Erratum: Stability and symmetry breaking in the general three-Higgs-doublet model

: Stability, electroweak symmetry breaking, and the stationarity equations of the general three-Higgs-doublet model (3HDM) where all doublets carry the same hypercharge are discussed in detail. Employing the bilinear formalism the study of the 3HDM potential turns out to be straightforward. For the case that the potential leads to the physically relevant electroweak symmetry breaking we present explicit formulae for the masses of the physical Higgs bosons.


JHEP10(2015)149
Abstract: Stability, electroweak symmetry breaking, and the stationarity equations of the general three-Higgs-doublet model (3HDM) where all doublets carry the same hypercharge are discussed in detail. Employing the bilinear formalism the study of the 3HDM potential turns out to be straightforward. For the case that the potential leads to the physically relevant electroweak symmetry breaking we present explicit formulae for the masses of the physical Higgs bosons.

Introduction
T.D. Lee has shown decades ago that in the general two-Higgs-doublet model (THDM) CP violation is possible in the Higgs sector [2]. Meanwhile a lot of effort has been spent to investigate the THDM; see for instance the review [3] and references therein. In particular, some progress could be made employing the bilinear approach. The bilinears appear naturally in the Higgs potential in any n-Higgs doublet model (nHDM), since only the gauge-invariant scalar products of the Higgs-boson doublet fields may appear in the potential. The bilinear formalism was developed in detail in [4,5] and independently in [6].
Initiated by these works, many aspects of the THDM and the general nHDM were considered within this formalism. For instance, CP-violation properties of the THDM were presented in [6,7]. Different symmetries of the THDM and the general nHDM were considered in some detail employing bilinears; see for instance [8][9][10][11][12][13][14]. Relations between vacua of different properties in multi-Higgs-doublet models were derived in [15].
In this work we will focus on the three-Higgs-doublet model (3HDM). Many of the properties of this model are direct generalizations of the THDM, but there appear also new aspects. As we will see in detail, the space of Higgs-boson doublets does, in terms of bilinears, not correspond to the forward light cone space, as in case of the THDM [5], but to a certain subspace; see [6,7,16]. Driven mainly by the quark-and neutrino mixing data, several 3HDM's have been proposed; see for instance [17][18][19][20]

JHEP10(2015)149
In an analogous way to the study of the THDM in [5] we will discuss in the following, in sections 2 to 7, stability, electroweak symmetry breaking, and the stationarity points of the potential for any 3HDM. In section 7 we discuss the potential after symmetry breaking, section 8 presents our conclusion. Throughout the study we will illustrate the general results by two simple illustrative 3HDM examples. In appendix A we give mathematical relations concerning bilinears. In appendix B we discuss an explicit non-trivial example of a 3HDM, based on an O(2) × 2 symmetry [21].

Bilinears
We consider the tree-level Higgs potential of models with three Higgs-boson doublets satisfying SU(2) L × U(1) Y electroweak gauge symmetry. This is a generalization of the case of two Higgs-boson doublets which were discussed in detail in [5].
We assume that all doublets carry hypercharge y = +1/2 and denote the complex doublet fields by ; i = 1, 2, 3. (2.1) In the most general SU(2) L × U(1) Y gauge invariant Higgs potential the Higgs-boson doublets enter solely via products of the following form: It is convenient to discuss the properties of the Higgs potential such as its stability and its stationary points in terms of gauge invariant bilinears.
First we introduce the 3 × 2 matrix of the Higgs-boson fields in the following way, We arrange all possible SU(2) L × U(1) Y invariant scalar products into the hermitian 3× 3 matrix

