Vacuum Stability Conditions for Higgs Potentials with $SU(2)_L$ Triplets

Tree-level dynamical stability of scalar field potentials in renormalizable theories can in principle be expressed in terms of positivity conditions on quartic polynomial structures. However, these conditions cannot always be cast in a fully analytical resolved form, involving only the couplings and being valid for all field directions. In this paper we consider such forms in three physically motivated models involving $SU(2)$ triplet scalar fields: the Type-II seesaw model, the Georgi-Machacek model, and a generalized two-triplet model. A detailed analysis of the latter model allows to establish the full set of necessary and sufficient boundedness from below conditions. These can serve as a guide, together with unitarity and vacuum structure constraints, for consistent phenomenological (tree-level) studies. They also provide a seed for improved loop-level conditions, and encompass in particular the leading ones for the more specific Georgi-Machacek case. Incidentally, we present complete proofs of various properties and also derive general positivity conditions on quartic polynomials that are equivalent but much simpler than the ones used in the literature.


I. INTRODUCTION
Since the experimental discovery of a Standard Model (SM)-like Higgs particle at the LHC [1,2] and the lack so far of any direct evidence for physics beyond the standard model (BSM) 1 , one might ask whether the properties of the discovered 125 GeV scalar particle being so much close to the SM predictions (see e.g. [4]) leaves any room for BSM physics to reside below the TeV or at the nearby few TeV scale. If new physics is present in the electroweak symmetry breaking sector it should either be very heavy (almost decoupled) or light but having very weak mixing with the SM-Higgs. For the latter case, extensions of the scalar sector of the SM by complex or real SU (2) L triplets, or further extensions comprising Left-Right symmetric gauge groups, or possibly higher representation multiplets, are appealing possibilities. A typical example is the Type-II seesaw model for neutrino masses [5][6][7][8][9][10], for which an essentially SM-like physical Higgs state is unavoidable, a consequence of the very small mixing between the doublet and triplet neutral components being set off by the tiny (Majorana) neutrino mass scale as compared to the electroweak scale. Another example is the Georgi-Machacek model [11,12] with one complex and one real triplet such that a tree-level custodial symmetry is preserved in the scalar sector through a global SU (2) R .
These scenarios have triggered various activities both on the phenomenological level, (including left-right symmetric or not, supersymmetric or not, scenarios) see e.g. among the recent works [13][14][15][16][17][18][19][20][21][22][23][24][25] (and references therein), and in experimental searches at the LHC for neutral, charged, and in particular doubly-charged scalar states that are specific to such class of models decaying either to same-sign leptons or W boson pairs [26,27], [28,29]. As for any extension of the SM, and in the absence of a unifying ultraviolet completion, these models have an increased number of free parameters and thus a large freedom in particular for the physical spectrum of the scalar sector. Theoretical conditions such as the stability of the potential, a consistent electroweak vacuum, unitarity bounds, etc., are thus welcome as a guide together with the experimental exclusion limits to narrow down future search strategies. 1 possible indirect "evidence" notwithstanding [3] 3 The present paper focuses on the potential stability issue for three models: the Type-II seesaw model, the Georgi-Machacek model, and a generalized two-triplet model. The aim is to address as thoroughly as possible the theoretical determination of necessary and sufficient (NAS) conditions on the scalar couplings that ensure a physically sound bounded from below (BFB) potential. The NAS BFB conditions have already been considered in the corresponding literature. Inspired by the approach of [30] initially proposed for the general two-Higgs doublet potential, the strategy consists in a change of parameterization of the field space reducing it to a minimal set of variables corresponding to positive-valued ratios of field magnitudes and to field orientations varying in compact domains. It is then found that in contrast with the general two-Higgs doublet case, the general doublet-triplet potential leads to a simplification that allows a fully analytical solution. A complete answer was given first in [31] and [32] for the Type-II seesaw model. Following the same approach the NAS BFB conditions were provided for the Georgi-Machacek model in [33]. We will nevertheless reexamine the issue for these two models, supplementing with complete proofs, for reasons that will become clear in the course of the study. Encouraged by the success of the approach, we extend it in the present paper to a generalized two-triplet model, that we will dub precustodial, for which we provide novel results by deriving the full NAS BFB conditions. Some stability constraints have already been given for this model in [34] and [35]corresponding however to specific directions in the field space, thus to a subclass of necessary conditions. This pre-custodial model can be of phenomenological interest by itself, but can also serve as a guideline for the effective potential beyond tree-level in the Georgi-Machacek model.
The main issue of the analysis will be to cast the conditions in a form as close as possible to a fully resolved one. By 'fully resolved' we mean an analytical expression that depends solely on the couplings with no reference to orientations or magnitudes in field space. A fully resolved form, when possible, is an ideal result both technically, since no scan over the field configurations is needed, and physically, as consistency constraints are expressed directly in terms of the (physical) couplings. This was the case for the conditions derived in [31], [32] while in [33] the conditions were resolved with respect to only one parameter, thus remaining in a partially unresolved form albeit with a residual field dependence reduced to a compact domain. As we will see, similar configurations arise in the pre-custodial model where the resolving occurs at different stages with respect to different parameters. A hindrance in the way of reaching fully resolved conditions emerges whenever dealing with a quartic polynomial 4 that cannot be reduced to a biquadratic one. This fact motivated us to investigate further a rather mathematical question, the positivity of general quartic polynomials, for which we determine NAS conditions that are simpler than the ones found in the literature.
A word of caution is in order here: The NAS BFB conditions we are considering are obtained by requiring the tree-level potential not be unbounded from below in any direction in the field space. It is only in that sense that they are necessary and sufficient. Obviously they might be only necessary in a wider physical sense when taking into account the structure of the vacua. Moreover, going beyond tree-level would modify these conditions. As alluded to above and will be briefly discussed towards the end of the paper, the tree-level conditions can, however, encapsulate in some cases the leading loop corrections.
Several methods to treat the stability of the potential have been conceived in the literature, e.g. specifically for multi-Higgs-doublets models [36][37][38] including elegant geometric approaches [39], or more general methods relying on copositivity [40][41][42] or on other powerful mathematical techniques [43] (and references therein). As attractive as it may seem, the ability of the latter systematic methods to treat in principle any model through readyto-use packages [44], can yet in practice run into technical difficulties when dealing with extended scalar sectors as noted in [43]. Also to the best of our knowledge a model with one triplet has been treated using copositivity [41] but for which only specific directions in field space where considered in agreement with [31], while [45] obtained with this method the all-directions conditons for the Type-II seesaw model in a form different from that of [31]. It should however be stressed that the copositivity method cannot always be applied to potentials with extended Higgs multiplets when all 4-dimensional operators allowed by the gauge symmetries and renormalizability are considered. This was the case for the doublet extensions studied in [36][37][38], and will be the case here for the extensions with two Higgs triplets. 2 Thus, the more pedestrian and somewhat mathematically lowbrow approach we adopt in this paper remains in our opinion an efficient way of tackling the stability problem specifically for the models under consideration with two triplets.
The paper is organized as follows. In Section II we revisit the derivation of the NAS-2 More precisely, the potential cannot always be cast in a bilinear form involving positive-definite independent vector components and an optimal space dimension to make the method advantageous; more on this at the end of Section V. Models with increased symmetries can be more tractable, see e.g. [46], [47].

