Softly-broken $A_4$ or $S_4$ 3HDMs with stable states

We study multi-Higgs models with a triplet of $A_4$ or $S_4$ symmetry that is softly-broken. Focusing on parameters that softly break the respective symmetries but preserve the directions of the vacuum expectation values of the triplet we analyse the masses of physical states and discuss their decays. In contrast with previously studied symmetries, we find some cases with stable states that are protected by residual symmetries left unbroken by both the soft-breaking parameters and by the vacuum.


Introduction
A very natural extension of the Standard Model (SM) is adding scalars to the field content (see e.g. the reviews [1,2,3,4]). The class of models where extra SU(2) doublets are introduced, Multi-Higgs Doublet Models with N doublets (NHDM) is rather appealing as it enables possibilities such as scalar CP violation and dark matter candidates. The 2HDM has been widely studied and 3HDMs, introduced by [5], are being actively explored. The most general 3HDM has the issue of proliferation of physical parameters, so the typical approach is to consider additional symmetry that makes the model more predictive. The discrete symmetries that can be considered in the 3HDM have been classified in [6] (see also [7,8]), and their CP properties studied in [9,10,11,12,13], with the minima studied in [14,15].
With large non-Abelian symmetries such as Σ(36), the predictive power is such that the model is not viable, so an interesting possibility is that the large symmetry is present but softly-broken. This idea was discussed in [16], and methods were developed to apply it to general 3HDMs, and exemplified through the study of the Σ(36) case. In the present work, we apply the same methodology to softly-broken A 4 and S 4 symmetric 3HDMs. In the potentials considered here, we find some qualitatively different results, of particular relevance that there are situations where there are unbroken residual symmetries that stabilize mass eigenstates against decay -which enables these as Dark Matter (DM) candidates.
In Section 2 we review the A 4 and S 4 potentials, considering the mass spectrum for the possible vacuum expectation values (vevs). In Section 3 we discuss the parameters that softly break the symmetry, the deviations induced in the masses, and analyse residual symmetries together with associated stablet states. In Section 4 we conclude.
2 A 4 and S 4 symmetric 3HDMs

Potentials
In this work we focus on the S 4 and A 4 symmetric potentials, due to the close relationship between these groups.
This fact leads to both potentials being written in a common form, according to the notation in [14] To obtain the S 4 invariant potential one just needs to set Λ 4 = 0, meaning that any results obtained for the A 4 invariant model carry over to S 4 by means of this substitution, further justifying our dual analysis. As such, the A 4 and S 4 models have 6 and 5 free parameters, respectively.
These models have 4 different types of vev directions that can lead to a global minimum, [6,15] (1, 0, 0) with α fixed by the relations The presence of these 4 classes manifests itself in 4 possibilities for the predictions of these models, with a set of inequalities on the Λ i selecting which one is the global minimum. These inequalities are also important to take into consideration when taking the limit of Λ 4 = 0 in (3), since they enforce that Λ 1 − Λ 2 > 0 and thus select α = π/2 as the global minimum for S 4 .

Fully-symmetric mass spectra
The spectra for A 4 and S 4 have been studied in [14], and here we summarize those results. Throughout this work, H i and h i refer to the heaviest and lightest pair of additional scalars, while H ± i refers to the lightest and heaviest of the charged pairs (the index i can take the values i = 1, 2). The Higgs-like boson, H SM , always has a mass given by m 2 (1, 0, 0): For this alignment (1, 0, 0) to be the global minimum, one must have that This guarantees this is the favored direction, and the magnitude of the vev is then given by The spectra is then expressed as (1, 1, 1): In the region of parameter space given by the inequalities we have the alignment (1, 1, 1) as the global minimum, with magnitude The spectra has less degeneracies (note the charged Higgses) (1, e iα , 0): When the parameters of the potential obey the inequalities we obtain (1, e iα , 0) as the alignment with magnitude where we have definedΛ The corresponding spectra is where in this case there are no degeneracies and the four m 2 H i are separated by commas.
(1, ω, ω 2 ): In the region of parameter space where (1, ω, ω 2 ) is the global minimum, with magnitude The masses are given by the expressions where we defined 3 Softly-broken phenomenology 3

