Conditions for global minimum in the A4 symmetric 3HDM

There has been a great interest in three Higgs doublet models (3HDM) symmetric under an exact A4 symmetry. We provide the complete analytic necessary and sufficient conditions for a point to be the global minimum along the neutral directions in such models, discussing many of the subtleties involved. We also present a number of numerical examples, to highlight those issues. We then turn to the directions which break electric charge, presenting a safe analytical sufficient condition for bounded from below (BFB) potentials. Based on extensive numerical simulations, we discuss one conjecture on BFB along charge breaking directions.


Introduction
Since the discovery of a scalar particle with 125GeV at the LHC [1,2], there has been a strong theoretical effort and experimental search for extra scalar particles. A particularly interesting framework is provided by N Higgs doublet models (NHDM). For example, models with three Higgs doublets (3HDM) supplemented by the A 4 discrete symmetry have long been explored in connection with flavour physics; see, for example, [3,4,5,6,7,8,9,10].
But, it was only in 2012 that Degee, Ivanov and Keus carefully addressed the identification of viable candidates for minima in the A 4 3HDM [11]. 1 They provide conditions for only two of the solutions of the stationarity equations to be minima. Ref. [13] revisited this issue, proposing conditions for two other solutions to be minima. We have found that some conditions are incorrect, while most only guarantee that the minimum is local, not global. And, in some cases, such local minima occur for potentials which are in fact unbounded from below. Indeed, a different but related issue concerns the conditions for the potential to be bounded from below (BFB). This turns out to have an interesting impact in our analysis.
As pointed out in Ref. [14], one must be careful to distinguish conditions for the potential to be bounded from below along the directions of neutral vacua (BFB-n), from the conditions that guarantee that the potential is bounded from below along the directions of charge breaking vacua (BFB-c). It turns out that the necessary and sufficient BFB-n conditions for the A 4 have been conjectured in [15] and proven in [16]. These hinged on the fact that Ref. [11] fully mapped the orbit space of the neutral directions of the A 4 3HDM. At the moment, there is no such analysis for the charge breaking (CB) directions.

.1 Potential and candidates for global minimum
The potential for the A 4 symmetric three Higgs doublet model (3HDM) is, in the notation of [11], Setting Λ 4 = 0, it reduces to the S 4 invariant 3HDM potential. Therefore, the A 4 and S 4 models have 6 and 5 free parameters, respectively. Contrary to what seemed implicit in [11], and became explicit in subsequent papers using their result -including, for example, [17,13] -there are 5 (and not 4) different vev directions that can lead to a global minimum. They are: C : v √ 6 (±1, η, η −1 ) , where η = e iπ/3 , ω = η 2 = e 2iπ/3 , and α is fixed by the relations The minima have been normalized such that v 2 1 +v 2 2 +v 2 3 = v 2 /2. Notice that A 4 contains Z 2 ×Z 2 , and that one has invariance under global hypercharge. As a result, the potential is invariant under independent sign changes of φ 1 , φ 2 , and φ 3 , as can be checked from direct inspection of (1). Thus, the ± in C and C ′ is immaterial.
Part of the confusion in the literature as to the number of minima appeared because C ′ is not defined in [11], although it is clear from their figure 3 that there are two distinct minima; C and C ′ . Moreover, [17] renames C ′ as C and does not mention a second possibility.
Why is this relevant? It is relevant because, for a given set of Λ's, C and C ′ do not yield the same value for the potential at that minimum. That is V (C) = V (C ′ ); one is the true global minimum and the other is not. Adding to the confusion, starting from C: Since an overall phase is irrelevant, it would thus seem that one could get to C ′ simply by transposing φ 1 ↔ φ 3 . However, transpositions such as this are not in A 4 . Indeed, performing φ 1 ↔ φ 3 in eq. (1), the potential does not remain invariant. Fortunately, the transformed potential is obtained from the initial one by simply changing Λ 4 → −Λ 4 . Thus, the formulae obtained in [11] for C can be adapted to C ′ by simply changing the sign of Λ 4 . Still, we emphasize that, for chosen Λ's, Part of the confusion may be due to the fact that transpositions such as φ 1 ↔ φ 3 are a part of S 4 . In that case, the potential has Λ 4 = 0 and the distinction between C and C ′ disappears.

