The three- and four-Higgs couplings in the general two-Higgs-doublet model

We apply the unitarity bounds and the bounded-from-below (BFB) bounds to the most general scalar potential of the two-Higgs-doublet model (2HDM). We do this in the Higgs basis, i.e. in the basis for the scalar doublets where only one doublet has vacuum expectation value. In this way we obtain bounds on the scalar masses and couplings that are valid for all 2HDMs. We compare those bounds to the analogous bounds that we have obtained for other simple extensions of the Standard Model (SM), namely the 2HDM extended by one scalar singlet and the extension of the SM through two scalar singlets.


Introduction
In order to unveil the detailed mechanism of electroweak symmetry breaking it is crucial to measure the self-couplings of the boson with mass 125 GeV discovered in 2012 at the LHC [1,2]. In this paper we call that boson h 1 . The Standard Model (SM) predicts h 1 to be a scalar and predicts its cubic and quartic couplings g 3 and g 4 , which we define through L = · · · − g 3 (h 1 ) 3 − g 4 (h 1 ) 4 , (1.1) JHEP12(2018)004

g 3 and g 4 in the SM
The Standard Model has only one scalar doublet φ 1 . We write it where v is the VEV, which is real and positive, and G + and G 0 are (unphysical) Goldstone bosons. In the SM H coincides with the observed scalar h 1 . The scalar potential is (1. 3) The minimization condition of V is µ 1 = −λ 1 v 2 . Therefore, in the unitary gauge where G ± and G 0 do not exist, The second term in the right-hand side of equation (1.4) indicates that the squared mass M 1 of the observed scalar is given by M 1 = 2λ 1 v 2 . Therefore, It should be noted that the sign of g 3 implicitly depends on the sign of h 1 . We fix that sign by noting that the covariant derivative of φ 1 gives rise to a term Thus, the coupling W + µ W µ− h 1 , viz. g 2 v √ 2 , is positive.

JHEP12(2018)004 2 The Standard Model plus two singlets
We consider the Standard Model with the addition of two real SU(2)×U(1)-invariant scalar fields S 1 and S 2 . 1 We assume two symmetries S 1 → −S 1 and S 2 → −S 2 . We call this model the SM2S. 2 The scalar potential is (2.1c)

Unitarity condidions
We derive the unitarity conditions on the parameters of V 4 . 3 We follow closely the method of ref. [33]. We write where a and b are complex fields. Then, There are seven two-particle scattering channels (Q is the electric charge, T 3 is the third component of weak isospin): 1 In appendix A we treat the simpler case of the HSM, viz. the Standard Model with the addition of only one real gauge singlet. 2 The SM2S has already been mentioned in the literature as a model for Dark Matter, see refs. [27][28][29][30]. 3 Strictly speaking, the unitarity conditions derived and utilized in this paper are the ones valid in the limit of infinite Mandelstam parameter s. For finite s one must take into account the trilinear vertices that are induced from the quartic vertices when one substitutes one of the fields by its VEV. The unitarity conditions then become s-dependent and may be either more or less restrictive than the conditions in the limit of infinite s. See refs. [31,32].

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In order to derive the unitarity conditions one must write the scattering matrices for pairs of one incoming state and one outgoing state with the same Q and T 3 . Let the incoming state be xy and let the outgoing state by zw, where x, y, z, and w may be either a, a * , b, b * , S 1 , or S 2 . The corresponding entry in the scattering matrix is the coefficient of xyz * w * in V 4 , with the following additions: For each n identical operators in xyz * w * there is an additional factor n! in the entry.
If x = y there is additional factor 2 −1/2 in the entry.
If z = w there is additional factor 2 −1/2 in the entry.
One finds in this way that the scattering matrices for the channels 1, 2, 3, and 4 are The scattering matrices for the channels 5 and 6 are The scattering matrix for channel 7 is The matrix (2.6) is similar to the matrix The unitarity conditions are the following: the eigenvalues of all the scattering matrices should be smaller, in modulus, than 4π. Thus, in our case, should have moduli smaller than 4π.

