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Tree-level metastability bounds for the most general two Higgs doublet model

Within two Higgs doublet models, it is possible that the current vacuum is not the global minimum, in which case it could possibly decay at a later stage. We discuss the tree-level conditions which must be obeyed by the most general scalar potential in order to preclude that possibility. We propose a new procedure which is not only more general but also easier to implement than the previously published one, including CP conserving as well as CP violating scalar sectors. We illustrate these conditions within the context of the Z2 model, softly broken by a complex, CP violating parameter.

Within two Higgs doublet models, it is possible that the current vacuum is not the global minimum, in which case it could possibly decay at a later stage. We discuss the tree-level conditions which must be obeyed by the most general scalar potential in order to preclude that possibility. We propose a new procedure which is not only more general but also easier to implement than the previously published one, including CP conserving as well as CP violating scalar sectors. We illustrate these conditions within the context of the Z2 model, softly broken by a complex, CP violating parameter. The discovery of a fundamental scalar particle [1,2] prompts the search for other fundamental scalars. The simplest theoretical possibility consists in adding a second doublet to the Standard Model (SM), forming a two Higgs doublet model (2HDM) -for reviews see, for example, Refs. [3,4].
There are several theoretical constraints that a viable 2HDM scalar sector must obey: it must be bounded from below [5], and it must conform to perturbative unitarity [6][7][8][9]. In addition, the scalar potential of the 2HDM may have simultaneously two neutral minima, both CP conserving or both CP-violating [10][11][12][13][14]. In those cases, if one were in the local metastable minimum, there would be the possibility of decaying later into the global minimum: the true vacuum. We dub this possibility the panic vacuum.
When building a model with a scalar sector allowing for several distinct minima, one is free to take a metastable minimum as physically acceptable, provided it is sufficiently long-lived, or to disregard it altogether. But one definitely cannot neglect this issue. Typically, this would be investigated by reconstructing the potential from the input parameters (vacuum expectation values, masses, etc. . . ), finding the second vacuum numerically and comparing the potential depths in the two vacua. This is time consuming, especially if one performs extensive scans over parameter space, and it could give a false positive. Luckily, within the 2HDM there is an efficient way to study these issues without trying to solve the minimization problem. Following the works [12][13][14], a discriminant for panic vacua has been studied in the context of the 2HDM with a softly broken U (1) symmetry [18], which we denote by "U (1)-2HDM", and in the context of a Z 2 -symmetric 2HDM, softly broken by a real parameter [19], which we denote by "real 2HDM". The discriminant in this case is a compact combination of the parameters of the potential and the vacuum expectation values (vevs) calculated at a single minimum, whose positive sign guarantees tree-level stability and negative sign signals the presence of a deeper minimum.
A technique applicable in the case of a scalar potential without CP violation was also presented in Ref. [19]. The technique had five steps: i) perform a numerical search for the eigenvalues of a certain 4 × 4 matrix (Λ E ); ii) perform a numerical calculation of its eigenvectors; iii) combine the eigenvectors into a rotation matrix, used to transform the parameters of the potential; iv) determine whether a certain rotated quantityM 0 were positive or negative; v) apply the discriminant, distinguishing between the metastable vacuum and the global minimum (at tree level).
Those studies are not applicable, for example, to the 2HDM with a complex soft breaking of the Z 2 symmetry, usually known as the C2HDM [20][21][22][23][24][25][26][27]. There has been renewed interest in this model, in part because it is the simplest model allowing for the 125 GeV Higgs to have a mixture of scalar and pseudoscalar components [28][29][30][31][32][33]. Remarkably, all the data available is still consistent with the possibility that its couplings to the up quarks are mainly scalar while its couplings to the down quarks are mainly pseudoscalar [32]. So far, no simple criteria to avoid metastable vacua has been devised for potentials with CP violation, either spontaneous or explicit.