5
BFB conditions for the Type-II seesaw model finding equivalence with the conditions of [32] that corrected [31], but stress that the conditions of [31] do remain valid necessary and sufficient when one of the couplings is negative. Adding one real SU (2) triplet, the approach is extended to the general pre-custodial model in Section III, including the Georgi-Machacek model as a special case. This section contains the bulk of the new results. We recall some useful ingredients of the two models potentials in Sections III A and III B. In Section III C we first identify six field dependent variables that provide a reduced parameterization of the field space suitable for the BFB study, four of which, dubbed α-parameters, vary in compact domains. We then investigate the NAS-BFB conditions following a procedure where the resolving with respect to these six field-dependent variables is performed step-bystep. Section III D deals with the analytical determination of the domains of variation of the α-parameters as well as all 2,3,4-dimensional analytical correlations between them. In Section III E we derive the main results identifying the fully and partially resolved branches of the NAS-BFB conditions. The special case of the Georgi-Machacek model is reconsidered in Section III F where we relate the reduced parameters to those of the pre-custodial model and provide a proof of their domain of variation that was only conjectured in the literature.
Section IV illustrates an unexpected feedback of the Georgi-Machacek model on the precustodial one. A wrap-up with further illustrations, comments and a user's guide, is given in Section V and we conclude in Section VI. Further material and detailed proofs, either missing in the literature for known properties, or for the new results found in this paper are given in appendices A -F. Special attention is payed, in appendices G and H, to the mathematical issue of deriving simple forms for the NAS positivity conditions of quartic polynomials .
We have used the 2 × 2 traceless matrix representation for the triplet and wrote the two multiplets in terms of their complex valued scalar components and indicated a choice of electric charges with the conventional electric charge assignment for the doublet and following Q = I 3 + Y ∆ 2 with I 3 = −1, 0, 1 and Y ∆ = 2 for the triplet. σ 2 denotes the second Pauli matrix. The potential V (H, ∆) is invariant under SU (2) L × U (1) Y field transformations H → e iα U L H and ∆ → e i2α U L ∆U † L where U L denotes an arbitrary element of SU (2) L in the fundamental representation. Since we are only interested in the issue of boundedness from below of the potential, we need not go further here into the details of the dynamics of spontaneous electroweak symmetry breaking, the structure of the physical Higgs states and the generation of Majorana neutrino masses.

