# Vacuum stability conditions of the economical \(3-3-1\) model from copositivity

## Abstract

By applying copositivity criterion to the scalar potential of the economical \(3-3-1\) model, we derive necessary and sufficient bounded-from-below conditions at tree level. Although these are a large number of intricate inequalities for the dimensionless parameters of the scalar potential, we present general enlightening relations in this work. Additionally, we use constraints coming from the minimization of the scalar potential by means of the orbit space method, the positivity of the squared masses of the extra scalars, the Higgs boson mass, the \(Z'\) gauge boson mass and its mixing angle with the SM *Z* boson in order to further restrict the parameter space of this model.

## 1 Introduction

Models addressing open questions concerning the standard model of particle physics (SM) usually resort to the use of new symmetries and/or the addition of extra particles. As a first example, we can mention models implementing different see-saw mechanisms (type I, II and III) which introduce bosonic or fermionic degrees of freedom in order to explain tiny neutrino masses and their mixings [1, 2, 3, 4]. In combination with that, new abelian and non-abelian symmetries are also invoked in order to obtain highly predictive scenarios where not only neutrino masses are fixed but also further correlations between neutrino oscillation parameters appear [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. A series of models with a matter content larger than the one of the SM are those dealing with the impressive observation that almost thirty percent of the energy content of the Universe is due to dark matter (DM) [17]. Arguably, the simplest model providing a DM candidate is that which extends the SM only by a real scalar transforming in a non trivial way under a stabilizing \({\mathbb {Z}}_2\) discrete symmetry [18, 19]. However, other well-motivated models based on supersymmetry [20, 21, 22, 23], extra dimensions [24], the \(B-L\) symmetry [25, 26, 27, 28] and those with an Axion/ALP [29, 30, 31, 32, 33, 34, 35] have been widely considered (see [36] for a review). All of them have additional symmetries and extra particles in their physical spectrum.

In the same vein, the so-called \(3-3-1\) models are interesting extensions of the SM. The fundamental idea behind all these models is to extend the \({\mathrm {SU}}(2)_{L}\) gauge group to the \(\text {SU}(3)_{L}\) one, in other words, the total gauge group of these models is \(\text {G}_{331}\equiv \text {SU}(3)_{C}\otimes \text {SU}(3)_{L}\otimes \text {U}(1)_{N}\). Here, *C* and *L* stand, as in the SM, for color and left chirality, respectively. However, *N* stands for a new charge different from the SM hypercharge, *Y*, and its values are assigned to obtain the latter after the first spontaneous symmetry breaking. More specifically, the values of *N* together with an embedding parameter *b* determine the electric charges of the matter content in these models since the electric charge operator is \(Q=T_{3}-bT_{8}+N{\mathbf {1}}_{3\times 3}\) [37], where \(T_{3,\,8}\) are the diagonal Gell-Mann matrices, \({\mathbf {1}}_{3\times 3}\) is the \(3\times 3\) identity matrix and the parameter *b* can take two values: \(1/\sqrt{3}\) or \(\sqrt{3}\).

In this paper, we consider the \(3-3-1\) model with \(b=1/\sqrt{3}\) and the simplest scalar sector as proposed in Refs. [38, 39] after a systematic study of all possible \(3-3-1\) models without exotic electric charges. This model is known in the literature as the “economical \(3-3-1\) model” and has some appealing features that turn it arguably the most interesting \(3-3-1\) model. Among these properties we can mention that right-handed neutrinos, \(N_{a}\), are in the same \(\text {SU}(3)_{L}\) multiplet as the SM leptons, \(\nu _{a}\) and \(e_{a}\). This is possible because the fundamental representation of the \(\text {SU}(3)_{L}\) gauge group is larger than the \(\text {SU}(2)_{L}\) one and the parameter \(b=1/\sqrt{3}\) allows the \(\text {SU}(3)_{L}\) multiplet to have two electrically neutral components: \(\nu _{a}\) and \(N_{a}^{c}\). This property allows for massive neutrinos at tree level. Yet, agreement with experiments is reached only when one-loop contributions to neutrino masses are considered [40]. Other no less important features of this model are the possibility of implementing the Peccei-Quinn mechanism in order to solve the strong CP problem [41] and the existence of axion dark matter [35]. Needless to say, this model also shares some appealing features with other versions [42, 43, 44, 45, 46, 47, 48, 49] such as the capability to shed some light on the family replication issue of the SM.