In this article, we solve that problem, thus completing the study of the 2HDM. We introduce a new discriminant which, using very simple criteria, distinguishes between a metastable vacuum and the global minimum. Besides devising a discriminant applicable to the most general 2HDM, we also improve upon previous analysis of the U (1)-2HDM and the real 2HDM by presenting the discriminant in terms of physical masses, mixing angles, and couplings rather than the parameters of the potential. In the most general 2HDM, our procedure improves over the one suggested in [19] for the CP conserving 2HDM in two ways. First, we build it only from eigenvalues of Λ E and do not ask for the calculation of the eigenvectors nor for explicit basis transformation. Second, we prove that our method is safe even in pathological cases of potentials unbounded from below with matrix Λ E having complex eigenvalues.
The paper is organized as follows. In section II we present our notation, used in section III to develop the discriminant D and the method which disentangles the metastable vacuum from the true vacuum. In section IV, we discuss in detail the U (1)-2HDM, the real 2HDM, and the C2HDM, presenting the discriminant in terms of physical parameters, such as masses and mixing angles, instead of parameters of the potential. In these cases, there is a simple condition forcing the potential to be bounded from below, and the method to avoid panic vacua involves a single application of a discriminantD. We draw our conclusions in section V and relegate some detailed proofs to the appendices.

II. NOTATION
The Higgs potential of the most general 2HDM is usually written as [4] where hermiticity forces all couplings to be real, except m 2 12 , λ 5 , λ 6 , and λ 7 . If all couplings are real, then there is explicit CP conservation.
For the study of minima, an alternative formulation in terms of quantities bilinear in the fields is useful [10][11][12][13][14][15][16][17]: where 1 and while Λ µν is a symmetric matrix. For our purposes, it is more useful to consider the mixed symmetry tensor Λ µ . ν = g µα Λ αν , viewed as a matrix in Euclidean space: Of course, one can change the basis of the fields by Henceforth, using a roman lowercase letter in a four dimensional vector refers implicitly to the spatial components: r k , k = 1, 2, 3. Choosing the matrix U in U (2) guarantees that the kinetic terms retain their canonical form. In that case, r 0 is invariant and r k suffers a O(3) change. However, as hinted in [10,11] and explored extensively in Refs. [12][13][14], if one wishes to study exclusively the properties of the potential, one can advantageously take any general transformation U (unitary or not), in which case r µ suffers a SO(1, 3) change. The advantage of these more general transformations is that, for potentials bounded from below (BFB) in the strong sense [10], one can diagonalize and the eigenvalues satisfy [12] Note that the signs of Λ k are arbitrary.
In the light of future discussions, it is instructive to see what changes within this formalism if the potential is unbounded from below. Although this situation is unphysical, one might accidentally run into it when scanning over the parameter space. In order to be sure that the discriminant we develop below for the general 2HDM does not produce false positive results, we must extend the bilinear formalism to this situation.
First, since Λ E is not symmetric, it might happen that some of its eigenvalues are complex. There can be only one pair of complex and mutually conjugate eigenvalues, see Appendix C. This leads to a potential unbounded from below, as proved in the Erratum of Ref. [12]. Note that in this case we can still have a local minimum; the simplest example being the inert 2HDM with λ 2 < 0. Second, if all eigenvalues are real and some of them are degenerate, it is possible that Λ E is not diagonalizable at all. This can happen for a potential whose quartic part has flat directions, so that it is not bounded from below in the strong sense. Third, even if Λ E is diagonalizable (7), it might happen that its eigenvalues do not satisfy (8), in which case the potential is also unbounded from below; yet a local minimum can exist. Thus, a discriminating procedure must be guaranteed to cut off these cases without referring to the diagonalization of Λ E .