Vevs, mass spectra, and values of the potential
The spectra for A 4 has been studied in [11], and here we summarize those results. We denote the charged scalar pairs by H ± i (i = 1, 2), and the new neutral scalars (besides the lowest lying SM-like Higgs) by H k (k = 1, 2, 3, 4). In cases where a mass ordering of the latter is clear, we denote by h i (H i ) the masses of the lightest (heaviest) neutral scalars.
The SM-like Higgs boson, H SM , always has a mass given by m 2 Thus, M 0 > 0 for all cases. This is related to the following fact. One can use a unitary 3 × 3 transformation in the space of doublets (the so-called Higgs-family space), to go into some other basis. A particularly useful basis 2 is the Higgs basis [18], where only the first doublet carries a vev [19,20,21]. In this basis, the imaginary part of the neutral component of the first doublet corresponds always to the neutral would-be Goldstone boson G 0 . The corresponding real part of the neutral component of the first doublet corresponds, in general, to a neutral Higgs field that will mix with other four neutral components into the physical (mass eigenstate) neutral scalars. However, in the particular case of the 3HDM with an exact A 4 symmetry, for each of the five points of the A 4 3HDM parameter space identified in [11] as candidates for global minima, there is always a transformation into a Higgs basis where that neutral scalar decouples completely from the remaining four fields. Of course, this would not be true for minima of a general 3HDM (in which case, even in the Higgs basis, the five neutral fields other than G 0 mix through a 5 × 5 mass matrix); but it is true for the case discussed in [11].
A : (1, 0, 0) For the (1, 0, 0) solution of the stationarity condition, the magnitude of the vev is given by We will sometimes use Λ A = Λ 3 . The spectra is then expressed as The value of the potential at the minimum is B : (1, 1, 1) For the (1, 1, 1) solution of the stationarity condition, the magnitude of the vev is given by We will sometimes use Λ B = Λ 1 . The charged Higgs spectra has less degeneracies than at point A, and we find The value of the potential at the minimum is C : (1, η, η −1 ) For the (1, η, η −1 ) solution of the stationarity condition, the magnitude of the vev is given by where The masses are given by where The value of the potential at the minimum is This case can be obtained from the previous one through Λ 4 → −Λ 4 . The vev becomes, where The charged Higgs masses are given by where The value of the potential at the minimum is D : (1, e iα , 0) For the (1, e iα , 0) solution of the stationarity condition, the magnitude of the vev is given by where we have definedΛ and The corresponding spectra is Notice that the definition ofΛ in Eq. (32) guarantees thatΛ+Λ 1 > 0,Λ+Λ 2 > 0, and that 2Λ+Λ 1 +Λ 2 ≥ |Λ 1 − Λ 2 | > 0. This implies that the second Eq. (35) is always positive. In this case there are no degeneracies in the expressions for the masses of the neutral scalars, and we have separated by commas the four m 2 H i . The value of the potential at the minimum is Alternatively, we may write

The necessary distinction between C and C ′
As mentioned in the introduction, the five minima A, B, D, C, and C ′ appear in figure 3 of [11]. But the corresponding C ′ masses and conditions are not presented and the fact that the two minima are distinct is absent from [17] , and subsequent literature, including [13]. Imagine that one is looking at a particular point in parameter space, corresponding to the values After some preliminary numerical calculations, one might suspect that C ′ : (1, ω, ω 2 ) (unfortunately renamed C in [17,13]) is the lowest minimum, with the value of the potential at the minimum given by But, it is not. The correct minimum is C : (1, η, η −1 ) and the value of the potential at the minimum is not V (C ′ ) but, rather, given by Eq. (23) as It is true, that given the Λ 4 → −Λ 4 interchange between C and C ′ , the roles of C and C ′ will be reversed in the case But that is not the issue. The problem with ignoring C (or C ′ , as the case may be) is that, for a given set of parameters of the potential, the two candidate vacua give different values for the potential; one of the vacua is the global minimum; the other is not. This issue is even more troublesome if one is looking for the minima in a completely softly-broken A 4 potential, where such a parameter symmetry is absent, and one is trying to use the results of the exact A 4 -symmetric potential as a guide.