A. The BFB conditions
In order to cope generically with the shape of V (H, ∆) along all possible directions of the 10-dimensional field space, we adopt a reduced parameterization for the fields that will turn out to be particularly convenient to entirely solve the problem analytically. Following [31] we define:

3)
H † H ≡ r 2 cos 2 γ, (2.4) T r∆∆ † ≡ r 2 sin 2 γ, (2.5) Obviously, when H and ∆ scan all the field space, the radius r scans the domain [0, +∞) and the angle γ ∈ [0, π 2 ]. With this parameterization it is straightforward to cast the quartic part of the potential, denoted hereafter by V (4) and given by the second line of Eq. (2.1), in the following simple form, We stress here that the crux of the matter is the existence of a parameterization, Eqs (2.3 -2.7), which allows to scan all the field space and in the same time recasting the relevant part of the potential into a biquadratic form in tan γ. It is the concomitance of these two facts that allows a tractable and complete analytical solution for the necessary and sufficient boundedness from below conditions. Indeed, the absence of linear and/or cubic powers of tan γ in Eq. (2.8) is anything but generic. (For instance, in a similar parameterization initially proposed in [30] to study two-Higgs-doublet models such terms do remain, hindering an easy fully analytical treatment.) One can thus consider only the range 0 ≤ tan γ < +∞ in accordance with the above stated range for γ. Boundedness from below is then equivalent to requiring V (4) > 0 for all tan γ ∈ [0, +∞) and all ξ, ζ in their allowed domain. The γ-free necessary and sufficient conditions on the λ i 's have already been given in [31] 3 : Note that the second inequality above is non-strict. This accounts rigorously for the only possible equality among the NAS conditions that is compatible with requiring V (4) to be strictly positive. 4 These inequalities are a subset of the general necessary and sufficient (NAS) positivity conditions for a quartic polynomial (see Appendix G). We stress here that Eq. (2.9) answers fully the question of (tree-level) boundedness from below in the totality of the 10 -dimensional field space. There remains however the dependence on ξ and ζ that parameterize the relative magnitudes of the dimension four gauge invariant operators in Eq. (2.1) that are not controlled solely by r and γ.
One can, however, show that 0 ≤ ξ ≤ 1 and In [31] the authors relied on this allowed range and on the monotonic dependence on (ξ, ζ) in Eq.(2.9) to obtain equations (4.21),(4.22) and (4.23) of [31] reproduced in Appendix B 0 b for later discussions. The authors of [32] rightly observed that [31] had actually overlooked the fact that (ξ, ζ) being correlated, cannot reach an arbitrary point in the rectangle defined by Eq.(2.10). Starting from Eq. (2.9) and using the constraint they showed that the set of conditions Eqs. (B14 -B16) established in [31], although sufficient in all field space directions, are in fact not necessary, even though deviation from absolute necessity is typically at the few percent level. Although we totally agree with their general observation, we will see that despite the correlation between ξ and ζ the conditions Eqs. (B14 -B16) do remain sufficient and necessary whenever λ 3 < 0; the modification will come only for λ 3 > 0. We will come back to this point in more detail later on in Appendix B.
For now, we just add that, as shown in Appendix A 0 c, it is possible to cast the ξ and ζ parameters as follows with c 2H , c 2∆ two independent cosines taking any value in their allowed domain [−1, 1]; note also that Eq. (2.11) comes as a direct consequence of these equations.
Altough the authors of [32] wrote a correct form of the necessary and sufficient BFB conditions, they only sketched a proof of their result. In Appendix B, we provide a detailed proof through a careful study of Eq. (2.9) leading to an alternative form of the fully resolved NAS BFB conditions. The latter reduce to: and Note also that Eq. (2.17) implies λ 3 > 0 and 2λλ 3 − λ 2 4 > 0 so that the B 2 part is relevant only when these conditions are satisfied simultaneously.
The above constraints are in fact totally equivalent to [32] although they have a slightly different form. Indeed the equivalence is not straightforward as the two involved Boolean forms are in general not equivalent to each other. However, they become equivalent due to the implication given by Eq. (B13). The above constraints: • constitute an independent check of the results of [32].
• are written explicitly as a union of domains one of which, B 1 , is a necessary consequence of constraints Eqs. (B15 -B16).
• allow to understand why in some regimes the previous constraints Eqs. (B15 -B16) would exclude only a very small part of the allowed parameter space. This is the case in particular in the regimes where λ 4 1 or λ 2 4 2λλ 3 .
• allow to see analytically that our previous constraints Eqs. (

III. GENERALIZATION ADDING ONE EXTRA REAL TRIPLET
Such a generalization can be of phenomenological interest by itself, but is also motivated by the structure of the Georgi-Machacek model beyond the tree-level [48].
A. The pre-custodial potential where the dimension-2, -3 operators are collected in and the dimension-4 operators in where U L denotes an arbitrary element of SU (2) L in the fundamental representation. This potential was written in [48] and later on in [34] with which we agree up to different normalizations and notations 5 . All other dimension-3,-4 gauge invariant operators are either vanishing or can be expressed in terms of the ones listed above. (For completeness we give a proof of this in Appendix C.)

B. The Georgi-Machacek potential
This model [11,12], a special setup of the model presented in the previous subsection, allows to extend the validity of the SM tree-level (approximate) custodial symmetry in the presence of SU (2) L triplet scalar fields. In particular the potential reads where we followed the notations of [33]. 6 We hat the λ's to distinguish them from those of Sec. II, and define the scalar bi-doublet and bi-triplet as 5 with the field correspondence as given by Eq. (3.10) and couplings correspondence: 3 and λ ABH = 4σ 4 . Note that our normalization factors for the various couplings are chosen such that they cancel out for at least one vertex originating from each operator when symmetry factors are taken into account in the Feynman rules. 6 In Eqs. (3.7, 3.8) τ a = σ a /2 with σ a the Pauli matrices are the usual SU (2) generators in the fundamental representation, t a the generators in the triplet (adjoint) representation, with a = 1, 2, 3, and U some rotation matrix about which we skip here the details (see [49] and [33]) as Eq. (3.7) will not be relevant to our study. so that the normalization of the VEVs are the same as in [33]. Note also the sign difference in a ++ and b + between Eq. (3.1) and Eq. (3.10). The potential V p-c is then mapped onto V G-M through the following correspondence among the couplingŝ provided, however, the following correlations hold for the pre-custodial potential couplings: AH . (3.12) The potential V G-M enjoys an increased symmetry as compared to that of V p-c , Eq. (3.5), with an invariance under an extra global SU (2), where U (n) L,R denotes n-dimensional representation of SU (2) L,R . The correlations given by Eq. (3.12) can thus be viewed as encoding the tree-level constraints imposed by the SU (2) R global symmetry on the potential. We come back to this point in Sec. V when discussing briefly quantum effects. References [48], [34] considered such correlations. 7