4)
A basis for the 3 × 3 matrices is given by The decomposition of K (2.4) reads now where the real coefficients K α are given by With the matrix K, as defined in terms of the doublets in (2.4), as well as the decomposition (2.8), (2.9), we immediately express the scalar products in terms of the bilinears, (2.10) In the following we shall frequently use also (2.11) From (2.10) follows therefore, we have the inequalities Furthermore, we see from (2.11) that K + = 0 implies ϕ 1 = ϕ 2 = 0 which gives with (2.10) Further discussion of the basis change from α = 0, . . . , 8 to +, 1, . . . , 7, − is given in appendix A. The matrix K (2.4) is positive semidefinite which follows directly from its definition. This in turn gives The hermitian matrix K (2.4) is constructed from the Higgs field matrix, K = φφ † . Therefore, the nine coefficients K α of its decomposition (2.8) are completely fixed given the Higgs-boson fields.

JHEP10(2015)149
Since the 3 × 2 matrix φ has trivially rank smaller or equal 2, this holds also for the matrix K. On the other hand, any hermitian 3 × 3 matrix with rank equal or smaller than 2 which clearly has then vanishing determinant, det(K) = 0, determines the Higgsboson fields ϕ i , i = 1, 2, 3 uniquely, up to a gauge transformation. This was shown in detail in [5] in their theorem 5 for the general case of n-Higgs-boson doublets. In appendix A we show that the gauge orbits of the three Higgs fields (2.1) are characterised by the following set in the 9-dimensional space of (K 0 , . . . , K 8 ): Here G αβγ are completely symmetric constants defined in (A.31), (A.32). That is, to every gauge orbit of the Higgs-boson fields corresponds exactly one vector (K α ) satisfying (2.16) and vice versa. The first two relations of (2.16) are analogous to the light cone conditions of the THDM; see (36) of [5]. The determinant relation, trilinear in the K α , is specific for the 3HDM. Further discussions of the matrices K with rank 0, 1, 2 are presented in appendix A. Based on the bilinears we shall in the following discuss the potential, basis transformations, stability, minimization, and electroweak symmetry breaking of the general 3HDM.