.1 Managing expectations
The procedure for introducing symmetry breaking parameters in this way was described in [16]. Here we summarize both it and its derivation, for convenience. Given a potential V 0 with a quadratic term V 2 = −m 2 Φ † i Φ i and an otherwise generic quartic part V 4 , if one considers as independent variables Φ i and Φ † i , the usual extremization conditions result in the set of equations implying that at any critical point we have the equality Now, we break the symmetry by considering the potential The extremization conditions pick up extra terms, namely The solutions to these equations are in the general case different from the fully symmetric model, and (19) ceases to be valid. However, if we now impose that a given vev alignment should remain a solution to these equations, up to a scaling factor, that is, v| V = ζ · v| V 0 , we can nevertheless state that, at the scaled critical point, due to the fact that V 2 and V 4 are homogeneous functions of degree 2 and 4, respectively. Introducing this in (20) we obtain that M must satisfy The upshot from this is that a given vev alignment is only preserved as a critical point if and only if it is an eigenvalue of the soft-breaking parameter (SBP) matrix M. This gives us a simple way of constructing the most general matrix that preserves any given vev alignment: Within this formula, we take − → n 1 as the desired vev alignment, and thus its eigenvalue is Regarding the remaining terms, we are free to choose any two eigenvalues and any two eigenvectors, as long as we guarantee that the set { − → n 1 , − → n 2 , − → n 3 } is an orthonormal basis of C 3 . For that, one can start from two vectors { − → e 2 , − → e 3 } that satisfy this condition and write the most general pair { − → n 2 , − → n 3 } via a rotation: This leaves us with a total of five degrees of freedom for parametrizing the SBPs, three eigenvalues and two angular variables. For the sake of simplicity, we now impose for the rest of this work µ 1 = 0 and we are left with four parameters, which can assume any real value. By setting µ 1 = 0 we are only imposing that the vev magnitude itself is also preserved (the alignment is preserved regardless of µ 1 = 0). However, it is possible to restrict θ to lie in the first quadrant without loss of generality, due to the fact that any { − → n 2 , − → n 3 } that one obtains with θ ≥ π/2 can be brought to a configuration reachable by 0 ≤ θ < π/2 by application of the appropriate combination of sign flips (under which M is invariant) and − → n 2 ↔ − → n 3 , equivalent to µ 2 ↔ µ 3 . Thus, the minimal description of the SBPs is defined by the dimensionless parameters Here, M 0 is the coupling parameter of V 2 . This recasting of parameters was done in order to simplify the results to be shown in the following sections. Before proceeding, one important point regarding the ranges of admissible values for µ 2 and µ 3 needs to be addressed. From the form of (23) it is easily seen that if we choose µ 2 , µ 3 > 0, then, for a given vev alignment, M will not contribute to V for this direction and give a positive contribution to any other direction, making our selected point deeper by comparison. Thus, if we start in a region of the Λ parameter space that selects one of the four directions shown in (2) and choose Σ < 0, |∆| < Σ (equivalent to µ 2 , µ 3 > 0) then we are guaranteed that the selected direction continues to be the global minimum, and as such these are sufficient conditions to ensure that the critical point is still the global minimum. Nevertheless, it is possible to find points outside of this region and/or that break the inequalities in the Λ parameter space where our selected alignment continues to be the global minimum, if our deviations from them are not too large. Quantifying this magnitude analytically is not a straightforward task, however, and a numerical approach is better suited. We emphasize then that µ 2 , µ 3 > 0 are sufficient but not necessary conditions to have the selected direction remain a global minimum.
The upshot from this is that while it is always possible to choose a set of parameters that imposes a given direction from the fully-symmetric model as the global minimum, there is also some added freedom in the region of parameters, introduced by the SBPs, where one can explore new phenomenology.