Although the economical \(3-3-1\) model has several appealing features, it also introduces a considerable number of degrees freedom that turn it less predictive. For instance, its scalar potential has at least nineteen coupling constants. It is a large number when compared with two couplings in the SM scalar potential. Needless to say, a large number of extra Yukawa couplings is allowed by the \(3-3-1\) gauge group. Therefore, in this paper we search for constraints that allow to reduce this, in some sense, undesirable freedom. More specifically, we study vacuum stability at tree level, i.e. the conditions guaranteeing that the scalar potential is bounded from below in all directions in the field space as the field norms approach to infinity. It is well-known that in the SM, at tree level, it is enough to make the Higgs boson quartic coupling positive [50, 51, 52]. Nevertheless, in the case of the economical \(3-3-1\) model we face a more complicated problem even at tree level since we have to deal with nineteen coupling constants and the scalar fields belong to the fundamental and anti-fundamental representations of \(\text {SU}(3)_{L}\). However, the problem becomes simpler when a \({\mathbb {Z}}_2\) symmetry acting on some fields is considered. This symmetry is already used in the \(3-3-1\) literature [37, 38, 41, 53, 54, 55, 56, 57] with different motivations. In the present context, this symmetry not only reduces the number of coupling constants to fourteen, but also makes the quartic terms in the scalar potential to have a \(\lambda _{ij}\phi _i^2\phi _j^2\) form. Therefore, demanding that the scalar potential is bounded from below as the field norms approach to infinity is equivalent to ensuring that the \(\lambda _{ij}\) matrix is copositive (positive on nonnegative vectors) [58, 59, 60, 61]. In addition, to make the problem even more tractable, we use the method of the orbit space in Refs. [62, 63] which greatly reduces the number of variables. At the end, the problem of vacuum stability is reduced to study the copositivity of a \(3\times 3\) matrix. This provides seventeen inequalities, for the ten quartic couplings, that at first sight seem too complicated in order to provide useful analytical relations. However, combining these inequalities with constraints coming from the first and second derivative tests for a minimum of the scalar potential, we manage to find enlightening analytical constraints for these coupling constants.

Finally, with the aim of restricting the rest of the scalar couplings we turn our attention to the scalar mass spectrum, since all of the squared scalar masses must be positive in general. However, these masses also depend on the vacuum expectation values (VEVs) of the scalar fields. Thus, we use the experimental limits on the mass of an extra neutral gauge boson \(Z'\) [64, 65, 66, 67], and the bound on the \(Z-Z'\) mixing angle in this model [68] to estimate the VEVs. Doing so, we find relations for three of the remaining four scalar potential couplings. Also, the experimental limit on the Higgs mass [64, 65, 66, 67] is used to constrain even more some couplings.

The rest of the paper is organized as follows. In Sect. 2 we introduce the generalities of the economical \(3-3-1\) model with a \({\mathbb {Z}}_2\) symmetry that allows a scalar potential with quartic terms in a biquadratic form of the field norms. In Sect. 3, taking advantage of this property, we search for constraints on the scalar potential couplings imposing the vacuum stability conditions at tree level. Specifically, we use the method of the orbit space to simplify the application of the first and second derivative tests together with the copositivity criterion. After finding clear and useful relations for the values of some scalar potential parameters, in Sect. 4, we go further applying the positivity of the scalar masses and the experimental Higgs mass in order to constrain more scalar parameters. Finally, we present our conclusions in Sect. 5.