The 3HDM potential and basis transformations
In terms of the bilinear coefficients, K 0 , K a , a = 1, . . . , 8 we can write the general 3HDM potential in the form where the 54 parameters ξ 0 , ξ a , η 00 , η a and η ab = η ba are real. The potential (3.1) contains all possible linear and quadratic terms of the bilinears -corresponding to all gauge invariant quadratic and quartic terms of the Higgs-boson doublets. Terms higher than quadratic in the bilinears should not appear in the potential with view of renormalizability. Any constant term in the potential can be dropped and therefore (3.1) is the most general 3HDM potential. We also introduce the notation (3.2) We can then write the potential (3.1) in the compact form Let us now consider a change of basis of the Higgs-boson fields, (3.5) and for the bilinears The matrix R(U ) has the properties (8). But the R(U ) form only a subset of SO (8).
For the bilinears a pure phase transformation, U = exp(iα)½ 3 , plays no role. We shall, therefore, consider here only transformations (3.4) with U ∈ SU(3). In the transformation of the bilinears (3.6) R ab (U ) is then the 8 × 8 matrix corresponding to U in the adjoint representation of SU(3).
The Higgs potential (3.1) remains unchanged under the replacement (3.6) if we perform an appropriate transformation of the parameters In the pure 3HDM potential, that is the model without fermions, we can use (3.9) to bring e.g. ξ to a standard form. Consider the hermitian matrix Applying a transformation U ∈ SU(3) we get with (3.7)-(3.9) With a suitable transformation U we can, therefore, diagonalise Λ ξ . That is, we always can achieve the form The number of relevant parameters of the general 3HDM potential is, therefore, 54 − 6 = 48. (3.13) Note that instead of ξ we could have chosen η in the above argument. Note also the slick proof of (3.12) and (3.13) employing the bilinear formalism. Let us remark on the basis transformations with respect to the 3-Higgs-doublet model. In a realistic model we have to consider, besides the Higgs potential, kinetic terms for JHEP10(2015)149 the Higgs-boson doublet fields as well as Yukawa terms which provide couplings of the Higgs-boson doublets to fermions. Under a basis transformation, that is, a transformation of the Higgs-boson doublets of the form (3.4), or equivalently, in terms of the bilinears, a transformation of the form (3.6), the kinetic terms of the Higgs doublets will remain invariant. However, we emphasize that, in general, the Yukawa couplings are not invariant under such a change of basis.
In order to illustrate the use of the bilinears we will consider two simple examples of explicit 3HDM Higgs potentials, (3.14) Here µ 2 > 0 is a parameter of dimension mass squared and λ > 0 is dimensionless. Employing (2.10) we write this potential in terms of the bilinears as This corresponds to the general form (3.1) with parameters, , and all other elements zero. (3.17) where we require Here, in the basis +, 1, . . . , 7, − (see (A.34) to (A.38)) only the following elements of ξ and E are non zero Let us now analyse stability of the general 3HDM potential (3.1), given in terms of the bilinears K 0 and K on the domain determined by (2.16). This can be done in an analogous way to the THDM; see [5]. The case 3/2K 0 = ϕ † 1 ϕ 1 + ϕ † 2 ϕ 2 + ϕ † 3 ϕ 3 = 0 corresponds to vanishing Higgs-boson fields and V = 0. For K 0 > 0 we define Due to (2.16) we have for k the domain D k : The domain boundary, ∂D k , is characterised by where we introduce the functions J 2 (k) and J 4 (k) on the domain (4.2). A stable potential means that it is bounded from below. The stability is determined by the behaviour of V in the limit K 0 → ∞, that is, by the signs of J 4 (k) and J 2 (k) in (4.4), (4.5). For a model to be at least marginally stable, the conditions for all k ∈ D k , that is, all k satisfying (4.2) are necessary and sufficient, since this is equivalent to V ≥ 0 for K 0 → ∞ in all possible allowed directions k. The more strict stability property V → ∞ for K 0 → ∞ and any allowed k requires V to be stable either in the strong or the weak sense. For strong stability we require for all k ∈ D k ; see (4.2). For stability in the weak sense we require for all k ∈ D k J 4 (k) ≥0, To check that J 4 (k) is positive (semi-)definite, it is sufficient to consider its value for all stationary points on the domain D k . This holds because the global minimum of the continuous function J 4 (k) is reached on the compact domain D k , and the global minimum is among those stationary points.

JHEP10(2015)149
To obtain the stationary points of J 4 (k) in the interior of the domain D k we add to J 4 (k) the second condition in (4.2) with a Lagrange multiplier u. The stationary points are then obtained from provided the gradient matrix of the constraint equation has rank 1. This can easily be checked. With k 0 = 1 we have from (A.30) and (A.33) In our case the corresponding matrix M = M α λ α /2 has rank 1, see (A.26), therefore, M 0 and at least one element M a with a ∈ {1, . . . , 8} have to be non zero. That is, ∂g(k)/∂k a has rank 1 as required.
For the stationary points on the boundary ∂D k we have two constraints, see (4.2), (4.3), Here the gradient matrix reads (4.14) For K as in (4.10) but now of rank 1 we find from (A.26), M = 0 and, therefore, M a = 0, a = 1, . . . , 8. Thus, the gradient matrix has here only rank 1 and not the required rank 2 for the application of the Lagrange multiplier method in an analogous way to (4.9). We turn, therefore, to the parametrization of the rank 1 matrices K of (A.14) to (A.16): This gives, according to (2.9) and (4.1), (4.5),