Bounded-from-below conditions
One may write , and Z = S 2 2 are positive definite quantities independent of each other. In order for V 4 to be positive the square matrix in equation (2.10) must be copositive [34]. A real symmetric matrix M is copositive if x T M x > 0 for any vector x with non-negative components. A necessary condition for a real n × n matrix to be copositive is that all its (n − 1) × (n − 1) principal submatrices are copositive too. 4 Thus, the matrices In order for the full 3 × 3 matrix in equation (2.10) to be copositive an additional BFB condition is required [35]:

Procedure
Let the VEV of S 1 be w 1 and let the VEV of S 2 be w 2 . 5 Then, the vacuum stability conditions are (2.14c) 4 The principal submatrices are obtained by deleting rows and columns of the original matrix in a symmetric way, i.e. when one deletes the i1, i2, . . . , i k rows one also deletes the i1, i2, . . . , i k columns. 5 In appendix B we demonstrate that stability points of the potential with either w1 = 0 or w2 = 0 have a higher value of the potential and cannot therefore be the vacuum.

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Using equation (1.2) with G + = 0 and G 0 = 0, i.e. in the unitary gauge, together with S 1 = w 1 + σ 1 and S 2 = w 2 + σ 2 , one obtains One diagonalizes the real symmetric matrix M as where R is a 3 × 3 orthogonal matrix that may be parameterized as Here, c j = cos ϑ j and s j = sin ϑ j for j = 1, 2, 3. One has where the h j are the physical scalars, i.e. the eigenstates of mass; the scalar h j has squared mass M j . We assume that h 1 is the already-observed scalar. The interactions of the scalars with W + W − are given by equation (1.8b), i.e.
We define the sign of the field h 1 to be such that the coupling of h 1 to W + W − has the same sign as in the Standard Model. Thus, we choose −π/2 < ϑ 1 < π/2.

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According to equation (2.15), and The oblique parameter T is given by [36] (2.24) In our numerical work we use as input the nine quantities v, w 1 , w 2 , M 1 , M 2 , M 3 , ϑ 1 , ϑ 2 , and ϑ 3 , which are equivalent to the nine parameters of the scalar potential µ 1 , m 2 1 , m 2 2 , λ 1 , ψ 1 , ψ 2 , ψ 3 , ξ 1 , and ξ 2 . We input equations (1.6) and choose arbitrary values for M 2 > 0 and M 3 > 0 such that M 2 ≤ M 3 (this represents no lack of generality, it is just the naming convention for h 2 and h 3 ). We enforce no lower bound on M 2 and M 3 , in particular we allow them to be lower than M 1 = (125 GeV) 2 . The VEVs w 1 and w 2 are chosen positive; this corresponds to the freedom of choice of the signs of S 1 and S 2 . The angle ϑ 1 is in either the first or the fourth quadrant, with cos ϑ 1 > 0.9, (2.25) so that the h 1 W + W − coupling is within 10% of its Standard Model value. The angle ϑ 2 is in the first quadrant; this corresponds to a choice of the signs of the fields h 2 and h 3 . The angle ϑ 3 may be in any quadrant. We firstly compute T according to equation (    solid line in figure 1 was obtained through a random scan of the parameter space; it is not an analytical bound.
In figure 2 we display the predictions for g 3 and g 4 . In order to produce that figure we have randomly generated √ M 2 , √ M 3 , and the VEVs w 1 and w 2 in the range 0 to 10 TeV. One sees that g 3 is always positive but below its SM value when M 2 > M 1 ; when M 2 < M 1 the allowed range for g 3 becomes much wider. When the masses of the new scalars get higher, g 3 takes values closer to the SM value. An important point is that g 3 remains of the same order of magnitude as in the SM, but g 4 may reach 15 times its SM value.
In the left panel of figure 3 one sees that when cos ϑ 1 → 1 the coupling g 3 necessarily approaches its SM value. This behaviour is because of equation (2.21e) and c 1 > 0.9, which implies |s 1 | c 1 . On the other hand, g 4 is not correlated with cos ϑ 1 , as one sees in the right panel of figure 3.