Comparison with the literature
The expressions for the vevs and masses in cases A, B, D, and C are contained in Ref. [11]. We agree with their results, except that the Λ 4 term in their Eq. (B.10) -their Eq. (43) on the arxiv version -for the charged Higgs masses in case C must be changed from 3) in order to agree with our Eq. (18).
The expressions for the vevs and masses in cases A, B, D, and C ′ are repeated in Ref. [13], but with the notational change that C ′ is renamed C and the appropriate change is made in the sign in front of Λ 4 . We have found a number of misprints in their Eqs. (12), (15), and (16).
The conditions that guarantee that a given vacuum is indeed the absolute minimum are discussed in the next section. It turns out that the conditions identified in [11,13] to guarantee that B is the absolute minimum are necessary but not sufficient to that effect. In addition, equations to guarantee that C ′ and D are the global minimum are not presented in [11]. They are presented in [13], where they are wrong. This is what we turn to next.

Bounded from below conditions (neutral directions)
The full conditions that guarantee that the exact A 4 symmetric 3HDM potential is bounded from below (BFB) are not known. The problem is that the form of the orbit space for the charged solutions (solutions of the vev that break electric charge) is not known. And, as illustrated in [14,15], guaranteeing BFB only along the neutral directions does not guarantee that the potential is indeed BFB.
The conditions for BFB of the A 4 3HDM along the neutral directions (BFB-n) have been conjectured in [15], and proved to hold in [16]. They are: But, the left hand side of Eq. (44) is equal to 4(Λ 0 + Λ D )/3. Thus, the second BFB condition may be traded for which holds if and only if Eq. (31) representing v 2 at point D is positive. Similarly, Eq. (45) with k = 3 may be written as which according to Eq. (11) holds if and only if the expression for v 2 at point B is positive. Finally, Eq. (45) with k = 1 and k = 2 may be written, respectively, as which according to Eqs. (16) and (24) hold if and only if the expressions for v 2 at points C and C ′ are positive, respectively. Said otherwise, the BFB-n conditions of Ref. [16] may be written compactly as These equations guarantee that the right hand side of equations (6), (11), (16), (24), (31) representing v 2 at points X are all positive.
To be specific, when solving the stationarity equations, one finds the trivial (0, 0, 0) solution. One also finds possible solutions at points P = A, B, C, C ′ , D, whenever the right hand side of is positive. Of course, the Λ parameters might be such that Λ 0 +Λ C > 0, while Λ 0 +Λ C ′ < 0. In that case, C is a critical point but C ′ is not. When we mention in the text that we require v 2 > 0 at a given point P , what we mean is that we require Λ 0 + Λ P > 0. Eq. (49) can be taken to mean that the potential is BFB-n if and only if all points P = A, B, C, C ′ , D are critical points simultaneously. The physical explanation is discussed in detail in [15]; it hinges on the fact that, for the exact A 4 potential, there is a connection between the BFB conditions and the value of the quartic part of the potential at the candidates for global minima.