Softly-broken mass spectra
Applying the procedure described above we studied the mass spectra predicted by the softlybroken A 4 potential (the spectra for the specific case of S 4 can be consulted in Appendix A). Before proceeding to the full set of results, we provide an overview of some common properties displayed between the vev directions. Starting by the completely general properties, we have that: • Due to the way in which the symmetry was broken (and by setting µ 1 = 0) the SM-like part of the spectrum behaves in exactly the same way as in the fully symmetric model, regarding both the expressions for m 2 H SM = 2 √ 3 M 0 and the scalar alignment, meaning that H SM is the only scalar that couples to the gauge bosons.
• All the mass degeneracies present in the parent model can be lifted by an appropriate choice of the soft breaking parameters.
Then, there are also properties shared among three out of the four alignments: • The mass splittings for the neutral bosons obey a specific pattern. Following the previously described nomenclature for the extended scalar sector, we have that Not observed for the vev alignment (1, e iα , 0) in A 4 and (1, i, 0) in S 4 .
• The mass expressions depend only on a specific combination of the angular parameters, effectively rendering them dependent on only three independent SBPs, meaning that one can travel along a certain direction in parameter space without affecting the values of the masses.
It is important to mention that all of these properties were also found in the Σ(36) symmetric model [16]. The fact that they do not hold for all the vev alignments in A 4 /S 4 can be attributed to the fact that Σ(36) is a more symmetric model than the ones we have at hand here.
Lastly, some remarks about the notation used are in order. The results are written in terms of a set of auxiliary parameters, Γ i , which are alignment dependent linear combinations of the Λ i , given below for each case. Through them, we always have that We also make use of the following definitions, which hold for all alignments, x = 2 sin 2θ sin ξ (1, 0, 0): Here we chose Then, with the alignment specific parameters the masses of the neutral bosons are given by with s defined as Regarding the charged pairs, we have In this alignment, the charged splittings are simply linear with µ 2 and µ 3 .
(1, 1, 1): We chose as orthogonal vectors The alignment specific parameters are It is important to highlight the approximate invariance between this set of Γ i and the previous set under the interchange Λ 1 ↔ Λ 3 . This approximate invariance was observed in the fullysymmetric model and can be seen as an inherited trait. The masses for the neutral sector are and for the charged scalars (1, e iα , 0):

Our choice of basis is
In this case we were unable to obtain the analytical expressions for the masses of the neutral bosons, due to the added algebraic complexity introduced by the parameter dependent alignment. We turned to a numerical approach. We note however that the analytical results for the limiting case of S 4 were obtained and are discussed in Appendix A.
For the numeric analysis, we chose a benchmark set of parameters as For the SBPs, (40) should be interpreted as the values chosen when the relevant parameter is not the independent variable being analyzed, meaning that, for example, in Figure 1, θ = ξ = π/7 and ∆ varies in the range shown. With this, the results are as follows:  Figure 1: Numerical dependence of mass ratio with ∆.  Figure 2: Numerical dependence of mass ratio with θ and ξ.
Numerically, we were able to verify that (like the corresponding S 4 case) the dependence on Σ is merely an overall shift of all the masses and therefore we omit its presentation here. Likewise, the dependence on ∆ modulates the masses of the distinct eigenstates as a prefactor as in S 4 , see Figure 1. The dependence on θ and ξ appears through cos 2θ and sin 2ξ as seen in Figure 2. This last dependence is a new feature of A 4 , and numerically we observe that the amplitude of the cos 2θ dependence is considerably greater than that of the sin 2ξ, as shown in the scales of the respective plots.
Regarding the charged scalars, defining the Γ i as we have that the masses are given by (1, ω, ω 2 ): For this alignment, we define ω = e 2iπ/3 , and with this our basis vectors are The alignment specific parameters are through which one can write the masses of the neutral bosons as and the charged part of the spectrum as