JHEP12(2018)004 3 The two-Higgs-doublet model
We next consider the model with two scalar gauge-SU(2) doublets φ 1 and φ 2 having the same weak hypercharge. This is usually known as 2HDM. The scalar potential is given by equation (2.1a), where where µ 1,2 and λ 1,2,3,4 are real. The ten (real) coefficients in V 4 may be grouped as [37] η 00 = λ 1 + λ 2 + 2λ 3 , Under a (unitary) change of basis of the scalar doublets, η 00 is invariant while where O is an SO(3) matrix. Only quantities and procedures that are invariant under the transformation (3.3) are meaningful.

Unitarity conditions
We write
Channel 5 produces the scattering matrix A similarity transformation transforms the matrix (3.6) into the direct sum of two 4 × 4 matrices (3.7b) -12 -

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Here, is invariant under a change of basis of the doublets. It is obvious that the eigenvalues of the matrices (3.7) are invariant under such a change too.
Channel 4 produces the scattering matrix which may readily be shown to be similar to M 1 . Channel 3 produces the scattering matrix The matrix (3.10) is similar to Thus, the unitarity conditions for the scalar potential of the 2HDM are the following: the eigenvalues of the two 4 × 4 matrices (3.7) and of the 3 × 3 matrix (3.12), and I in equation (3.8), should have moduli smaller than 4π. These conditions were first derived in refs. [38,39]. We emphasize that they are, as they should, invariant under a change of basis of the two doublets.

Bounded-from-below conditions
Necessary and sufficient conditions for the scalar potential of the 2HDM to be BFB were first derived in ref. [37]. Ivanov [40] and Silva [41] later produced other, equivalent conditions to the same effect. We have implemented numerically both the conditions of ref. [37] and those of ref. [41]. We have found that the Ivanov-Silva algorithm runs several times faster than the one of ref. [37]. We have also checked that all the points produced by either algorithm were validated by the other one.

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The points in our scatter plots were produced by using the algorithm of ref. [41]. That algorithm runs as follows. One constructs the 4 × 4 matrix and one computes its four eigenvalues. Then the potential is BFB if all the following conditions apply: • All four eigenvalues are real.
• All four eigenvalues are different from each other.
We may now return to inequality (3.23), which implies, in principle, many more necessary conditions for boundedness-from-below. Setting for instance sin θ = cos θ one concludes that which must hold for any h and α. Therefore [44], We have numerically analyzed the BFB conditions by giving random values to λ 1 , , and arg (λ * 6 λ 7 ) and then checking whether the BFB conditions are met. We have confirmed that the conditions (3.22), (3.25), and (3.27) always hold. 6

Procedure
We consider the most general 2HDM and purport to find out its ranges for g 3 and g 4 . We use the Higgs basis for the scalar doublets; in that basis only φ 0 1 has VEV and therefore φ 1 has the expression (1.2), while In equation (3.28), σ 1 and σ 2 are real fields and C + is the physical charged scalar of the 2HDM. We emphasize that using the Higgs basis represents no lack of generality, because both the unitarity and the BFB conditions are the same in any basis. Since only φ 1 has VEV, the vacuum stability conditions are µ 1 = −λ 1 v 2 and µ 3 = −λ 6 v 2 [46]. The coupling µ 2 in equation (3.1a) is unrelated to the parameters of V 4 ; one may trade it for the charged-Higgs squared mass M C = µ 2 + λ 3 v 2 . The mass terms of H, σ 1 , and σ 2 are given by line (2.15b), with [46] The matrix M is diagonalized through equations (2.17)-(2.19). 6 The BFB conditions worked out in this subsection are, clearly, the ones valid at tree level. At loop level the BFB conditions change, see ref. [45].