C. The pre-custodial BFB conditions
Being a polynomial in the fields, the tree-level potential has no singularities at finite values of the fields; it follows that boundedness from below means that the potential does not become infinitely negative at infinitely large field values. This is equivalent to requiring strict positivity of the quartic part of the potential, Eq. (3.4), for all field values in all field directions. The latter requirement is sufficient as it implies that at infinitely large field values, where |V (2,3) p-c | |V (4) p-c | in Eq. (3.2), the potential does not become infinitely negative. That it is also necessary might not seem obvious since the last term in Eq. (3.4) is linear 7 We agree with [34] except for a factor two difference on the right-hand side of the first equation of the second line of Eq. (3.12) as compared to the first equation of the second line of Eq. (10) of [34].

13
in A and in B, so that V (4) p-c might be negative for some finite values of the fields without being unbounded from below. That this does not happen, and the above requirement is indeed necessary, can be easily seen as follows: If there existed a point in field space where p-c ≤ 0, then scaling all the fields at that point by the same real-valued amount s would have lead to V (4) p-c ≡ s 4 V (4) p-c ≤ 0, implying unboundedness from below since s can be chosen infinitely large. Note finally that strict positivity is important here because a vanishing V (4) p-c at very large field values would generically lead to the dominance of V T rAA † ≡ r 2 sin 2 a cos 2 b, (3.17) T r(B 2 ) ≡ r 2 sin 2 a sin 2 b, (3.18) where r is a non negative number, and a ∈ [−π/2, +π/2] and b ∈ [−π, +π] two angles. It 14 will also prove useful to define the following real-valued quantities, Hereafter we will refer to the latter four parameters as the α-parameters. In terms of T, t and the α-parameters, the quartic part of the potential now reads where It is important to note that scanning independently over all values of the thirteen real-valued components of the fields A, B and H amounts to varying T, t and the α-parameters. The latter, however, do not all vary independently. For one thing, the α-parameters vary in bounded domains: α A and α AH are nothing but respectively ζ and ξ defined in Eqs. (2.6, 2.7). Hence see Appendix D for details. For another, the α-parameters are uncorrelated only locally.
But similarly to what was pointed out in [32] and discussed at length in sec. II A for the Type-II seesaw model potential, they are correlated globally in that they cannot reach the boundaries of their respective domains independently of each other. The actual domain in the 4-dimensional α-parameters space is certainly not the simple hyper-cube defined by Eqs. (3.26 -3.29). One can approach the true domain by considering the projected domains on the sub-spaces of these parameters taken two-by-two. This is not trivial to establish and will be carried out in full details in Sec. III D. The more difficult task of determining fully the true domain will be discussed in Section III D 7.
In contrast, the variables T and t vary in ∈ As noted previously, only the highest degree monomial coefficient can vanish. However, for the sake of simplicity we will consider in the sequel only the strict inequality c 0 > 0. It is convenient to recast the above inequalities in the following equivalent form that disposes of the (less tractable) square root: which simplifies further to Note that the second inequality in Eq. (3.34) and the first inequality in Eq. (3.35) depend solely on α A or on α AB . They can be easily resolved since the dependence on these parameters is monotonic; if required to be valid ∀α A , α AB in the domains given by Eqs. (3.26, 3.28), they become equivalent to requiring them simultaneously at the two edges of these domains, namely: A > 0, (3.36) for the first, and λ (1) AB > 0, (3.37) for the second. Equation (3.35)-(II) needs more care due to the nontrivial global correlation between α A and α AB (see next section and Fig. 2), and will be kept in its present form for the time being. One will also have to tackle a further complication involving the two inequalities of Eq. (3.35). Indeed, due to the 'or' structure of Eq. (3.35), none of the two corresponding inequalities need to be necessarily valid for all α A , α AB in their domains; it suffices that one of the two inequalities be satisfied in a given subset of α A , α AB , and the other inequality satisfied in the complementary subset. In particular, Eq. (3.37) is only sufficient. To reach the NAS conditions one will have to consider all possible coverings of the domain by two subsets for which such a configuration holds. This issue will be solved explicitly in Sec. III E 1.
We turn now to the two inequalities of Eq. (3.33). The first is quadratic in t, see Eq. (3.24), but could in principle be treated as a biquadratic polynomial in √ t, since t ∈ [0, +∞). The second, 4a 0 c 0 − b 2 0 > 0, is a general quartic polynomial in this same variable. This is the first place where we encounter the issue of positivity conditions for a general quartic polynomial. Relying on a classic theorem about single variable polynomials that are positive on (−∞, +∞), we derive in Appendix G a relatively tractable form of the corresponding NAS conditions for a quartic polynomial. However these conditions are not directly applicable to the case at hand since the relevant variable here, t, is in [0, +∞). In this case the NAS conditions would obviously be less restrictive, see for instance [50,51] for recent reviews. 8 Relying on these theorems we extend the results of Appendix G to the domain [0, +∞) in Appendix H.
However, this is not the full story. Similarly to what we stated above in subsection III C 1 regarding Eq. (3.35), the 'or' structure of Eq. (3.33) implies that it is sufficient for the two inequalities b 0 > 0 and 4a 0 c 0 − b 2 0 > 0 to be separately valid in two complementary subsets of the allowed t and α-parameters domains. The NAS conditions will then be obtained by investigating all possible coverings of these domains for which this happens.
The upshot is that the possibility of varying freely t with respect to the α-parameters is not sufficient anymore. Indeed, a given subset of the α-parameters where for instance b 0 > 0 (or 4a 0 c 0 − b 2 0 > 0) will be necessarily correlated with t. A strategy for an explicit resolution will be given in Sec.III E 2.
Although it will prove unavoidable to deal with positivity conditions of quartic polynomials on sub-domains of (−∞, +∞), it will still be useful for the subsequent discussions to replace from the onset t ∈ [0, +∞) by a variable on (−∞, +∞) if possible. This is indeed the case if one considers the variable Z defined as since α ABH can take either signs, cf. Eq. (3.29). However, in order to apply safely the NAS positivity conditions on a polynomial in Z, one should make sure that Z is not correlated with the other parameters, α A , α AH and α AB appearing in the inequalities, even though these parameters are globally correlated with α ABH .
It is obviously the case for |Z| since t is uncorrelated with the other parameters and allows to scan independently of the value of |α ABH | the full [0, +∞) range. However, the sign of Z is controlled by α ABH which is globally correlated with α A , α AH and α AB . It is thus crucial to check that the sign of α ABH is not correlated with the latter parameters. That this is indeed the case is easily seen by recalling that all the inequalities are required to be valid ∀A, B, H in the field space, and noting that α A , α AH and α AB remain unchanged, while α ABH flips sign, at the two field space points A and −A (or equivalently at B and −B, or H and iH), see Eqs. (3.21, 3.22). It follows that one can change freely the sign of α ABH for any given configuration of α A , α AH and α AB . (As we will see in the next subsection, Figs. (3 -5), this translates into domains symmetrical around α ABH = 0.) The variable Z ∈ (−∞, +∞) is thus genuinely uncorrelated with the other field dependent reduced parameters.