After spontaneous symmetry breaking (SSB), the fields acquire vacuum expectation values: where, without loss of generality, we have taken φ 1 to be real. Thus, where Then, v 1 and v 2 depend only on tan β = v 2 /v 1 . Notice that r µ refers to bilinears in fields, while r µ refers to their vevs. Given our definitions, the allowed r µ satisfy corresponding to the forward lightcone. The SU (2)×U (1) symmetric vacuum lies at the apex, the surface corresponds to neutral vacua, while any point in the interior of the lightcone represents charge breaking vacua.

III. DISCRIMINATING THE GLOBAL MINIMUM FROM METASTABLE VACUA
Phenomenological analyses of 2HDM usually start with the following procedure. One assigns values to physical observables, such as masses, mixing angles and couplings, which obey current (and simple) experimental constraints. For example, one imposes that one of the Higgs masses equals 125 GeV. The fact that masses squared are (of course) chosen positive, implies that that point in parameter space corresponds to a minimum; it may be local or global, but it is guaranteed to be a minimum. Further, since either or both vevs are nonzero, one is guaranteed to be in a minimum which breaks SU (2) × U (1). But one is still unprotected against unpleasant situations where one is sitting in the panic vacuum or when the potential is unbounded from below. From the input values, one extracts the parameters of the potential. One can then impose theoretical constraints: boundedness from below, perturbative unitarity, and avoidance of the panic vacuum.
When λ 6 = λ 7 = 0, Λ E is block diagonal, its eigenvalues can be found analytically, and the conditions of bounded from below in the strong sense can be written simply in terms of the parameters of the potential as: When either λ 6 or λ 7 differ from zero [12], one needs to construct the matrix Λ E , diagonalize it numerically -c.f. Eqs. (5) and (7) -and then impose Eqs. (8). It turns out that, for a general potential, this part of the procedure is more difficult than it seems. The point is that, even after the diagonalization leading to four eigenvalues Λ α (α = 0, 1, 2, 3), one must identify which one of the four is Λ 0 . Ref [19] addressed this issue in the context of the CP conserving two Higgs doublet model by resorting to the identification of eigenvectors, a rotation matrix, and some rotated vectors.
Here we propose a procedure valid for the most general potential and involving only the eigenvalues. Imagine that we have identified the four eigenvalues Λ α , that they are real, and we wish to know which one to ascribe the subindex 0. We construct the projection operators:P Then, will be positive for only one value of α; the corresponding eigenvalue is the time-like Λ 0 . This assertion is proved in Appendix A. The method proposed here to avoid panic vacua uses exclusively the stationarity conditions and the diagonalization of Λ E already needed to impose boundness from below. The former can be obtained by minimizing the auxiliary potentialV with respect to r µ and ζ, yielding Any component of this equation can be used to determine ζ. Typically, equating ζ obtained from two different components yields some m 2 ij (in M µ ) in terms of the quartic coefficients λ k (in Λ µ . ν ). The quantity ζ calculated at any neutral stationary point has a direct physical interpretation [11,12]: the charged Higgs mass squared is equal to ζv 2 . Our new discriminant is given by where 1 is the four dimensional identity matrix. Writing Λ E in the diagonal basis, With these definitions, we introduce the following Method : 1. Determine ζ from Eq. (16). (a) If Λ 0 < Λ k for some k = 1, 2, 3, then the potential is not bounded from below and the point must be discarded.
(b) If Λ 0 > Λ k for all k = 1, 2, 3, then we must continue. i. If ζ > Λ 0 then we are in a global minimum (there is no other minimum). ii. If ζ < Λ 0 then we are in the panic vacuum (metastable minimum).
As mentioned, in cases other than softly broken Z 2 the analysis of boundedness from below requires the diagonalization of Λ E , while D in Eq. (17) can be calculated in any basis. And, in many cases of interest it will turn out that our point has D > 0. This explains why we compute D before checking whether the potential is bounded from below. When D > 0, we are in the global minimum and we automatically know that the potential is bounded from below. There exists no other situation in which D > 0 is compatible with a minimum. This is proved in Appendices B and C. Thus, if D > 0. metastability is avoided and the phenomenological analysis of this point in parameter space can continue immediately. The diagonalization of Λ E is only performed when D < 0. This procedure solves completely the identification of the global minimum in the most general 2HDM.