JHEP10(2015)149
Now we can determine the stationary points of J 4 (w † , w) subject to the constraint (4.16).
With u a Lagrange multiplier we get Here the gradient matrix of the constraint is of rank 1 as required and we get explicitly All stationary points obtained from (4.9) and (4.19) have to fulfill the condition J 4 (k) > 0 for stability in the strong sense. If for all stationary points we have J 4 (k) ≥ 0, then for every solution k with J 4 (k) = 0 we have to have J 2 (k) > 0 for stability in the weak sense, or at least J 2 (k) = 0 for marginal stability. If none of these conditions is fulfilled, that is, if we find at least one stationary direction k with J 4 (k) < 0 or J 4 (k) = 0 but J 2 (k) < 0, the potential is unstable. In our explicit example I, with V I from (3.15), the functions J 2 (k) and J 4 (k) read Obviously, J 4 (k) is always positive for λ > 0 in any direction k, therefore, the potential is stable in the strong sense. That is, stability is here guarantied by the quartic terms of the potential alone. For example II from (3.18), (3.19), we get We have V 4 > 0 for K − > 0 but V 4 = 0 for K − = 0. Thus, we have to investigate V 2 for K − = 0: Due to (2.13) and (2.14) we have that is, for K + = 0. But this implies K = 0. Thus, the potential V II from (3.18), (3.19) is stable in the weak sense.

JHEP10(2015)149
5 Electroweak symmetry breaking of the 3HDM Suppose now that the 3HDM potential is stable, that is, bounded from below. Then the global minimum will be among the stationary points of V . In the following the different types of minima with respect to electroweak symmetry breaking are discussed and the corresponding stationarity equations are presented.
As we have discussed in section 2, the space of the Higgs-boson doublets is determined, up to electroweak gauge transformations, by the space of the hermitian 3 × 3 matrices K with rank smaller or equal 2. Since the rank of the matrix K is equal to the rank of the Higgs-boson field matrix φ (2.3) we can distinguish the different types of minima with respect to electroweak symmetry breaking as follows. At the global minimum, that is, the vacuum configuration, we write the 3 × 2 matrix of the Higgs-boson fields as In the case this matrix has rank 2, we cannot, by a SU(2) L × U(1) Y transformation, achieve a form with all charged components v + i , i = 1, 2, 3 vanishing. This means that the full SU(2) L × U(1) Y is broken. In case we have at the minimum a matrix φ with rank one, we can, by a SU(2) L × U(1) Y transformation, achieve a form with all charged components v + i vanishing. The unbroken U(1) gauge group can then be identified with the electromagnetic gauge group. Therefore, a minimum with rank one corresponds to the electroweak-symmetry breaking SU(2) L × U(1) Y → U(1) em . Eventually, a vanishing matrix at the minimum, φ = 0, corresponds to an unbroken electroweak symmetry. Of course, only a minimum with a partially broken electroweak symmetry is physically acceptable.
We study now the matrix K v corresponding to φ (5.1) For an acceptable vacuum φ , K v must have rank 1. From (A.20) we see that K v has rank 1 and is positive semidefinite if and only if By a suitable U(3) transformation (3.4) we can bring the vacuum value φ of rank 1 to the form

JHEP10(2015)149
In a realistic model v 0 must be the usual Higgs-boson vacuum expectation value, v 0 ≈ 246 GeV. (5.5) With (5.4) we get in this basis a particularly simple form for K v respectively K vα : Another possible choice for the vacuum expectation value, obtainable by a suitable transformation (3.4) Here we get In the cases where φ of (5.1) has rank 2 or rank 0 also the matrix K v , (5.2), has rank 2 or zero, respectively. The corresponding conditions for K v are given explicitly in (A. 19) and (A.21), respectively. We can, therefore, summarise our findings for the vacuum values to a given potential V as follows.
Let φ be the vacuum expectation value of the Higgs-boson field matrix to a given, stable, potential V and K v = φ φ † = K vα λ α /2. The gauge symmetry SU(2) L × U(1) Y is fully broken by the vacuum if and only if We have the breaking SU Of course, we always have with G αβγ defined in (A.31).