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The three invariants of M are We input parameters λ 1,2,··· ,7 that satisfy both the unitarity conditions and the BFB conditions of subsections 3.1 and 3.2, respectively. 7 We also use the values of M 1 and v in equations (1.6). The two equations are quadratic in M C . By affirming the fact that both quadratic equations (3.31) must hold for the same value of M C , one is able to compute both M C and cos 2 ϑ 1 . We thus get to know the full matrix M , hence its eigenvalues M 2 and M 3 and its diagonalizing matrix R.
We require cos ϑ 1 > 0.9. We also compute the oblique parameter where T singlets is given by equation (2.23). We require −0.04 < T < 0.20. We have applied the method devised in ref. [41] to guarantee that our assumed vacuum state is indeed the state with the lowest possible value of the potential. The method may be described as follows. Let the matrix Λ E in equation (3.16) have four eigenvalues Λ 0,1,2,3 . We already know, from the BFB conditions, that those eigenvalues must be real and different from each other; let us order them as Λ 0 > Λ 1 > Λ 2 > Λ 3 . Let the charged-Higgs squared mass be M C ; define ζ ≡ 2M C v 2 . Then, the assumed vacuum state is the global minimum of the potential if either ζ > Λ 0 , or Λ 0 > ζ > Λ 1 , or Λ 2 > ζ > Λ 3 . This test led us to discard about 10% of our previous set of points.
The four-Higgs vertex is given by The three-Higgs vertex is given by We also want to consider the h 1 C + C − vertex, which may be relevant in the discovery of the charged scalar. That vertex is given by where, in the 2HDM, (3.36)

Results
As we know from subsections 3.1 and 3.2, in general λ 1 can take any value in between 0 and 4π/3. Once the constraint cos ϑ 1 > 0.9 is imposed, however, λ 1 can be no larger than ∼ 1; this is illustrated in figure 7. The closer cos ϑ 1 is to 1, the closer λ 1 must be to its SM value M 1 2v 2 = 0.258; note that λ 1 is almost always larger than its SM value when cos ϑ 1 > 0.9; the minimum value that we have obtained for λ 1 is 0.2135. If cos ϑ 1 0.99, then the masses of the new scalar particles of the 2HDM, namely √ M C , √ M 2 , and √ M 3 can be no larger than ∼ 700 GeV; if cos ϑ 1 0.95, they can be no larger than ∼ 550 GeV. When cos ϑ 1 becomes close to 1, the masses of the new scalar particles may reach O(TeV); this is illustrated in figure 8.
One sees in figure 9 that √ M C and √ M 2 differ by at most ∼100 GeV unless 200 GeV < √ M C < 500 GeV. (Remember that by convention M 2 is always smaller than M 3 , but they may be smaller than M 1 .) JHEP12(2018)004  We now come to the predictions for g 3 and g 4 in the 2HDM, which are depicted in figure 10. One sees that g 3 in the 2HDM has a range only slightly larger than in the SM2S, while g 4 in the 2HDM is much more restricted than in the SM2S; g 4 g SM 4 4 in the 2HDM but g 4 g SM 4 15 in the SM2S. An interesting feature is that g 3 may be zero or even negative, i.e. it may have sign opposite to the one in the SM. (We recall that the sign of g 3 is measured relative to the sign of c 1 ; we arrange that c 1 is always positive.) On the other hand, g 4 is always positive because of the boundedness from below of the potential.
In figure 11 we depict the coupling g 1CC of the 125 GeV neutral scalar to a pair of charged scalars in the 2HDM. One sees that that coupling is in between −200 GeV and 1,700 GeV. The expression for g 1CC in equation

The two-Higgs-doublet model plus one singlet
We consider in this section the two-Higgs-doublet model with the addition of one real SU(2) × U(1)-invariant scalar field S. We assume a symmetry S → −S. As a shorthand, we shall dub this model the 2HDM1S (other authors use just 2HDMS [47]). The quartic part of the scalar potential is