Since ζ = m 2 H ± /v 2 , we can also make another curious observation. Suppose we know completely the quartic part of the potential and we know that it is bounded from below. Then, the local vs. global minimum ambiguity is decided just by the value of the charged Higgs mass through Eq. (17).

IV. APPLICATION TO A SOFLTY BROKEN Z2 POTENTIAL
This section is dedicated to potentials with a softly broken Z 2 symmetry, where λ 6 = λ 7 = 0 in Eq. (1). We allow m 2 12 to be complex (C2HDM) or real (real 2HDM). In addition, we may have λ 5 = 0 (U (1)-2HDM). In such cases, Λ E in Eq. (5) is block diagonal and its eigenvalues are easy to find: This formulas lead from Eqs. (8) to Eqs. (12). Moreover, using a basis where both vevs are real 2 , we find In going from the first to the second line we have used the the stationarity conditions. Some relevant additional formulas will be presented in section IV A. Since both Eqs. (8) and (18) depend on Λ α , there is an intimate connection between boundedness from below and the discriminant for panic vacua. In cases other than soflty broken Z 2 (real or complex), including in particular the most general CP conserving potential, there is no simplified compact form to assure that the potential is bounded from below written in terms of the original parameters of the potential λ 1 -λ 7 . Thus, as proposed in our Method of section III, it is best to start by computing D (a determinant which can be calculated in any basis) and only later perform the diagonalization of Λ E which enables the application of Eqs. (8), guaranteeing that the potential is bounded from below. In contrast, for soflty broken Z 2 potentials (λ 6 = λ 7 = 0, m 2 12 real or complex), the bounded from below conditions can be applied directly in the form of Eqs. (12). In that case, one can start by imposing these conditions and later study the discriminant Then: 2 One can always find such a basis. The phase will appear as part of m 2 12 and λ 5 .
We are in a global minimum if and only ifD > 0.
We stress that the use ofD is only relevant after one has imposed boundedness from below. This is simple for softly broken Z 2 models, but requires the diagonalization of Λ E otherwise. In those more general cases, one would be better served using D before diagonalization of Λ E ; if D > 0 the point corresponds to the global minimum (the potential is guaranteed to be bounded from below) and the phenomenological analysis may continue; only if D < 0 should the diagonalization of Λ E proceed in order to determine whether the point corresponds to a global minimum or not, along the lines of the Method of section III Using Eqs. (19) and (20), it is easy to write both D andD in terms of the parameters of the potential. We will shortly use this to describe D in terms of physical parameters. However, the presence of the square root √ λ 1 λ 2 in Λ 0 makes it impossible to write simple expressions forD in terms of physical parameters. In contrast, it is possible to write D in terms of physical parameters. This is what we turn to next.