## 2 The model

*a*takes the same values as in Eq. (1), \(s=1,\dots ,4\) and \(t=1,\dots ,5\).

*a*,

*b*,

*s*,

*t*are in the same range as in Eqs. (1–4). It is also straightforward to write down the most general scalar potential consistent with gauge invariance and renormalizability as

## 3 Minimization and vacuum stability

Now, we turn our attention to find constraints on the \(\mu _i-\) and \(\lambda _i-\)scalar parameters coming from minimization and vacuum stability. The general minimization of the scalar potential in Eq. (16) is a difficult task due to the large number of free parameters in the scalar potential (14 free parameters), the large number of components of the scalars triplets in the model (18 components for the \(\rho ,\,\eta ,\,\chi \) triplets) and the degeneracies of the extremal points of the potential required by the \(\text {G}_{331}-\)invariance. Fortunately, there is a powerful tool to simplify this problem which consists in working with the norm of the fields and orbit parameters. This method to minimize scalar potentials, also known as the method of the orbit space, is detailed in Refs. [60, 61, 62, 63] in the context of spontaneous symmetry breaking. It has been used, for instance, in models with \(\text {SU}(5)\) and \(\text {SO}(10)\) gauge symmetries when scalars belong to different representations [60, 72, 73, 74]. The crucial observation of the method is that working with the norm of the fields, \(|\phi |\) (\(|\phi |^2 \equiv \phi ^{*}_{k}\phi _k\) - where a sum over repeated indices is implied) and the invariant orbit parameters \(\varvec{\theta }\)’s (generically defined by \({\varvec{\theta }}=\frac{f_{ijkl}\,\phi ^{*}_{i}\phi _{j}\phi ^{*}_k\phi _l}{|\phi |^4}\)) contain all the information needed to determine the minimum of the potential and, in addition, greatly reduce the number of variables.

- (1)
\(|\varvec{\theta }_{1,2,3}|=0\), if \(\lambda _7-2|\lambda _{10}|\ge 0\), \(\lambda _8\ge 0\) and \(\lambda _9\ge 0\);

- (2)
\(|\varvec{\theta }_{1,2}|=0\) and \(|\varvec{\theta }_{3}|=1\), if \(\lambda _7-2|\lambda _{10}|\ge 0\), \(\lambda _8\ge 0\) and \(\lambda _9<0\);

- (3)
\(|\varvec{\theta }_{1,3}|=0\) and \(|\varvec{\theta }_{2}|=1\), if \(\lambda _7-2|\lambda _{10}|\ge 0\), \(\lambda _8<0\) and \(\lambda _9\ge 0\);

- (4)
\(|\varvec{\theta }_{2,3}|=0\) and \(|\varvec{\theta }_{1}|=1\), if \(\lambda _7-2|\lambda _{10}|<0\), \(\lambda _8\ge 0\) and \(\lambda _9\ge 0\);

- (5)
\(|\varvec{\theta }_{1,2,3}|=1\), if \(\lambda _7-2|\lambda _{10}|<0\), \(\lambda _8<0\) and \(\lambda _9<0\).

- (1)
\(\lambda _5\sqrt{\lambda _1}+\lambda _6\sqrt{\lambda _3}>0\), if \(\lambda _8\ge 0\) and \(\lambda _9\ge 0\).

- (2)
\(\lambda _5\sqrt{\lambda _1}+(\lambda _6+\lambda _9)\sqrt{\lambda _3}>0\), if \(\lambda _8\ge 0\) and \(\lambda _9< 0\).

- (3)
\((\lambda _5+\lambda _8)\sqrt{\lambda _1}+\lambda _6\sqrt{\lambda _3}>0\), if \(\lambda _8<0\) and \(\lambda _9\ge 0\).

- (4)
\((\lambda _5+\lambda _8)\sqrt{\lambda _1}+(\lambda _6+\lambda _9)\sqrt{\lambda _3}>0\), if \(\lambda _8<0\) and \(\lambda _9<0\).