Bounded-from-below conditions
Deriving necessary and sufficient BFB conditions for even a rather simple potential like the one in equation (4.1) is a notoriously difficult problem [48]. If V 4 were negative for some possible values of S 2 , φ † 1 φ 1 , φ † 2 φ 2 , and φ † 1 φ 2 , then V 4 would tend to −∞ upon multiplication of those four values by an ever-larger positive constant. Therefore, we want V 4 to be positive for all possible values of S 2 , φ † 1 φ 1 , φ † 2 φ 2 , and φ † 1 φ 2 . In order to guarantee this, we proceed in the following fashion.
Necessary condition 1. When S 2 = 0, equation (4.1) reduces to its first two lines, i.e. to the quartic potential of the 2HDM. Therefore, one must require the fulfilment of the conditions of subsection 3.2, viz. the four conditions in between equations (3.16) and (3.18).
Necessary condition 2. When φ † 1 φ 2 = 0, Since φ † 1 φ 1 , φ † 2 φ 2 , and S 2 are positive definite quantites, we must require [34,35] ψ > 0, (4.3a) After enforcing the necessary condition 1, we know that V 4 > 0 when only the first two lines of the potential (4.1) exist; after enforcing the inequality (4.3a), we know that V 4 > 0 when only the third line exists. If we guarantee that the fourth line of the potential (4.1) is always positive too, then we will be sure that V 4 is always positive. We therefore have the following 8 Sufficient condition. If, besides the two necessary conditions, then V 4 is BFB. Among the sets of parameters of the potential (4.1) that we have randomly generated, there were some that met both the two necessary conditions and the sufficient conditions (4.4); we have used those sets of parameters. There were many other sets that JHEP12(2018)004 satisfied the two necessary conditions but did not meet the sufficient conditions (4.4); for those sets, we have numerically found the absolute minimum of V 4 . We have done this by using S 2 = 1 together with equations (3.19) and by minimizing V 4 in the domain r 2 > 0, 0 ≤ θ ≤ π/2, 0 ≤ h ≤ 1, and 0 ≤ α < 2π. If the minimum of V 4 is positive, then the set of input parameters is good, else the set of input parameters is bad and one must discard it.

Unitarity conditions
There are the same five scattering channels as in the 2HDM, cf. subsection 3.1; but the channel Q = T 3 = 0 has an additional scattering state S 2 . Additionally, there are two extra scattering channels: • The channel Q = 1, T 3 = 1/2 with the two states aS and cS.
• The channel Q = 0, T 3 = −1/2 with the two states bS and dS.
Both these channels produce a scattering matrix Thus, the unitarity conditions for the 2HDM1S are the following: both |I| and the moduli of all the eigenvalues of the 2×2 matrix M 4 , of the 3×3 matrix M 3 , of the 4×4 matrix M 1 , and of the 5 × 5 matrix M 2 must be smaller than 4π.

Procedure
Just as in the previous section, we utilize the Higgs basis for the two doublets, i.e. equations (1.2) and (3.28). We also write S = w + σ, where w is the VEV of the scalar S and σ is a field. The mass terms of the scalars are cf. equation (3.29). One diagonalizes M as where R is a 4 × 4 orthogonal matrix. The squared mass M 1 is given by equation (1.6a). Without loss of generality, M 2 < M 3 < M 4 . Just as in the previous sections, we require The expression for the oblique parameter T is [36] and we demand −0.04 < T < 0.20. We input random values for the 15 real parameters M C , λ 1,2,3,4 , |λ 5,6,7 |, ψ, ξ 1,2 , |ξ 3 |, arg (λ * 5 λ 6 λ 7 ), arg (λ * 6 λ 7 ), and arg (λ * 6 ξ 3 ). We moreover input M 1 and v 2 given in equations (1.6). Then, 1. We require the input parameters to satisfy the BFB conditions of subsection 4.1this may imply a numerical minimization of V 4 to check that V 4 > 0.
2. We require the input parameters to satisfy the unitarity conditions written after equation (4.6).
3. We compute the VEV w from the condition that M 1 should be an eigenvalue of the matrix M .