A. The C2HDM
We consider here the C2HDM [20][21][22][23][24][25][26][27][28], for which λ 6 = λ 7 = 0 and arg(λ 5 ) = 2arg(m 2 12 ), ensuring that there is explicit CP violation in the scalar potential. We follow the notation of Refs. [31,32]. In particular, a basis is chosen such that both vevs are real -δ = 0 in Eq. (9). The minimization conditions are: where λ 345 = λ 3 + λ 4 + Re (λ 5 ). We may parametrize the original fields as where G + and G 0 are the Goldstone bosons and H + is charged Higgs with mass m H ± . Thus forth c angle (s angle ) refers to the cosine (sine) of that angle. The neutral mass matrix M 2 is diagonalized by the orthogonal transformation such that where the neutral scalar masses are ordered as m 1 ≤ m 2 ≤ m 3 . We parametrize the matrix R by [22] where i is short for α i . The limit of the real 2HDM is obtained by setting α 2 = α 3 = 0 and α = α 1 − π/2. The parameters of the potential may be traded for physical observables through with The observables are m 1 , m 2 , m H ± , α 1,2,3 , β, and Re(m 2 12 ), while m 3 is determined from We can now write the discriminant D in terms of physical parameters. Substituting Eqs. (29) into Eqs. (19) and (20), and those into Eq. (18), we find where B. The real 2HDM and the U (1)-2HDM We now consider a real softly broken Z 2 potential, where λ 6 = λ 7 = 0, and λ 5 , m 2 12 , and the vevs are real. In this case, CP is conserved, η 3 = A is CP odd, while the neutral CP even scalars are denoted by H (heavy) and h (125 GeV). In terms of the notation in the C2HDM, the angles become α 2 = α 3 = 0 and α = α 1 − π/2. In this case, the expression for D in terms of physical parameters simplifies considerably and can be programmed directly into any phenomenological analysis. We find The discriminant introduced in reference [19] for the case of the real 2HDM is where This quantity can also not be written directly in terms of physical parameters. The problem, as explained above, is related to the presence of √ λ 1 λ 2 in the Λ 0 of Eq. (19), which is the origin of k in Eq. (35). Although this expression is written in terms of the parameters of the potential and not the physical observables, it can be recovered in our formalism in the following way. We first calculate D using (19), and then relate it to D Z2 via where each factor is positive definite. Combining the pre-factors in Eqs. (35) and (38), we that D depends only on k 4 and, thus, we understand why it can be written in terms of physical parameters in a simpler way. Things get even simpler in the case of the potential with a softly broken U (1) symmetry. The discriminant for this case was introduced in Ref. [18] as For this model, 2m 2 12 = m 2 A s 2β > 0, and the discriminant may be taken as V. CONCLUSIONS When studying models with two Higgs doublets, it is possible that the vacuum chosen is not the global one, and that there is another vacuum lying below. In that situation, the vacuum is metastable, there could be a later transition into the global vacuum, and we dub this situation the panic vacuum. We have developed a method to avoid panic vacua, involving the discriminant D in Eqs. (17)- (18), which is applicable to any 2HDM potential. In particular, we proved that D > 0 guarantees that we are staying in the global minimum and, automatically, the potential is bounded from below. If D < 0, more steps are needed to discriminate between the global minimum from a panic vacuum or a potential unbounded from below. Our method is not computer time consuming and it be easily implemented in phenomenological studies in which extensive scans over 2HDM parameter space.
We have shown how D can be written in terms of physical parameters for the C2HDM, the real 2HDM and the U (1)-2HDM, and we have shown how it is related with the discriminants previously presented for the later two cases [18,19].
If the potential has a softly broken Z 2 symmetry, a simpler strategy is possible. Indeed, in that case, the bounded from below conditions are easy to implement -c.f. Eq. (12). Then one need only applyD in Eq. (23) and one has a global minimum if and only ifD > 0.
This article completes the identification of panic vacuua for all 2HDM, including the previous unsolved cases where CP violation is present in the scalar potential.
Similarly, k ν = (P k ) νµ µ (A5) for k = 1, 2, 3 lies along the eigenvector of Λ k , and, thus, it is spacelike: This is true for any vector µ , apart, of course, from the case when the vector is accidentally chosen to be orthogonal to some eigenvector. Choosing the simplest case µ = (1, 0, 0, 0), means that reach a b (ζ) = 0, and the ellipsoid collapses onto the (m a , m c ) plane. This is shown as the vertical line on the right panel of Fig. 1. Along this process, some points may be passed twice (as happens, for example, for the illustrated by the black dot), meaning that one can have at most two stationary points. The analysis for ζ ∈ [Λ b , Λ c ] follows the same lines and, again, one concludes that there can be at most two stationary points in this region. Finally, taking ζ ∈ [Λ c , Λ 0 ], the ellipsoid grows without bound and all points in the (m a , m b , m c ) space are crossed exactly once. As shown in Refs. [10,11], larger ζ correspond to smaller values for the potential. In our notation, if r µ and r µ are two stationary points with the corresponding values ζ > ζ , then This means that the largest ζ for which Eq. (B3) holds will be the global minimum (provided the potential is bounded from below). In the case discussed here, this would occur for ζ ∈ [Λ c , Λ 0 ]. Due to the positivity ofr 0 , the above description holds forM 0 > 0. IfM 0 < 0, then ζ can vary from Λ 0 to +∞. If ζ starts from +∞ and decreases, then the ellipsoid starts from the unit sphere and monotonously grows to infinity as ζ approaches Λ 0 . This process covers exactly once all points in the (m a , m b , m c ) space outside the unit sphere. We get a single extremum in this case, which must be the global minimum.