Conditions for global minimum
Recall that requiring that the 125GeV scalar has a positive mass squared implies that M 0 > 0 for all cases. Let us assume that we found that a given point P is a local minimum. Then, in a notation that should be obvious from Eqs. (10), (15), (37), (23), and (30), we conclude that, at that point P one must have Λ 0 + Λ P > 0. Let us continue to imagine that P is a local minimum. We wish to know whether V (X) could be lower at some other point X = P . But The prefactor on the right hand side of this equation is guaranteed to be positive by the BFB conditions in Eq. (49). Thus, V (X) < V (P ) can only be attained if it is possible to have together with Λ 0 + Λ P > 0, the other conditions needed for P to be a local minimum, and the conditions for BFB-n potential. 4 A : (1, 0, 0) Using the geometrical minimization technique, Ref. [11] identifies the conditions for (1, 0, 0) to be "stable" as 5 It is interesting that (53) can be re-obtained in the following way. Requiring that m 2 (7) is positive, we obtain Λ 3 = −|Λ 3 | is negative. Thus, from (6), Λ 0 > |Λ 3 |. Adding (8) and (9), we conclude that We wish for Λ 2 4 = 0; Λ 4 = 0 would lead to the S 4 symmetric 3HDM. This coincides almost exactly with (53). Defining the previous conditions require y 1 +y 2 ≡ Σ > 0 and y 1 y 2 > 0. But this forces both y 1 and y 2 be larger than 0 (and, although irrelevant to our argument, smaller than Σ, related through (8)-(9) with m 2 h i + m 2 H i ). Thus, |Λ 3 | > −Λ 2 , −Λ 1 , and we have recovered all the conditions in (53). A curious fact is that in this argument we did not have to require that V (A) lie lower than the values of the potential at the other solutions of the stationarity equation; V (B), V (D), V (C), V (C ′ ). Indeed, one can show that requiring that, at the vacuum A, v 2 > 0 and that all squared masses are positive 6 guarantees that the potential is BFB-n and that V (A) < V (other). Said otherwise, the conditions (53), which guarantee that A is a local minimum, also guarantee that it is the global minimum.
Thus, conditions (53) imply Λ C,C ′ > Λ A . Similarly and where the last inequality follows from the last Eq. (53). This completes the proof that asking that A be a local minimum, as in Eqs. (53), automatically guarantees that the potential is BFB-n and that A is also the global minimum. We will prove next that this very simple and favorable situation does not occur for the remaining points of interest.
B : (1, 1, 1) Using the geometrical minimization technique, Ref. [11] identifies the conditions for (1, 1, 1) to be "stable" as 7 As in case A, we can recover these conditions by requiring that, at point B, v 2 > 0 and that all expressions for the squared masses are positive. Thus, these conditions ensure that B is a local minimum. These conditions are also enough to prove that V (B) < V (A). However, contrary to what we saw in case A, here the conditions (66) are not enough to guarantee that V (B) < V (C). Indeed, we have shown that if a point satisfies (66) but, in addition, satisfies also Λ 0 + Λ C > 0 , and Λ 4 < 0 , and then it has V (C) < V (B). A numeric example is Λ 0 = 8.8427 , Λ 1 = −7.36842 , Λ 2 = −3.62124 , which leads to Λ C = −8.0366 (and, thus, Λ 0 + Λ C > 0), and |Λ 4 | − √ 3(Λ 2 + |Λ 1 |) = 1.5431 > 0. As a result, in this case, Finally, we can find points for which Eqs. (66) are satisfied but still lead to V (B) > V (D). Those are points for which, in addition to (66), C : (1, η, η −1 ) Ref. [13] claims that, in the region of parameter space where (1, η, η −1 ) is the global minimum. 8 This is incorrect. We find that these conditions are also not the correct necessary and sufficient conditions for C to be a local minimum. To find the correct equations for C to be a local minimum, we proceed as follows. The first Eq.
The second Eq. (18) implies that The last two equations imply that a result which can also be obtained by adding the first and second equations (18). For Eq. (16) to yield a positive vev squared, we must have Adding the two expressions in Eq. (19), we find For the first equation (19) to yield a positive mass squared, we also need that A 2 > B 2 . After some calculations, this requires that c < c − or c > c + , where However, one can show that c − < c ab < c + .
Thus, the solution c < c − is impossible, and, moreover, the simultaneous requirement that c > c ab and c > c + reduces to c > c + always .
The conditions for C to be a local minimum are, thus, We point out that, even after imposing (82), Λ 1 , Λ 2 , and Λ 4 can appear with either sign. It is true that, under the conditions that A be the global minimum in (53), it is guaranteed that V (A) < V (C). So, if C is to be the global minimum, one or more of the conditions in (53) must be violated. We are now interested in what happens to V (A) − V (C) under the conditions (82), that, as we have just seen, guarantee that V (C) is a local minimum. Requiring V (A) < V (C) is only possible if Λ 0 + Λ 3 > 0, Λ 3 < 0, and Λ C − Λ 3 > 0. But the latter condition forces c < −(a + 2b), which is impossible given that c > c + . We conclude that, under conditions (82), V (C) < V (A) always.
The discussion here parallels that on C. One may merely define c ′ = + √ 3Λ 4 , performing the substitution c → c ′ in the formulae of case C. Recall, however, that, given a numerical point, C and C ′ are not "equivalent". Indeed, for a given point with Λ 4 = 0, V (C) is always different from V (C ′ ).
Point C ′ here is what is named point C in Ref. [13], where it is claimed that the conditions in (72) are necessary and sufficient for C ′ to be a global minimum. We have shown that, for example, the point corresponds to a potential which is bounded from below, where C ′ is both the local and global minimum and which (in particular because it has Λ 2 > 0) violates (72).
D : (1, e iα , 0) Ref. [13] claims that, in the region of parameter space where (1, e iα , 0) is the global minimum. This is incorrect. We find that these conditions are also not the correct necessary and sufficient conditions for D to be a local minimum. Indeed, for example, the point Λ 0 = 1.178820 , Λ 1 = −0.137928 , Λ 2 = 1.458710 , corresponds to a potential which is bounded from below, where D is both the local and global minimum and which (in particular because it has Λ 2 > 0) violates the conditions (90). Conversely, the point passes all conditions in (90), but it is the point C (not the point D) which is the global minimum of the potential.
The correct necessary and sufficient conditions for the point D to be a local minimum can be cumbersome to write 9 . To simplify their appearance, we introduce the notation Recall that from the definition ofΛ in Eq. (32), ℓ 1 > 0, ℓ 2 > 0 and, thus, the second Eq. (35) is automatically satisfied. The first Eq. (34) requiresΛ > 0, the second Eq. (34) requires ℓ 3 < 2Λ, and the first Eq. (35) requires ℓ 3 > 0. Eq. (36) may be rewritten as Requiring that these masses squared be positive imposes ℓ 3 < L, where and the right hand side is always positive, as one can easily check. Finally, one must ensure that Λ 0 +Λ D > 0. As a result, The conditions for D to be a local minimum are, thus, One practical form to simulate points in this approach is to choose Λ 1 , Λ 2 , and Λ 4 such that the first equation holds. This determines ℓ 1 and ℓ 2 , to substitute in Eq. (95). Then, choose some ℓ 3 (which determines Λ 3 ) satisfying the next two equations, and finally taking Λ 0 to obey the last equation.

