++h.c.,where Vl and Uν are the mixing matrices for the charged leptons and neutrinos, with νL=(νeL,νμL,ντL) and lL=(eL,μL,τL). There is also a term involving the doubly charged gauge boson, but it is not relevant for our reasoning [40].We have gathered all ingredients to now discuss the neutrinoless double beta decay in the minimal 3-3-1 model.


## 3. The Neutrinoless Double Beta Decay

Neutrinoless double beta decay is a landmark process in particle physics. It is defined as the transition of a nucleus into another nucleus with an atomic number larger by two units and the emission of two electrons only:(15)A,Z⟶A,Z+2+2e-.Since there are no leptons in the initial state, but two in the final state, the observation of neutrinoless double beta decay constitutes an irrefutable proof that lepton number is violated by nature. In the past decades, there has been a substantial improvement on the bound over the half-life of the neutrinoless double beta decay [45]. These bounds can be later translated into upper limits on the effective majorana mass defined as(16)mββ=∑i=e,μ,τUei2mνi.The effective majorana mass term grows inversely with the half-life. A stronger lower bound on the half-life of the neutrinoless double beta decay implies a stronger bound on the effective majorana mass term (see Figure2). The strongest bound today comes from KamLAND-Zen [46], t1/2>2.6×1025 yrs, implying that(17)mββcurrent<0.15eV.Figure 2
Updated prediction onmββ from neutrino oscillations as a function of the lightest neutrino mass according to the current neutrino data. In the figure, both normal and inverted hierarchy schemes for neutrino masses are displayed. Figure taken from [45]. In the figure is also visible the current limit from KamLAND-Zen.In the foreseeable future, CUORE is expected to achieve130Te>9.5×1025 yrs [45] which translates into(18)mββCUORE<0.073eV.A projected limit further into the future with nEXO oft1/2>1027 yrs for Xe136 is expected, which then would yield(19)mββnEXO<0.01eV.Now from the current and projected experimental sensitivity we will approach the theoretical aspects of this observable. That said, the amplitude in Figure1 is proportional to(20)A1∝g4mββMW4p2cθ4=32GF2mββp2cθ4,where 〈p2〉 is the average of the four-momentum transfer squared, which is approximately (100MeV)2.Moreover, one should observe that the amplitude for the diagram in Figure3 is proportional to(21)A2∝32GF2MWMW′2cθ3sθp2.Figure 3
Novel 331 contribution for double beta decay due to light massive neutrinos.Bearing in mind thatθ should be small in order not to alter the SM W properties, then A2/A1<1. This results in(22)A2A1=MWMW′2p2mββtan⁡θ<1.Therefore we get the lower mass bound:(23)MW′>2.540TeV0.1eVmββ1/2×tan⁡θ10-61/2.In Figure4 we exhibited this lower mass bound on the W′ mass. It is quite visible that even for small tan⁡θ the bound on W′ is rather strong. It is important to make it clear that the 3-3-1M models give us other contributions beyond the one depicted in Figure 3, as pointed out by [47], but they are of much smaller intensity and therefore do not give new better results.Figure 4
Temperature plot of the lower mass bound on theW′ mass as function of mββ and tan⁡θ. The black curve is drawn simply to guide the eye and check that, indeed for tan⁡θ~10-6 and mββ~0.1eV, we find the lower mass bound MW′≳2.5TeV.However, equation (23) is not very useful to us because the W′ mass also appears in tan⁡θ. Therefore, these quantities are not independent; they are strongly correlated. Bearing in mind that we are working in the regimen where θ≪1, then tan⁡θ~θ and we can solve for MW′ to find(24)MW′>50.4TeV0.1eVmββ1/4MWW′21/4.This equation can give rise to more robust bounds on theW′ mass since the parameters that go into it are the majorana effective mass mββ and MWW′2 term, which depends on vevs of the scalar fields σ1 and σ2. Since the Standard Model W mass does not come mostly from vevs of these scalar fields as long as they are much smaller than vSM, then our choices for vσ1 and vσ2 are in principle completely arbitrary.To have a clear vision of what (24) represents, we show the lower mass bounds on the W′ mass in Figure 5. There we select vσ2=3GeV; larger values are ruled out by the ρ-parameter [39, 48] and vary vσ1. Notice that, depending on the value adopted for vσ1, neutrinoless double beta decay might yield very strong constraints on the W′ mass. In particular for mββ~0.1eV we get a lower mass bound on the W′ mass that varies from 300 GeV to about 9.5 TeV. The latter is achieved if vσ1=10-3 GeV.Figure 5
Lower mass bound on theW′ mass from the nonobservation of the neutrinoless double beta decay with different choices for the vev of scalar field σ1 which controls the parameter MWW′2 in (24). The gray solid area denotes Landau pole upper limit [12, 50] and light red denotes the KamLAND-Zen exclusion [46].One shortcoming of our lower mass bound on theW′ mass is its dependence on the choice for vevs. Anyways, these constitute an independent bound on the W′ mass which can be much stronger than the limit on the W′ arising from colliders searches. Notice that our W′ does not couple only to ordinary quarks; therefore it cannot be produced resonantly at the LHC via s-channel processes. Nevertheless, one can use LHC limits on the Z′ mass to convert into a lower mass limit on the W′ mass, since both masses are dictated by the same quantity, vχ, the scale of symmetry breaking of the 3-3-1 symmetry. Indeed one can find that MW′~0.32vχ and MZ′~0.4vχ. Limits from dilepton searches at the LHC at 14 TeV with ~23fb-1 exclude MZ′<4 TeV (see Figure 9 inside [49]). With a projection with 100fb-1 of data, extrapolating the luminosity effect, one would exclude MZ′<4.7 TeV. These two limits translate into MW′>3.2 TeV and MW′>3.7 TeV, respectively.Therefore, we can conclude that the neutrinoless double beta decay offers a complementary probe to collider physics and, depending on choices forvσ1, it can offer the most restrictive bound on this gauge boson mass. We emphasize that the mixing angle θ is quite small since it is dictated by MWW′2/MW′2, rendering our conclusions robust.