D. Global correlations among the α-parameters
In this section we first determine the allowed domains of the α-parameters taken two by two, then combine the resulting six global correlations to obtain an analytical approximation of the full 4D domain. Since the α-parameters are ratios of gauge invariant quantities, cf.
Eqs. (3.21,3.22), it is convenient to choose a gauge that reduces the dependence on the set of components fields of the A, B and H multiplets. Apart from the treatment of α A versus α AH , we carry all the discussion in this section assuming a gauge that diagonalizes the (hermitian and traceless) B multiplet as defined in Eq. (3.1), which then takes the form It follows that the dependence on b cancels out in α AB and, up to a global sign, in α ABH .

α A versus α AH
These parameters are identical respectively to ζ and ξ that were defined and studied in detail in Section II A and Appendix A 0 c. We just recall here the corresponding domain:

α A versus α AB
With no particular gauge choice but using the fact that the parameter α A is a ratio, one can recast it in terms of reduced parameters in the following form: where we defined |a 0 | = a cos ϕ sin θ, |a + | = a cos θ, |a ++ | = a sin ϕ sin θ, ρ = arg(a 0 ) − 2 arg(a + ) + arg(a ++ ), To determine the boundary of the allowed domain one can for instance study the variation of α A in Eq. (3.46) as a quadratic function of x ≡ sin 2ϕ in the domain 0 ≤ x ≤ 1 to identify the set of maximal and minimal possible values of α A for a given α AB depending on cos ρ. The maximum is reached for x = α AB 1−α AB cos ρ which lies in the allowed domain only if cos ρ ≥ 0 and α AB ≤ 1 2 . Otherwise, the maximum is reached at one of the boundary values x = 0 or x = 1. We find that the boundary of the domain is given by the following four curves: (I) : α AB = 0 and

α AH versus α ABH
Similarly to the preceding case, we recast α AH and α ABH in terms of reduced parameters and in the gauge where Eq. (3.39) holds: (1 + cos 2ϕ cos 2ψ sin 2 θ) + 1 2 √ 2 (cos ϕ cos θ 3 + sin ϕ cos θ 4 ) sin 2ψ sin 2θ, where θ and ϕ are as previously defined and A numerical parametric scan over the various angles allows to guess the boundary of the α AH versus α ABH domain. The result turns out to be very simple given by the two curves: Fig. 3. The proof for the upper boundary (3.51) is simple: It suffices to exhibit particular configurations of the various angles for which α AH saturates its upper bound while α ABH scans all its allowed domain. An example is ϕ = ψ = θ = π 2 , keeping all the θ i 's free.
This gives α AH = 1 and α ABH = √ 2 cos θ 2 , which proves the above statement. The lower boundary (3.52) is much more difficult to establish analytically. The proof is somewhat involved and will be relegated to Appendix D 0 c.