JHEP10(2015)149
6 Stationary points Following the study of stability and electroweak symmetry breaking in the last two sections we shall now present the stationarity equations. We suppose again that the potential is stable. Then the global minimum is among the stationary points of V . We classify the stationary points by the rank of the stationarity matrix K. In the following we use the conditions for K having rank 0, 1, 2, or 3 as given in appendix A; see (A.18) -(A.21).
All stationarity matrices K = K α λ α /2 of rank 2 are obtained from the following system of equations where u is a Lagrange multiplier: Explicitly we get here, using (3.3), The gradient matrix of the constraint is given by (see (A.30) and (A.33)) and has rank 1 as required. This holds since for K of rank 2, M has rank 1, as we see from (A.26), implying, for instance, M 0 = tr(M λ 0 ) > 0. For the stationarity matrices K = K α λ α /2 of rank 1 we cannot use the Lagrange multiplier method in an analogous way to (6.1). The two constraints for rank 1, g 1 = 2K 2 0 − K a K a = 0 and g 2 = det(K) = 0 yield a gradient matrix of rank 1 for K of rank 1. However, the Langrange multiplier method requires that this matrix has rank 2 in this case. The stationarity equations of the potential V (K) for K of rank 1 follow from a parametrization of the matrix K of rank 1, as given in (A.16):

JHEP10(2015)149
This gives Taking into account the constraint (6.5) with a Lagrange multiplier u we have to determine the stationary points of under variation of K 0 , and w † , w. The gradient matrix of the constraint has rank 1 as required and we get with (3.3) the following system of equations The stationarity matrix K = K α λ α /2 with the lowest value of V (K 0 , . . . , K 8 ) gives the global minimum K v of the potential. Note that in general there may be degenerate global minima with the same potential value. Systems of equations of the kind (6.2), (6.8) can be solved via the Groebner-basis approach or homotopy continuation; see for instance [22,23].

The potential after symmetry breaking
In this section we discuss the potential after symmetry breaking and the procedure to calculate the physical Higgs-boson masses and self couplings in the 3HDM. We will assume that the potential is stable and leads to the desired electroweak symmetry breaking SU(2) L × U(1) Y → U(1) em . In particular, the global minimum is then a solution of the set of equations (6.8). In this case we can, in the unitary gauge, by an electroweak gauge transformation and a U(3) rotation (3.4) always achieve the form (5.4) for the vacuum expectation value of the Higgs-field matrix. For the original Higgs fields expressed in terms of the physical fields we get then with v 0 real and positive, neutral fields H 0

JHEP10(2015)149
In general, however, the physical fields of definite mass are linear combinations of the fields in (7.2). Obviously, the 3 original complex doublets of any 3HDM, corresponding to 12 real degrees of freedom, yield 5 real fields and 2 complex fields, with the 3 remaining degrees of freedom absorbed via the mechanism of electroweak symmetry breaking. Throughout this section we shall work in a basis where K v has the form (5.6). Representing K as in (6.4) we get where e 1 , e 2 , e 3 are the three-dimensional Cartesian unit vectors. That is, K 0 = v 2 0 / √ 6 and w = e 3 must be solutions of (6.8). We get and define for α = 0, . . . 8 where the ± components are defined in (A.34) ff. Inserting all this in (6.8) with w = e 3 we get from the first equation there The ζ α are all real, therefore, we have from (7.7) The second equation of (6.8) gives ζ − = 0 (7.9) and thus, from (7.8), u = 0. To summarise, in the basis where φ and K v have the forms (5.4) and (5.6), respectively, we have ζ α = 0 for α = 4, 5, 6, 7, −. (7.10) The next task is to expand φ, K, and V in terms of the physical fields (7.2). For φ we write (7.11)

JHEP10(2015)149
From this we get for K(x) and K α (x) the following with K v and K vα given in (5.6) 14) For the potential we have the following expansion in the order of the physical fields β (x).
Since we are expanding around the global minimum we must have that V (0) is below or at most equal to V (K α = 0) = 0. From (7.20) this implies Furthermore, the mass squared matrices M 2 ch and M 2 n must be positive semidefinite. This implies, for instance, from (7.23) that we must have