On the other hand, in the right panel of Fig. 3, the maximum and the minimum excluded regions for \(\lambda _6\) as a function of \(\lambda _9\) are shown. The maximum excluded region of \(\lambda _6\) is characterized by two different bounds. \(\lambda _6\le 2\sqrt{\lambda _1\lambda _2}-\lambda _9\) are excluded if \(\lambda _9<0\) and \(\lambda _6\) values satisfying \(\lambda _6\le 2\sqrt{\lambda _1\lambda _2}\) are excluded if \(\lambda _9\ge 0\). That region is reached when \(\lambda _4 + \left( \lambda _7 -2 |\lambda _{10}|\right) \theta _\text {H}\left( 2|\lambda _{10}| -\lambda _7\right) \) \(+2\sqrt{\lambda _1\lambda _{3}}=0\) and \(\lambda _5 +\lambda _8\,\theta _\text {H}\left( -\lambda _8\right) +2\sqrt{\lambda _2\lambda _{3}}=0\). Finally, the minimum excluded region also has two different bounds. \(\lambda _6\le -2\sqrt{\lambda _1\lambda _2}-\lambda _9\) are excluded if \(\lambda _9<0\) and \(\lambda _6\) values satisfying \(\lambda _6\le -2\sqrt{\lambda _1\lambda _2}\) are excluded if \(\lambda _9\ge 0\). That region is reached when \(\left( \lambda _4 + \left( \lambda _7 -2 |\lambda _{10}|\right) \theta _\text {H}\left( 2|\lambda _{10}| -\lambda _7\right) +2\sqrt{\lambda _1\lambda _{3}}\right) \sqrt{\lambda _2}\) \(+\left( \lambda _5 +\lambda _8\,\theta _\text {H}\left( -\lambda _8\right) \right. \left. +2\sqrt{\lambda _2\lambda _{3}}\right) \sqrt{\lambda _1} >4\sqrt{\lambda _1\lambda _2\lambda _3}\).

## 4 Scalar mass spectrum

*h*, \(H_2\) and \(H_3\), from which we define

*h*as the SM Higgs boson. Their square masses can be written as:

*g*is the coupling constant of the \(\hbox {SU}(2)_L\) group. Using \(v=246\) GeV, \(g=0.65\), \(\sin ^2 \theta _W\simeq 0.22 \), \(M_{Z_1}\simeq 91.19\) GeV [67, 84] and imposing \(-3.98\times 10 ^{-3}\lesssim \tan \phi \lesssim 1.31\times 10^{-4}\) we find the allowed region in the \(v_\chi -v_\eta \) plane. As we can see in Fig. 5, \(v_\chi \gtrsim 100\) TeV is needed to obtain significant deviations from the zero-mixing value, \(v_\eta \simeq 197.34\) GeV.

*h*, \(H_2\) and \(H_3\). Their square masses are obtained from the eigenvalues of the matrix \(m_{ij} = \frac{1}{2}\frac{\partial ^2 V}{\partial \phi _i \partial \phi _j} \bigg |_{\phi =\text {min}},\) where \(\phi _i\), \(i=1,2,3\), are the real parts of the fields \(\eta _1^0\), \(\rho _2^0\) and \(\chi _3^0\), respectively. In that basis, \(m_{ij}\) coincides with \(\frac{1}{2}H_0\), where \(H_0\) is the Hessian matrix given in Eq. (21). A perturbative analysis in powers of \(v/v_\chi \) shows that \(m^2_h\propto v^2\) and \(m^2_{H_{2,3}}\propto v_\chi ^2\). For this reason, we have that \(m^2_{H_{2,3}}\gg m^2_h\) since \(v_\chi ^2\gg v^2\). This observation allows us to calculate an analytical expression for \(m^2_h\). In order to do so, it is useful to write the characteristic polynomial of \(\frac{1}{2}H_0\),