α A versus α ABH
Here again a numerical parametric scan over the various angles helps guessing the boundary of the α A versus α ABH domain. However, one still needs for that to admit ad hoc that the whole boundary is obtained when sin θ = 1. The analytical proof is quite involved and is given in Appendix D 0 d for completeness. We find that the boundary is determined by the following: where Y (defined in Eq. two curves:

α AH versus α AB
The boundary of the allowed domain in the (α AB , α AH ) plane is given by: We will proceed differently here by constructing an analytical approximation of the true α-parameters domain from a back-projection using only six planes. Obviously any point in the true domain should have its projections on the six planes lying within the six domains determined above. This necessary condition can be characterized by the interior of a four dimensional convex domain that we will refer to as the 4D potatoid. To determine explicitly this 4D potatoid we first express separately in the form of a logical (inclusive) disjunction each of the six domains of Figs. 1-6, then form the logical conjunction of these disjunctions. The resulting Boolean expression is somewhat involved but, interestingly enough, it eventually simplifies to the following form: This form is non-trivial in that it does not display explicitly all six correlations among the four α-parameters; in particular, the correlation between α A and α ABH does not appear explicitly and, depending on α AB , either only three or five of the six correlations are explicitly needed. These features will prove useful when resolving the constraints in Section III E 2. It is also informative to partially visualize the 4D potatoid by considering its 3D projections along each of the four directions. This amounts to combining the domains three by three which leads after some simplifications to: Since relying on continuity arguments one does not expect holes in the interior of the true domain, that would leave no imprint in the projections on the six planes, one concludes that differences between the potatoid and the true domain should be located on the boundaries of the former. We defer a detailed study showing that this is indeed the case till section IV.
There we will make use of an interesting feedback on the issue from the more constrained Georgi-Machacek model.

E. Resolved forms of the pre-custodial BFB conditions
For now we ignore the above subtleties and exploit in the present section the domains of the α-parameters, as determined so far, to push as much as possible an explicit resolving of the conditions given by Eqs. (3.32, 3.33) for the λ parameters themselves.
As stressed at the end of Section III C 1, in order to fulfill the 'or' structure of Eq.
AB ) will satisfy Eq. The task can seem daunting since there are a priori infinitely many ways of forming a covering of the domain. However, one can identify a clear procedure. Note first the obvious fact that, for a given (λ  to select or reject the considered point in λ-space.
Given the linear dependence on α AB in Eq. In the latter case it is required to be entirely contained in the "yes" region determined by the parabola.
Putting everything together, the problem becomes equivalent to solving for the following complementary conditions: where the numbering corresponds to that of Fig. 8.
We can now derive in a fully analytical way the resolved form of Eqs. The details are very technical and will not be described here. We give the final result in Fig. 8 where we have defined the following Boolean expressions 3 :  11 Note that an alternative approach to obtain these results is to start from the third inequality in Eq. (3.30) with no 'OR' structure rather than from Eq. (3.33). Its advantage is to avoid the use of partitions and coverings but necessitates the study of functions with square roots as in Appendix B leading though to more compact conditions. We have checked the agreement of the two approaches. The partitions/coverings approach we developped will nevertheless be unvoidable for the all-field-directions full analytical resolving of the pre-custodial model in the case λ ABH = 0, not treated in the present paper. § AB satisfying the inequalities given by Eqs.
AB space for λ We investigate now Eq. (3.33) that should be valid ∀Z, α A , α AB , α AH , α ABH in their allowed domains. (We use here the variable Z defined in Eq. (3.38) instead of t, and refer the reader to Section III C 2 for a discussion on the relevance of Z.) As argued repeatedly in Sections III C 1, III C 2 and discussed in detail in the previous subsection, the 'or' structure in Eq. (3.33) implies that the validity of the inequalities should be required for all possible coverings of the (Z, α-parameters) space. However, the situation is more complex here than in the previous subsection, since 4a 0 c 0 − b 2 0 , cf. Eq. (3.24), involves simultaneously all four α's and is a complete quartic polynomial in Z. Given the particularly involved NAS conditions for quartic polynomials, Eqs. (G30a -G30d), we do not expect to resolve completely this case in an explicit form similar to that given in Fig. 8. The aim here is to proceed as far as possible towards an explicit resolving, then deal with the rest through mere numerical scans on the α-parameters defined by Eq.  •B 8 : Viewing b 0 , Eq. (3.24), as a quadratic polynomial in Z, we denote by z ± its two roots.
Thus B 8 corresponds to the NAS condition for which z ± are not real-valued, that is to requiring the discriminant of this polynomial to be negative, for all α AH , α ABH in the domain given by Eqs. (3.76) Clearly then, the NAS conditions for the sufficient condition b 0 > 0 read, see Fig. 10, However, as will be discussed later on in Sec. IV, the condition on the left-hand side of Eq. (3.76) is in fact only sufficient to yield B 8 .