JHEP10(2015)149
To obtain the physical Higgs bosons of definite mass, the matrices (7.23) and (7.24) have to be diagonalised. We note that the field h 0 (x) is a mass eigenstate if E α− = 0 for α = 4, 5, 6.7. (7.28) In this case h 0 (x) is what is called aligned with the vacuum expectation value and its mass squared is given by In our example I, the 3HDM Higgs potential (3.15), we find stationary points for vanishing fields, corresponding to an unbroken EW symmetry, from the set (6.1) we get no solution with K 0 > 0, and from the set (6.8) we get one solution with The corresponding potential value is V (0) = −1/4 · (µ 2 ) 2 /λ and is the deepest stationary point and therefore the global minimum. From (5.6) we see that the global minimum corresponds to a vacuum expectation value v 0 = µ 2 /λ. For the mass matrices we get from (7.23) and (7.24) Turning to example II, V II of (3.18), we have as stationary points the trivial one, K α = 0 with V II (0) = 0, no stationary point from (6.1) and one point from (6.8). The latter is obtained again for Here we get from (5.6), (7.23), and (7.24) This simple example shows that the squared masses of the charged physical Higgs bosons need not be degenerate in a 3HDM having the correct electroweak symmetry breaking. In appendix B a nontrivial example of a 3HDM is discussed.

Conclusion
The three-Higgs-doublet model has been studied as a generalization of the THDM. Stability, electroweak symmetry breaking, and the types of stationary points of the potential have been investigated. Explicit sets of equations have been presented which allow to determine the stability of any 3HDM and, in case of a stable potential, to find the global minimum or the degenerate global minima in case the potential has such. For the case that the 3HDM has the physically relevant electroweak symmetry breaking SU(2) L × U(1) Y → U(1) em we

JHEP10(2015)149
have given explicit expressions for the mass squared matrices of the charged and neutral physical Higgs bosons. The use of bilinears turns out to be very helpful: in particular, irrelevant gauge degrees of freedom are avoided and the degree of the polynomial equations which are to be solved is reduced in this formalism. In general, the sets of equations which determine stability and the stationary points are rather involved. However, approaches like the Groebner-basis approach or homotopy continuation may be applied to solve these systems of equations in an efficient way. This has been demonstrated for a 3HDM based on a O(2) × 2 symmetry.

Acknowledgments
The work of M.M. was supported, in part, by Fondecyt (Chile) Grant No. 1140568.

(A.2)
Employing the properties of the Gell-Mann matrices (2.7) we can write the second trace condition in the form (tr(K)) 2 − tr( Suppose now that the matrix K has rank 3, then, we have to have for all three κ i It follows immediately from (A.2) If, for the reverse, we have for a hermitian matrix K the conditions (A.5) fulfilled, then, using (A.2) we find that we must have all κ i > 0. That is, K has rank 3 and is positive definite. Suppose the matrix K has rank 2, then, without loss of generality, we can assume