Dark matter candidates
In the S 4 and A 4 fully-symmetric models each vev alignment preserves a (different) subset of symmetries of the potential, leading to a scalar sector with stable neutral states and therefore to potential dark matter candidates stabilized by those residual symmetries. Now, in general, when one introduces M into the potential some of the residual symmetries will be broken while others might still be conserved. If all the symmetries are broken then the previously stabilized fields can decay and the model ceases to have dark matter candidates. However, if one desires to retain this possibility then it is possible to look into the subset of symmetries preserved by the vev and see what conditions one must impose on M. Thus, in this section, for each vev alignment we first determine if there are any symmetries left intact by the general form of (23), and, if not, we determine the least restraining set of conditions on M so that at least one symmetry is preserved. The analysis applies to S 4 and A 4 , since the symmetry elements used are common to both. We verified all of our findings numerically.
(1, 0, 0): Here we focus on the symmetry element common to A 4 and S 4 . This element generates a Z 2 subgroup, acting only on the second and third doublets by flipping their signs. It is straightforward to check that this symmetry is conserved by the vev direction (1, 0, 0). Regarding the soft-breaking parameters, we have that ρ acts on the contractions of the doublets as and the remaining combinations are not shown since they map to themselves. These transformation properties force the elements of M to satisfy due to the hermiticity of M. Now, for every combination of the parameters (Σ, ∆, θ, ξ) one always has that M 12 = M 13 = 0, meaning that if one works in the region of parameters that selects this alignment as the global minimum all the SBP matrices generated by (23) are compatible with dark matter candidates, without adding further constraints. This feature contrasts with the findings in the Σ(36) model [16] where this vev direction led to a complete violation of the potential's symmetries (in Σ(36) this Z 2 is not a symmetry of the potential). It is interesting to consider how general is this feature, given this direction is a solution to potentials invariant under symmetries like ∆(54), where this same Z 2 given by ρ is not a symmetry of the potential, but there are other Z 2 symmetries. Consider ρ 23 : which is a part of the S 4 and also the ∆(54) group in the chosen basis. This element is preserved by the vev. The potential with the respective SBPs does not remain invariant under ρ 23 (as M 22 = M 33 in general). The SBPs we are considering depend on the vev (and not on the original symmetry of the potential), so we conclude that potentials invariant under groups containing ρ have dark matter candidates, whereas ρ 23 is not going to remain a residual symmetry due to the general SBPs.
(1, e iα , 0): In this case the fully-symmetric model only conserves two elements, which result from the composition of the U(1) symmetry with a general CP transformation and form two Z 2 subgroups: Both of them are broken by the general form of M. To preserve either of these symmetries, and proceeding analogously to the previous case, we must have that yielding the solutions which must be imposed as an additional constraint on the SBPs in order to have dark matter candidates.
(1, 1, 1), (1, ω, ω 2 ): Finally, we group these two directions due to their similar characteristics. In the fullysymmetric case both directions conserve multiple symmetries that are nevertheless completely broken by M.
In order to derive the least constraining relations we select the symmetry elements which require that ρ 13 : whose solutions are which must be imposed as an additional constraint on the SBPs in order to have dark matter candidates. Here, the positive solution corresponds to the condition obtained from ρ 23 and the negative solution from ρ 13 . Regarding the bounds on θ, they were obtained by combining the bounds on cos ξ with (25).

Conclusions
We have studied softly-broken A 4 and S 4 3 Higgs Doublet Models. We softly-break the symmetries in a controlled way such that the direction and magnitude of the symmetric vacuum expectations values are preserved. We present the mass spectra for each case after spontaneous symmetry breaking.
Having found cases with non-decaying mass eigenstates at leading order, we investigated in detail the residual symmetries left unbroken by the vacuum. In most cases these residual symmetries are broken by the soft-breaking terms and present the additional constraints required to preserve the residual symmetries. As an highlight, we have found one very appealing case where the most general vev-preserving soft-breaking terms preserve residual symmetries, without further constraints.
The residual symmetries, if left unbroken by both the vacuum and the soft-breaking parameters, lead to vanishing couplings between the different mass eigenstates and guarantee the stability of the lightest states charged under the symmetry. We conclude that in these cases there are stable dark matter candidates, which can occur in the softly-broken A 4 and S 4 3 Higgs Doublet Models without additional symmetries being imposed.

Acknowledgments
with the following Γ i      Focusing on the neutral bosons, one notices a different structure from all the remaining vev alignments, which leads to the curious situation when one sets sin 2θ = 0 there is always a degenerate pair, but the specific pair depends on the chosen parameters. More specifically, defining if (a > 0, b > 0) the degeneracy is between (h 2 , H 1 ), if (a > 0, b < 0) the degeneracy is between (h 1 , h 2 ) and if (a < 0, b > 0) the degeneracy is between (H 1 , H 2 ). The remaining case can not occur, since it would imply that Λ 3 > Λ 1 , which cannot happen if we are to have (1, i, 0) as the global minimum (if we work in the region of parameter space allowed in the fully-symmetric model).
Finally, it is important to emphasize that this "shifting" degeneracy does not influence the ordering of the neutral bosonic masses if, again, one obeys the inequalities that select this vev as a global minimum in the fully-symmetric model.