Condition (3.82) appears much less amenable to a resolved form as it involves all four
α-parameters simultaneously. One can however still resolve it partially but this will not be pursued further here. 12 The remaining conditions corresponding to Eqs. (G30b,G30c,G30d) will be treated numerically.  : In this case a nonlinear change of variable is used with ξ ∈ [0, +∞) before applying criterion (H13). Here too a simplication occurs for a 0 and a 4 after the change of variable. Up to a global positive definite denominator, they are expressed in terms of Eq. (3.87):  In [33] the authors provided a detailed study of the properties of the potential relying on a generalization of the parameterization used in [31]. They identified the two parameterŝ G-M =r 4 cos 4γ λ 1 + (λ 2 −ωλ 5 ) tan 2γ + (ζλ 3 +λ 4 ) tan 4γ , (3.92) Noting that T r(Φ † Φ) = 2H † H and T r(X † X) = 4T r(AA † ) + 2T r(B 2 ) one can relater and tanγ to the parameters defined in Eqs. (3.16 -3.19) to obtain, tan 2γ = (1 + cos 2 b) tan 2 a, andr 2 cos 2γ = 2r 2 cos 2 a.
, , 1] as already found in [33]. The allowed domain in the (ω,ζ) plane has been given in [33]. This was done stating that the boundary of the domain is obtained from the real valued components of the neutral field directions, that is keeping only Re χ 0 and ξ 0 and zeroing all the others in Eqs.
(note that ref. [33] used SU (2) L instead), and use SU (2) L to rotate away for instance χ ++ and the imaginary part of χ + , bringing the bi-triplet X in the form where u(≡ Re χ + ) denotes a real-valued scalar field. 13 With this choice of gaugeω andζ 13 One could be tempted to zero, on top of χ ++ , the (real-valued) ξ 0 entry rather than Im χ + . However one take the following form ω = 1 4 2 √ 2 cos θ 0 cos(arg(χ 0 )) + sin θ 0 sin θ 0 sin 2 θ + (3.100) where we defined the polar angles by Note that due to the invariance of V G-M under X → −X one can always fix uniquely either the sign of ξ 0 or that of u. In our parameterization ξ 0 > 0 while u can take either signs. In Eqs. (3.101, 3.103) we kept for simplicity only linear terms in u. We will come back to the exact contribution later on. Here we first concentrate on the 0 th order u contributions toω andζ, i.e. Eqs. (3.100, 3.102) which we dubω 0 andζ 0 . In Appendix E we give a detailed proof for the determination of the boundary in the (ω 0 ,ζ 0 ) domain, i.e. under the working assumption that u = 0 (= cot θ u ). We find that this boundary is defined by the following upper and lower curves: can show that this is not possible through a non infinitesimal SU (2) rotation. More generally, one cannot zero more than two entries of X through SU (2) L × SU (2) R rotations.
This reproduces exactly the boundary given in reference [33] as illustrated in Fig. 11 (note   FIG. 11: The boundary in the (ω 0 ,ζ 0 ) plane delimiting the allowed inner domain, in the limit u = 0. This agrees with reference [33].
however that we deal with the inverse function with respect to reference [33]). As shown in Appendix E 0 c the condition sin 2 θ + = 1, i.e. ξ + = 0, is sufficient and necessary for the determination of the (ω 0 ,ζ 0 ) boundary. In particular the necessity of this condition is a non-trivial result. From Eq. (3.100) one sees that sin 2 θ + = 0 could as well have defined a boundary. More importantly, the involved dependence on sin 2 θ + inζ 0 , Eq.
The key point is that the latterX has the same form as X given by Eq. (3.99) with u = 0.
We are then brought back to the same configuration that leads to the fact that the boundary is reached for ξ + = 0 and is given by Eqs. In the following we will refer to this domain as the ω-ζ-chips.  14 In practice this is achieved by scanning over the four α-parameters that satisfy Eq. (3.61) and following each trajectory (ω α (t),ζ α (t)) scanning over 0 ≤ u ≤ π 2 with t =  -One sees from Fig. 13 (a) that for α ABH −0.27 there are no exclusions by the ωζ-chips. In particular, the 2D section at α ABH = 0 corresponds to the full α A , α AB domain of Fig. 2 which is thus not reduced by the constraint from the ω-ζ-chips.
In fact this result could be easily retrieved once noted that the α A , α AB domain of  1]. If follows that the study in Sec. III E 1 that lead to the NAS conditions given by Fig. 8 remains valid, at least for the α AH = 1 2 , α ABH = 0 section. Moroever, since the domain of Fig. 2 is not only a projection but corresponds as well to the latter section of the α-potatoid, then the above mentioned NAS conditions are sufficient conditions for all other sections at fixed α AH , α ABH since by construction they all fall in the interior of the α A , α AB domain of Fig. 2. Obviously this holds even if these sections have portions excluded by the ω-ζ-chips, e.g. when α ABH < −0.27 as seen from Fig. 13 (a), since sufficiency is more constraining. We can thus safely conclude that the conditions given by Fig. 8 are NAS for the validity of Eq. (3.32) in all the α-potatoid.
-Along a similar line of thought, one deduces from Fig. 13 (b), where there are no exclusions by the ω-ζ-chips as soon as α ABH −0.06, and from the fact that the projected domain shown in Fig. 1 is also retrieved as a 2D section at α AB = α ABH = 0, that the conditions given by Eq. and (α AH = 1, α AHB = − √ 2), see Fig. 12 (b). The resulting truncated domain will however cease to be the largest section so that the obtained conditions are now only necessary for an extended fraction of the α-potatoid. As seen from Fig. 12 (d), the maximal section taking into account the ω-ζ-chips constraint does exist somewhere inside the 3D domain but would be difficult to determine analytically.