Finally, if it happens that Λ 0 is not the largest among the eigenvalues, the above construction is still valid up to obvious modifications and can be used to count extrema in each region. However the potential in this case is not bounded from below, and this picture does not allow one to spot the presence of this fact.
Within the bilinear approach, the neutral Higgs mass matrix was written in the Higgs basis in [11,12,17]. Despite being compact, those expressions do not provide insight on how the value of ζ with respect to Λ 0 and Λ k is related with minimum versus saddle point assignments. To gain it, we use instead the basis-invariant approach to Higgs masses [34], which allows us to switch to the Λ µν -diagonal basis. In this basis, the Hessian in the 4D space of neutral modes takes the following form: There are a few important remarks concerning this formula. First, all indices here refer to four dimensional Euclidean spaces, and H ab can be calculated as a usual product of three matrices. However, there are two different spaces involved here. The indices α, β refer to the same space of bilinears from which we erased its Minkowski space metric, while the indices a, b refer to the space of neutral scalar modes at the extremum. These two 4D spaces are shown in Fig.2. Second, the signature matrix S and the matrix Λ E − ζ1, whose determinant we calculate to get D, originate G 0 neutral physical Higgs space r α φ a R FIG. 2: The correspondence between the 4D spaces established by the transformation R in (C3). The neutral field space contains a would-be Goldstone G 0 , which is annihilated by R. Its complement, the physical Higgs space, shown in gray, is mapped onto a 3D plane tangent to the forward lightcone in the rα space at the extremum point. The direction orthogonal to this plane points inside the cone and it cannot be represented by any neutral scalar mode.
from the same Λ µν − ζg µν : the former is obtained by erasing the Minkowski metric; the latter is obtained by lowering one index. This leads to D = det S. However, as we will see below, in our discussion we will need not only det S but also its full signature; defined as the number of positive and negative eigenvalues. Third, since the diagonalization of Λ µν belongs in general, to SO(1, 3), it modifies the kinetic term of the Higgs fields, and the mass matrix is not equal to the Hessian (C3). However such a transformation, effectively rotating and stretching the Higgs space, leaves invariant the signature of the Hessian (the signs of its eigenvalues) -see proposition 1 in Ref. [12]. Therefore, the signature of H ab faithfully represents the signs of the masses squared of the four neutral scalar degrees of freedom.
The link between the signatures of H ab and of S is less trivial. The transformation matrix, R, is singular: det R = 0, which indicates that one of the four directions in the scalar field space ϕ a is, in fact, a would-be Goldstone boson. Orthogonal to it lies the 3D physical neutral Higgs space (thenceforth referred to as the "physical Higgs space"), which is mapped by R onto the 3D subspace in the r α space tangent to the lightcone at the extremum point. Therefore, when distinguishing a minimum from a saddle point, one must pay attention not to the full signature of S but to the signature of its restriction onto this 3D subspace.
This allows us to establish the following relation between the signatures of S and H ab .
• If S has the signature (+, +, +, +), then H ab is positive definite in the physical Higgs space, and we are at a minimum. Also, D > 0.