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4. We enforce the conditions in appendix C. They guarantee that the vacuum state with v = 174 GeV and w = 0 has a lower value of the potential than all the other possible stability points of the potential. 5. We compute the full matrix M , its eigenvalues M 2,3,4 , and its diagonalizing matrix R; we choose the overall sign of R such that R 11 ≡ c 1 > 0.
6. We impose both the condition (4.10) and the condition that the oblique parameter T is within its experimental bounds.

Results
In figure 12 we have plotted the differences among the masses of the scalars against the mass of the charged scalar. One sees that √ M C and √ M 3 cannot be more than ∼ 300 GeV from each other, but √ M 2 may be much smaller than both of them. In figure 13 we present a scatter plot of the mass of the lightest non-SM neutral scalar against c 1 . One sees that, contrary to what happens in the 2HDM (cf. figure 8), √ M 2 may reach 1 TeV even when c 1 is as low as 0.9.
We depict in figure 14 the three-and four-Higgs couplings g 3 and g 4 in the 2HDM1S. The main difference relative to the 2HDM (cf. figure 10) is that g 4 may be much higher, just as in the SM2S. In the 2HDM1S there is no clear correlation between g 3 and g 4 .  In figure 15 we have plotted the h 1 C + C − coupling g 1CC . That coupling in the 2HDM1S may be more than two times larger than in the 2HDM; very large values of g 1CC occur even for c 1 very close to 1. This is because the right-hand side of equation (4.14) may be dominated by its fourth term when w v. The first term displays the same behaviour as the corresponding term in the 2HDM, viz. it is usually positive and no larger than 1,500 GeV, but it is often overwhelmed by the fourth term.

Conclusions
In this paper we have emphasized that both the bounded-from-below (BFB) conditions and the unitarity conditions for the two-Higgs-doublet model (2HDM) are invariant under JHEP12(2018)004  a change of the basis used for the two doublets. Therefore, one may implement those conditions directly in the Higgs basis, viz. the basis where only one doublet has vacuum expectation value. This procedure allows one to extract bounds on the masses and couplings of the scalar particles of the most general 2HDM, disregarding any symmetry that a particular 2HDM may possess. We have focussed on the three couplings g 3 (h 1 ) 3 , g 4 (h 1 ) 4 , and g 1CC h 1 C + C − , where h 1 is the observed neutral scalar with mass 125 GeV and C ± are the charged scalars of the 2HDM. We have utilized the same procedure for two other models, namely the Standard Model with the addition of two real singlets (SM2S) and the two-Higgs-doublet model with the addition of one real singlet (2HDM1S), in both cases with reflection symmetries acting on JHEP12(2018)004 each of the singlets. We have found, for instance, that: • The coupling g 3 may, in both the 2HDM and the 2HDM1S, have sign opposite to the one in the SM. On the other hand, in any of the three models that we have studied, |g 3 | can hardly be much larger than in the SM.
• The coupling g 4 , which is always positive because of BFB, may for all practical purposes be equal to zero in all the three models. (As a matter of fact, g 3 = g 4 = 0 is possible in all three models.) But it may also be much larger than in the SM. A distinguished feature is that g 4 may be much larger (up to g 4 ∼ 0.5) in the models containing singlets than in the 2HDM, wherein it can at best reach g 4 ∼ 0.13.
• The coupling g 1CC may be of order TeV, but only when the mass of C ± exceeds 300 GeV; in general, a positive g 1CC may be larger for higher masses of C ± , but g 1CC may also be negative for any C ± mass. Moreover, g 1CC may be more than two times larger (either positive or negative) in the 2HDM1S than in the 2HDM.
A comparison of the predictions of the three models for g 3 and g 4 is depicted in figure 16.
We emphasize that our method may be used to obtain bounds and/or correlations among other parameters and/or observables of these models. Unfortunately, it may be difficult to generalize our work to more complicated models, both because they may contain too many parameters and because it is very difficult to derive full BFB conditions for even rather simple models.