JHEP10(2015)149
It follows immediately from (A.2) that tr(K) > 0, (tr(K)) 2 − tr(K 2 ) > 0, det(K) = 0. (A.7) If, for the reverse, we have for a hermitian matrix K the conditions (A.7) fulfilled, then, from the last equation in (A.2) at least one κ i = 0. Without loss of generality we can suppose κ 3 = 0. We have then which implies κ 1 > 0 and κ 2 > 0. That is, K has rank 2 and is positive semidefinite. Another way to characterise the positive semidefinite matrices of rank 2 is as follows. We set (A.9) Let w 1 and w 2 be orthonormal eigenvectors of K to κ 1 and κ 2 , respectively, then we have For 0 < χ < π/4 the w i are fixed up to phases, for χ = π/4 we may make arbitrary U(2) rotations of w 1 and w 2 . Clearly, every positive semidefinite matrix K of the form (A.10) has rank 2 and every positive semidefinte matrix K of rank 2 can be written in the form (A.10). Now, let us suppose the matrix K has rank 1, then, without loss of generality, we can assume κ 1 > 0, κ 2 = 0, κ 3 = 0. On the other hand, having the conditions (A.13) for a hermitian matrix K fulfilled, employing (A.2), the determinant condition requires that at least one κ i vanishes, for instance κ 3 = 0 without loss of generality. Then the second condition requires that another eigenvalue has to vanish, for instance κ 2 = 0. Eventually, the first condition then dictates that the remaining κ 1 > 0. Hence, K has rank 1 and is positive semidefinite. Let now w be eigenvector of K to the eigenvalue κ 1 with We set

JHEP10(2015)149
Then, any positive semidefinite matrix K of rank 1 can be represented as Conversely, any matrix of the form (A.16) with K 0 > 0 and w † w = 1 is a positive semidefinite matrix of rank 1. Clearly, w is fixed up to a phase transformation. Finally, suppose the matrix K has rank 0, then, clearly, all κ i have to vanish, corresponding to tr(K) = 0, (tr(K)) 2 − tr(K 2 ) = 0, det(K) = 0. (A.17) Vice versa, starting with the conditions (A.17) for a hermitian matrix K, the determinant condition requires that one eigenvalue, for instance κ 3 = 0 has to vanish, the second condition in turn requires that another, say κ 2 = 0, and the first trace condition that also the third κ 1 = 0. This means K = 0. Therefore, we have shown the following theorem.   [5]. But the determinant condition, trilinear in K α , is specific for the 3HDM.
To express also det(K) in terms of the expansion coefficients K α , α = 0, . . . , 8, we proceed as follows (see also [14]). We introduce, along with the matrix K, a matrix M = (M ij ): For a hermitian matrix K also M is hermitian. For any U ∈ U(3) we have the relation Using this we find easily that under a transformation ( Inserting here (A.22) we get the expression of M α in terms of the K β (2.9) as follows: Clearly, G αβγ is completely symmetric in α, β, γ. Explicitly we get

B Example of a 3HDM Higgs potential
Let us apply the developed formalism to a non-trivial 3HDM potential. We emphasize that any specific 3HDM can be treated along the following lines. We will apply the homotopy continuation approach to solve the systems of polynomial equations allowing us to discuss stability and the stationarity points of the model. Of course, other methods may be applied, like the Groebner-basis approach. These methods were successfully applied to Higgs potentials in the past; see for instance [22,23]. In these works brief introductions to Groebner-bases and homotopy continuation can also be found. The model we want to study was presented in [21] and is based on a O(2)× 2 symmetry involving three Higgs-boson doublets. All the elementary particles and in particular the The term µ m ϕ † 1 ϕ 2 + ϕ † 2 ϕ 1 breaks the U(1) symmetry softly (for details we refer to [21]). This model has nine real parameters and one complex parameter a 4 , corresponding to eleven real parameters in total.