Neutral global minima: summary
We are now ready to generate points guaranteed to be the local minimum in A 4 symmetric potentials, which are BFB (at least along the neutral directions). The Procedure is as follows: 2. Impose the BFB-n conditions as in Eq. (49).
3. Guarantee that the chosen point is indeed the global minimum by imposing the conditions Λ X −Λ P > 0, for all X = P .
This procedure guarantees that all points correspond to potentials that are BFB-n. But they do not guarantee that the potential is BFB along the charge breaking directions (BFB-c). This is what we discuss in the next section.
5 Protecting against global minima along charged directions

A lower bound for BFB-c
Using the gauge freedom, one can write the most general vacua of any 3HDM as [14] So far, we have discussed what happens to the potential (evaluated at the vacuum 10 ) along the neutral directions α 2 = α 3 = 0. But, as shown in [14,15], a potential can be bounded from below along the neutral directions (BFB-n), but not along the charge breaking directions (BFB-c). Unfortunately, the BFB-c condtions are not yet known for the A 4 3HDM. So, in this section we develop sufficient conditions for BFB-c, following the technique developed in [22].
It is convenient to define [14] z When evaluated at the vevs, it is always true that [14] 0 ≤ z ij ≤ r i r j .
It is also useful to follow the notation of Ivanov and Faro [14], adapted to the A 4 3HDM, and write the quartic potential as where and where α is defined by Eqs. (4). The relation with (1) is Since the problem of finding necessary and sufficient conditions for the A 4 potential is not yet solved for the charged directions, we follow the technique in [22] and seek a lower potential to bound the original potential. Using the parameterization in Eq. (97), we get Now we go back to V CB . Because of Eq. (99) we always have Now, using Eq. (106), we can finally write Therefore this potential bounds the original potential; if it is BFB, so it is the original one. We start by defining where the diagonal (A d ) and off-diagonal (A 0 ) matrix elements are given, respectively by To check the BFB for V Lower 4 , we just have to check the copositivity of the matrix A [23,24]. This leads to The addition of the last equation implies simply that A o ≥ −A d /2. Thus, the copositivity conditions are Such conditions will ensure sufficient conditions for the potential to be BFB, but they are not necessary. There will be good points in parameter space that are discarded by this procedure. We will come to this issue below when we compare the respective sets of points.