A The Higgs Singlet Model
The Higgs Singlet Model (HSM) is the Standard Model with the addition of one real scalar singlet S. We furthermore assume a symmetry S → −S. The scalar potential has just five parameters µ, m 2 , λ, ψ, and ξ. The bounded-from-below (BFB) conditions are The unitarity conditions are We assume that φ 1 has VEV v and S has VEV w. We write S = w + σ together with equation (1.2). The mass matrix for H and σ is where c ≡ cos ϑ and s ≡ sin ϑ. We assume |c| > 0.9. The oblique parameter must satisfy −0.04 < T < 0.20. The three-and four-Higgs couplings are given by In figure 17 we compare the predictions of the HSM and of the SM2S for g 3 and g 4 . One sees that there is no substantial difference between the two models.

B Other stability points of the SM2S potential
In this appendix we consider more carefully the various stability points of the potential of the SM2S in equation (2.1). The vacuum value of that potential is given by Equations (2.14) follow from the assumption that v, w 1 , and w 2 are not zero. Defining The mass matrix M of the scalars is real and symmetric and is given in equation (2.16). We assume that M has three positive eigenvalues M 1 , M 2 , and M 3 . It follows that all the JHEP12(2018)004 principal minors of M are positive. 9 (This is called 'Sylvester's criterion' [49].) Thus, These inequalities display some resemblance to the BFB conditions (2.12), (2.13).
We now consider other stability points of the potential where either v or w 1 or w 2 vanish.
Finally, there is another stability point with value of the potential. 9 The principal minors of a square matrix are the determinants of its principal submatrices.

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From inequalities (B.4c) and (B.4f) it follows that V (4) ≤ V (2) is equivalent to which in turn is equivalent to and this is obvioulsy true. One thus concludes that V (4) can never be larger than V (2) . In similar fashion one finds that Next consider the inequality V 0 ≤ V (4) . Because of (B.4f) and (B.4g), it is equivalent to This may be written as which is of course true. In similar fashion one obtains that

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We have thus demonstrated that, because of our assumption that all three eigenvalues of the matrix M are positive, V 0 is smaller than V 1,2,3,4,5,6 , viz. the stability point of V with nonzero v, w 1 , and w 2 is the vacuum.
This result may be easily understood in the following way. The potential (2.1) of the SM2S may be rewritten where V 0 is the vacuum expectation value of the potential given in equation (B.3a) and We assume that the point X = 0, 0, 0 T is a local minimum of the potential V . Then, since the potential in equation (B.17) is a quadratic form is X, the point X = 0, 0, 0 T must also be the global minimum of V . 10

C Global minimum conditions for the 2HDM1S
In the 2HDM1S, we define We define the column vector X = q 1 , q 2 , z, z * , q 3 T . The scalar potential of the 2HDM1S may then be written as The coefficients µ 1 , µ 2 , µ 3 , and µ 4 contained in the column vector Y have squared-mass dimension; µ 3 is in general complex while µ 1 , µ 2 , and µ 4 are real. The coefficients contained in the symmetric matrix Λ are treated by us as an input, cf. section 4.3. Since we study the 2HDM1S in the Higgs basis, where φ 2 has zero VEV, in the vacuum one has q 2 = z = z * = 0, q 1 = v 2 , and q 3 = w 2 ; the vacuum expectation value of the potential is (C.4) 10 We thank Igor Ivanov for presenting this argument to us. 11 Since we only analyze the potential at the classical level, we simplify the notation by treating the fields as c-numbers instead of q-numbers.