*P*, as

*P*when calculated from \(\det \left[ m^2\,\mathbf{{1}}_{3\times 3}-H_0/2\right] \). Now, since \(m^2_{H_{2,3}}\gg m^2_h\), we can write

Similar conclusions are obtained for the planes \(\lambda _6 - \lambda _{4}\) and \(\lambda _6 - \lambda _{5}\). Now, for the most general case, i.e. when \(|\lambda _{15}|\) is not negligible, Eq. (42) takes the most general form \(a\,\lambda _4^2+ 2 b\,\lambda _4\,\lambda _5 + c\,\lambda _5^2+2d\,\lambda _4+2f\,\lambda _5+g = 0\) where the coefficients \(a,\cdots ,g\) depend on \(\lambda _{1,2,3,6,15}\). Thus, for the general case the ellipse/hyperbola is not centered at the origin and the rotation angle also acquires a dependence on \(|\lambda _{15}|\). We are not going to consider that case in detail.

## 5 Conclusions

In this work, we find tree level constraints on the scalar potential couplings of the economical \(3-3-1\) model when considerations of vacuum stability and positivity of the squared scalar masses are taken into account. In particular, we consider the model with a discrete \({\mathbb {Z}}_2\) symmetry acting on \(\chi \), \(u_{4R}\), \(\,d{}_{\left( 4,5\right) R}\) fields in a non trivial way. Besides all the appealing features discussed in Sect. 2, this discrete symmetry makes the quartic terms in the scalar potential to have a biquadratic form of the norm of the fields. This allows us to apply copositivity criterion in order to guarantee that the scalar potential is bounded from below. When copositivity criterion is imposed in combination with the first and second derivative tests for the vacuum expectation values given in Eq. (12), ten of the scalar couplings are constrained. In more detail, \(\lambda _{1,2,3}\) need to be positive and the \(\mu _{1,2,3}\) parameters are completely determined, c.f. Eq. (20). Besides that, \(\lambda _{4,5,6}\) couplings are constrained from below by the copositivity and from above by the positivity of the principal minors of the Hessian matrix, Eq. (31). More interestingly, there is always an excluded region for \(\lambda _{4,5,6}\) which we called the minimum excluded region in Figs. 2 and 3, respectively. This region comes from the copositivity criterion and gives a lower bound for \(\lambda _{4,5,6}\). It is remarkable that the excluded region for these \(\lambda \) couplings also has a maximum provided all copositivity conditions are satisfied. \(\lambda _{7,8,9,10}\) play an important role in determining the form of both the minimum and the maximum excluded regions for \(\lambda _{4,5,6}\). On the other hand, copositivity does not have anything to say about the upper bound on \(\lambda _{4,5,6}\) and it is here that second derivative test is important. We analyse the role that \(|\lambda _{15}|\) (where we have applied a phase shift in the fields to make \(\lambda _{15}\) real and positive without loss of generality) has in determining that upper bound. As the \(|\lambda _{15}|\) coupling is technically small, we have studied the bound when \(|\lambda _{15}|\ll v_\eta ,v_\rho \) showing that the smallest upper bound on \(\lambda _{4,5,6}\) is always larger than \(2\sqrt{\lambda _1\lambda _3}\), \(2\sqrt{\lambda _2\lambda _3}\), \(2\sqrt{\lambda _1\lambda _2}\), respectively.

In order to constrain the rest of \(\lambda \) couplings, we turn our attention on positivity of the squared scalar masses. After finding general expressions for the masses of the charged and CP-odd scalars, we find constraints on \(\lambda _{7,8,9}\) given in Eqs. (34–35), respectively. Actually, we apply a stronger limit for the case of the masses of charged scalars since these, roughly speaking, must be heavier than 80 GeV. As the constraints on \(\lambda _{7,8,9}\) strongly depend on the VEVs, even in the case of \(|\lambda _{15}|\rightarrow 0\), we estimate the lower bounds on \(\lambda _{7,8,9}\) using the VEVs that satisfy the upper bound on the mixing angle between the two neutral gauge bosons in the model and the lower bound on the mass of \(Z'\) gauge boson. Doing that, we obtain the lower bounds in Eq. (39).