JHEP10(2015)149
We choose as an explicit numerical example the following values for the parameters, where we take only the quartic couplings from the reference point in [21]: Actually, we start with all quartic parameters as given in (B.3) and then fix the quadratic parameters by the condition of a vanishing gradient of the potential, employing K v0 = v 2 0 / √ 6 (see (5.6) with v 0 as given in (5.5)). In this way we ensure that there is at least one stationary solution which corresponds to the correct vacuum expectation value. Let us note that this procedure by no means guarantees that the corresponding potential is stable and has a global minimum with the correct partially broken electroweak symmetry -as we will see below.
The stability and stationarity equations are polynomial systems of equations as given in (4.9), (4.19) and (6.1), (6.8), respectively. In this example we apply for all the polynomial systems of equations the homotopy continuation approach as implemented in the PHCpack package [25]. We first look for solutions disregarding the inequalities and then select by hand all solutions which fulfill them. Technically, for the real indeterminants we take into account solutions with an imaginary part smaller than 0.001. With respect to our computations we remark that for the most involved cases of systems of equations we encounter about a minute of time consumption on an ordinary PC.
We start with studying stability of the potential; see section 4. To this end we separate the quadratic and the quartic terms of the potential. Inserting the parameters (B.2) into (4.4), (4.5), yields with the parameters given in (B.3). Now we have to find the stationary points of J 4 (k), that is, we have to solve the systems of equations (4.9) and (4.19), respectively. In case of (4.9) the invariants are the eight components of the vector k and one Lagrange multiplier u. in case of (4.19) the invariants are the vector components of w, w † , (the eigenvector of the matrix K, see (A.16)), as well as one Lagrange multiplier u. We decompose the vector components of w, w † into real and imaginary parts such that in this form all indeterminants, that is, Re(w 1 ), Im(w 1 ), Re(w 2 ), Im(w 2 ), Re(w 3 ), Im(w 3 ), u, are real. Then we split the system of equations (4.19) into its real and imaginary parts. Note that the last equation w † w − 1 = 0 has only a trivial imaginary part. Eventually, we end up with seven equations in seven indeterminants in this case.

JHEP10(2015)149
With respect to the system (4.9) we detect four solutions k and u fulfilling the inequality 2 − k 2 > 0. Plugging these solutions into J 4 (k) in (B.4) we find solely positive values. With respect to the systems (4.19) we find solely solutions which give, with respect to J 4 (k), also positive values. Stability in general requires that there is no stationary direction k with J 4 (k) < 0 or J 4 (k) = 0 but J 2 (k) < 0. In our example J 4 (k) is positive for all stationary directions k, therefore, the potential is stable in the strong sense for the chosen parameters.
Since the potential with parameters (B.3) is stable we proceed by studying the stationary points; see section 6. To this end we plug the parameters (B.2) into the potential (3.1). This gives Now we have to solve the systems of polynomial equations (6.1) (corresponding to solutions which break electroweak symmetry fully) and (6.8) (corresponding to solutions with break electroweak symmetry partially leaving the electromagnetic U(1) symmetry intact). For the set of equality equations (6.1) we find one real solution fulfilling the inequalities 2K 2 0 − K a K a > 0 and K 0 > 0, This solution corresponds to a potential value of V (K) = −1.83109 · 10 9 GeV 4 . Since this solution originates from the set (6.1) it corresponds to a stationary point with fully broken electroweak symmetry. Eventually, we write the set (6.8) again with explicit real and imaginary parts of the vectors w, w † and subsequently separate real and imaginary parts of the equations. We have here eight equations in eight indeterminants, K 0 , Re(w 1 ), Im(w 1 ), Re(w 2 ), Im(w 2 ), Re(w 3 ), Im(w 3 ), u. In terms of bilinears, employing (A.16), we encounter three real solutions fulfilling the inequality K 0 > 0:

JHEP10(2015)149
In addition, we always have the trivial solution with vanishing bilinears, corresponding to a vanishing potential. This solution corresponds to an unbroken electroweak symmetry.
The global minimum is given by the stationary point corresponding to the deepest potential value. In this example the deepest stationary point is given by (B.6) and corresponds to a fully broken electroweak symmetry which is physically not acceptable.
Our analysis clearly shows that requiring the potential to have a stationary point giving the desired electroweak symmetry breaking and vacuum expectation value does not guarantee that one has a viable model. In contrast, the study of stability and all stationary points reveals where the true global minimum of the potential is. Then one has to check if, at this global minimum, one has the desired partial electroweak symmetry breaking. As we have seen, our methods employing bilinears allow to perform these investigations in an efficient way.
Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.