Numerical simulations and conjectures
We have made a very large number of simulations of parameters obeying the BFB-n conditions and the conditions that point P (where P is either of A, B, C, C ′ , D) is a local minimum. As mentioned above, any point satisfying these conditions with P = A is guaranteed to have the global minimum at A. In contrast, points obeying BFB-n and local minimum at P ∈ {B, C, C ′ , D} can have the global minimum at any other point X, except X = A. To be specific, let us generate points obeying BFB-n in Eq. (49) and the conditions for local minimum at P = C in Eq. (82). Then, we will find points where C is indeed the global minimum. But, we will also find points where X ∈ {B, C ′ , D} is instead the global minimum. An example with P = C a local minimum and a global minimum at X = D below is given in Eq. (88). Moreover, BFB-n conditions and the conditions that point P (where P is either of A, B, C, C ′ , D) is a local minimum do not even preclude the existence of a charge breaking (CB) global minimum below. We found such examples when choosing P = D as a local minimum. One such example is Λ 0 = 2.92212 , Λ 1 = −0.276035 , Λ 2 = 2.58004 , leading to in units of M 2 0 /4. Thus, looking only at the neutral minima, D would indeed be the global minimum. However, in this case, there is a lower lying minimum at a CB point, yielding V CB = −0.38331 (M 2 0 /4). In fact, the point in Eq. (112) survives even after imposing the global minimum conditions Λ X − Λ D > 0 for all neutral X ∈ {A, B, C, C ′ }.
Starting from (112), one can generate an even worse potential by gradually lowering Λ 0 while keeping Λ 1,2,3,4 fixed. 11 For a Λ 0 (roughly) between 0.2844 and 0.313, all points pass the conditions for BFB-n and global-D, but, nevertheless, correspond to potentials which are unbounded from below along the CB directions. For example, the potential corresponding to Λ 0 = 0.300000 , Λ 1 = −0.276035 , Λ 2 = 2.58004 , obeys BFB-n and the requirement that D be the lowest lying neutral minimum but, still, is unbounded from below. This example disproves a conjecture in Ref. [15]. 12 . But we have found something quite interesting concerning BFB-c in our extensive simulation. We performed a simulation of over 10000 parameter points, followed by a numerical minimization searching for the global minimum. For each of these parameter points, the minimization was repeatedly attempted with 200 random seed points. Requiring only that some point P be a local minimum without requiring BFB-n did yield many cases where the potential was unbounded from below along the charge breaking directions. As (112) and (114) illustrate, we have also found some points which passed the stronger requirements of BFB-n with D the lowest lying neutral minimum, but which yielded potentials either with a lower-lying CB minimum or which were even unbounded from below along the CB directions.
However, in all cases we generated where BFB-n holds, P = A, B, C, or C ′ is a local minimum, and where the potential lies lower than at any other neutral point X = P , we found no lower lying CB minimum and no potential unbounded from below along CB directions. We propose this result here in the form of a Conjecture: The combined requirements that BFB-n holds and that point P (where P is either of A, B, C, C ′ ) is a global minimum guarantee that the potential is also BFB-c and also that there in no lower-lying CB global minimum.
The analytical proof of this conjecture remains as an open difficult problem.

Conclusions
We have revisited the conditions for global minimum in the exact A 4 3HDM, in light of the candidates for local minimum identified in [11] via the geometric minimization method. We stressed that C and C ′ are not equivalent, since V (C) = V (C ′ ). Given a set of Λ parameters, both must be probed as candidates for global minimum. For point A, we found that the conditions determined in [11] and repeated in [13] are indeed conditions that guarantee that A is the global minimum. In fact, we have found that, by themselves, they also imply that the potential is BFB-n. The conditions for local/global minimum at points B, C, C ′ , and D are either absent or wrong in the literature [11,13]. We present here the correct conditions for B, C, C ′ , and D to be the global minimum.
The clarification provided here as to the number of global minima candidates and the exact conditions for each global minimum is also important given recent developments proposing specific types of softsymmetry breaking which preserve de form (A, B, C, C ′ or D) of the critical points which are candidates for global minimum [25,13]. In a numerical simulation, the correct conditions with an exact symmetry can be used to gain confidence in the results with soft-symmetry breaking, as the A 4 soft-breaking terms are made smaller.
Finally, we address the issue of the charge breaking directions. We present a sufficient analytical condition for the potential to be BFB-c. We also perform an extensive numerical simulation, resulting in a conjecture about BFB-c, proposing that, after BFB-n and global-P , only points with P = D are at risk of allowing for a lower-lying CB (true) global minimum or unbounded from below potentials.