## 4. Validity of the Minimal 3-3-1 Model

It has been shown that the 3-3-1M is valid up to energies of5.7 TeV or so due the presence of a Landau pole [12, 50]. The collider bounds on the Z′ and W′ mentioned above already pose a strong tension on the model since these mass limits translate into vχ≈10 TeV and vχ≈12 TeV. With the addition of new exotic fermions that can contribute to the renormalization group equations, this tension can be alleviated. As can be seen in Figure 5, for mββ~0.1 eV and vσ1~10-6 GeV, the lower mass bound on W′ lies around 1.6 TeV, which implies that vχ>3.6 TeV.This would constitute a strong but valid claim, under the assumptions made. Neutrinoless double beta decay under certain assumptions already excludes the minimal 3-3-1 model, imposing that the scale of symmetry breaking of the model should lie beyond its validity (for previous bounds on this model see [51–59]).These findings are valid formββ~0.1eV, but if we consider the nEXO sensitivity which is expected to reach mββ~0.01eV, the impact that our study brings to the minimal 3-3-1 model is even more profound. Even for vσ1~10-6 GeV, we would already impose vχ≥6 TeV.In summary, neutrinoless double beta decay offers an orthogonal probe to the minimal 3-3-1 model. If thevevs of the fields in the scalar sextet are sufficiently small, neutrinoless double beta decay does not favor the original version of the minimal 3-3-1 model.


## 5. Conclusions

We discussed neutrinoless double beta decay in the context of the minimal 3-3-1 model. It features a minimal fermion content, arguably rendering it more predictive in comparison to other models based on this 3-3-1 gauge group. In the past decades, we have observed a significant improvement in the bound of the neutrinoless double beta decay. We exploit this fact to obtain a lower mass bound on theW′ boson that arises due to the enlarged gauge group. Our limits are based mostly on the charged current and the vev of a scalar field that generates neutrino masses and induces the W-W′ mixing. This mixing leads to a sizable contribution to neutrinoless double beta decay.We obtained a fully analytic expression that represents a lower mass bound on theW′ mass. Depending on the vev adopted for this scalar field (σ1), neutrinoless double beta decay might offer the most restrictive limit on the W′ surpassing those of collider probes. Moreover, since the W′ mass is directly connected to the scale of symmetry breaking of the model, a lower mass bound on the gauge boson represents also a lower bound on the scale of symmetry breaking. In particular, if vσ1>0.1 GeV, current limits on the half-life of the neutrinoless double beta decay strongly disfavored out the minimal 3-3-1 model.



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