Moreover, we find approximate formulas for the squared masses of the CP-even scalars in the model. If the \(v_\rho ,v_\eta \ll v_\chi \) hierarchy is satisfied (as assumed in this \(3-3-1\) model), the squared masses of the CP-even scalars different from the Higgs boson are proportional to \(v_\chi ^2\), which allows us to find a \(2\%\) accurate formula for the Higgs squared mass, c.f. Eq. (41). Using the fact that the Higgs mass must be \(m_h=125.18\pm 0.16\) GeV [64, 65, 66, 67], we find that \(\lambda _5-\lambda _4\), \(\lambda _6-\lambda _4\) and \(\lambda _6-\lambda _5\) satisfy the ellipse or the hyperbola general equations with coefficients determined by \(\lambda _{1,2,3,6,15}\) couplings. We outline the behaviour of such conics for the case \(|\lambda _{15}|\ll v_\eta ,v_\rho \) in Fig. 6. In that case, they are centered at the origin and their rotation angle strongly depends on \(\lambda _6\). Furthermore, we find equations for squared masses of the other two CP-even scalars in terms of \(m_h^2\) and the trace and determinant of the Hessian matrix, see Eq. (44). Because these squared masses are larger than \(m_h^2\) in our approach, their positivity do not bring new constraints on \(\lambda \) couplings.

Although the objective of this paper is to derive tree level conditions for the quartic couplings of the scalar potential coming from vacuum stability, the minimization of the scalar potential, the positivity of the squared masses of the extra scalars, the Higgs boson mass, the \({\hbox {Z}}^\prime \) gauge boson mass and its mixing angle with the SM Z boson in order to restrict the parameter space, it is also interesting to comment some modifications coming from the running of the coupling constants. Roughly speaking, we expect that major differences when compared with the SM are due to the presence of new particles, such as heavy quarks and leptons. For example, there are three additional quarks (an up-type quark and two down-type quarks) which are assumed to have masses in the scale of the \(\text {SU}(3)_L\) symmetry breaking, i.e. \(1-10\) TeV. For this reason, we expect that these quarks modify significantly the beta functions of the quartic couplings that involve the \(\chi \) scalar triplet because the \({\mathbb {Z}}_2\) symmetry acting on \(\chi \), \(u_{4R}\) and \(d_{4,5R}\) fields makes these new quarks gain masses mainly through the \(\chi \) scalar triplet. Thus, the beta functions of \(\lambda _{4,5}\) are expected to receive large contributions coming from the diagrams where the new quarks are running. However, other \(\lambda \) couplings can receive some contributions from these quarks due to the mixings between scalar mass eigenstates. Moreover, we expect that the beta function of the \(\lambda \) quartic couplings receive several positive contributions from the one-loop diagrams with scalars running in them. These positive contributions, roughly speaking, will reduce the allowed regions of the \(\lambda \) couplings in a similar way as in the triplet and inverse seesaw models [80, 86]. Nevertheless, in order to give a quantitative answer to this question all coupled one-loop renormalization group equations must be carefully studied for this model.

## Notes

### Acknowledgements

The authors are thankful for the support of FAPESP funding Grant No. 2014/19164-6. B.L.S.V. also thanks DRCC/IFGW at UNICAMP for their kind hospitality. G.G. is supported by CNPq Grant No. 141699/2016-7. C.E.A.S is grateful for the financial support of CNPq, under grant 159237/2015-7, and to the Abdus Salam International Centre for Theoretical Physics for its kind hospitality. The authors thank Renato M. Fonseca and Ana R. Romero Castellanos for useful discussions about the manuscript.

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