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It follows that Solving for v 2 and w 2 the system (C.5) and plugging the solution into equation (C.4), one obtains Moreover, in the Higgs basis In equation (C.7a), M C is the mass of the charged scalar; we treat it as an input, just as v and w. 12 By using equations (C.5) and (C.7) we find the values of µ 1 , µ 2 , µ 3 , and µ 4 from the input. We want to check that, for each set of input parameters (i.e. λ 1,...,7 , ξ 1,2,3 , ψ, v, w, and M C ) in our data set, the state that we assume to be the vacuum, characterized by q 2 = z = z * = 0, is indeed the global minimum of the potential. In order to do this we must consider all the other possible stability points of the potential and check that the value of the potential at each of those points is larger than V 0 in equation (C.6). The stability points may either be inside the domain defined by equations (C.1) or they may lie on a boundary of that domain. There is only one possible stability point inside the domain; deriving equation (C.2) relative to X, we find that it is given by For each set of input parameters, we have computed the column vector X (1) by using equation (C.8a). If that vector happened to be inside the domain, viz. if X 4 > 0, then we computed V (1) by using equation (C.8b). We checked whether V (1) > V 0 ; if the latter condition did not hold, then we discarded that set of input parameters.
Next we have considered the various possible stability points on boundaries of the domain. Firstly there is the boundary with q 3 = 0 but q 1 > 0, q 2 > 0, and |z| 2 < q 1 q 2 . In that case one has There is one possible stability point with For each set of input parameters, we have computed the column vectorX (2) by using equation (C.11a). Whenever that vector happened to fulfilX
Secondly we have checked a possible stability point with null q 1 (and z) instead of null q 2 (and z). In analogy with equations (C.5) and (C.6), in that case one has q 2 = −ψµ 2 + ξ 2 µ 4 ψλ 2 − (ξ 2 ) 2 , (C.12a) (C.12c) For each set of parameters, we have computed q 2 and q 3 through equations (C.12a) and (C.12b), respectively. Whenever q 2 and q 3 were both positive, we have computed V (3) through equation (C.12c); if V (3) < V 0 , then we discarded the set of parameters. Thirdly, we have considered the following possible stability points on boundaries of the domain: 1. The point q 1 = q 2 = z = q 3 = 0 has V = 0, Therefore, when V 0 > 0 we have discarded the set of parameters.
2. When q 1 = q 2 = z = 0 but q 3 = 0, there is a stability point featuring Whenever q 3 in equation (C.13a) happened to be positive and simultaneously V (4) in equation (C.13b) was smaller then V 0 , we have discarded the set of parameters.

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3. When q 1 = q 3 = z = 0 but q 2 = 0, there is a stability point featuring Whenever q 2 in equation (C.14a) happened to be positive and simultaneously V (5) in equation (C.14b) was smaller then V 0 , we have discarded the set of parameters.
4. When q 2 = q 3 = z = 0 but q 1 = 0, there is a stability point featuring Whenever q 1 in equation (C.15a) happened to be positive and simultaneously V (6) in equation (C.15b) was smaller then V 0 , we have discarded the set of parameters.

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For each set of parameters of the potential, we have searched for solutions, i.e. for q 1 > 0, q 2 > 0, q 3 > 0, and a phase θ satisfying the system (C.17) of four equations. (This proved to be a highly nontrivial task.) Whenever we found a solution, we computedV 0 through equation (C.16) and checked whetherV 0 < V 0 ; when that happened for at least one solution of (C.17), we have discarded the corresponding set of parameters.
(C.19c) IfṼ 0 < V 0 for any solution of equations (C.18), then we discarded the set of input parameters. By applying all the tests in this appendix, we have eliminated about half of our initial set of sets of input parameters. Thus, the tests in this appendix prove crucial in the correct analysis of the 2HDM1S.
We have also applied the tests in this appendix, with the necessary simplifications, to the case of the 2HDM [50]. In particular, in that case we do not have to solve the very complicated system of four equations (C.17), we only have to solve the much easier system of three equations (C.18). We have checked that the tests in this appendix yield, for the 2HDM, exactly the same result as the much simpler method described in the paragraph between equations (3.32) and (3.33).

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