V. PUTTING EVERYTHING TOGETHER: A USER'S GUIDE
It is time to recapitulate the various results we arrived at and then provide a roadmap for an optimal exploitation: • While studying the general pre-custodial potential we were lead automatically in sections III C 1 and III E 1 to constraints that involved only the A and B multiplets for which we provided the fully resolved NAS BFB conditions in analytical form, see See text for further discussions.
necessary BFB conditions for the full pre-custodial potential since they correspond to the potential in the H = 0 field direction.  one finds that the constraint Eq. (3.84) should be applied whenever λ BH < 0, thus retrieving the fully resolved NAS BFB conditions for the SM extended by one real SU (2) triplet.
• We give in Table I a roadmap for a user's implementation of the constraints following two alternative roads each made of two steps. Step 1 ○ is common and corresponds to the fully resolved necessary constraints that are also NAS if restricted to the A, B or H, A sectors. Note that these constraints are already stricter than the ones given in [34] under the assumption of two nonvanishing complex fields at once or the ones extended to the "custodial" direction in [35], as they are NAS in all directions within A, B or H, A. Also specifying to the Georgi-Machacek case we do retrieve the conditions found in [33]. Steps 2 ○ and 2' ○ are two technically different but theoretically equivalent ways to complete the NAS conditions. Note first that in both cases branches a Let us close this section with an outlook on some issues related to the subject of the present paper but lying beyond its scope: perturbative unitarity constraints. They typically bind the absolute magnitudes of the λ couplings and some of their combinations from above. These constraints should eventually be studied for the general pre-custodial model (see however [35]) and be combined with   At least one among the conditions B 3 , B 5 and B 7 is active in cases (ii), (iii) or (iv) of the flowchart of Fig. 8. We illustrate a few such configurations on Fig. 15. The domains shown in the figure are necessary but not sufficient; they can be reduced further when adding the rest of the NAS BFB conditions. Note that such a potential tension disappears in the limit of decoupling between the two triples (λ (i=1,2) AB → 0) in accordance with the unitarity/BFB conditions found in [31].
quantum corrections. They affect the tree-level constraints in various ways: -they modify the form of the constraints, introduce a notion of scale at which they should be satisfied and criteria for the validity of perturbativity, as treated for instance in [54], [35,55] -however, it is not often appreciated that combining perturbative-unitarity and stability requirements beyond the tree-level needs some further care because the physical meaning of the running couplings becomes different in these two classes of constraints. Since AH , λ BH ) domain with λ (1) the shape of the potential (see for instance [56, 57] 17 ). It thus appears that, in so far as replacing the tree-level couplings by their runnings in the tree-level conditions is a good approximation, the potential stability conditions need not be required at all 'scales', from the electroweak scale all the way up to some very high cut-off Λ (e.g. M GU T or M P lanck ) as often done in the literature [17,32,34], but only at that scale Λ which represents the largest value of the fields. Barring Landau poles, there is indeed no physical reason to require the improved quartic part of the potential to remain positive for intermediate values of the fields. (Obviously this is at variance with the unitarity constraints that should be satisfied already at the energy scale of a given scattering experiment.) Furthermore, a longstanding issue is how to improve the effective potential in the presence of several scalar fields (see [58] for a recent reappraisal, and references therein). As concerns the NAS BFB conditions of Table I, they can be used beyond the tree-level in two different ways: i) The quartic part, V p-c , Eq. (3.4), of the pre-custodial potential has the same form as the general counterterms needed to renormalize the Georgi-Machacek model accounting 17 where it was also stressed that even an additive constant becomes field dependent beyond tree-level. for a deviation from the tree-level correlations Eq. (3.12) due to the custodial symmetry breaking loop effect of the U (1) Y gauge couplings [48], [34]. One is thus guaranteed that the ten λ couplings of V (4) p-c will absorb the one-loop corrections of the Georgi-Machacek effective potential up to field dependent factors of the form log(M(φ i ) 2 /Q 2 ) − c, where M is typically a binomial function of the fields, Q is some renormalization scale and c a renormalization scheme dependent constant. It follows that satisfying the conditions of Table I on the λ's that absorb the one-loop induced quartic couplings, will also guarantee the stability of the full one-loop Georgi-Machacek effective potential at large field values Table I can also obviously be used as a seed for the loop corrected This was considered for the two-Higgs-doublet model [36], for extensions with two real scalar singlets [42], or for the most general potential in Left-Right symmetric models [59]. It is also the case for the pre-custodial model with λ ABH = 0 studied in this paper. The copositivity method ceases to be efficient in this case since one cannot write the potential in a bilinear form with vectors of dimension 3 or more having positive definite independent components.
(The fact that a two-dimensional such a form is still possible is not helpful as one hides the complexity of the conditions in the dependence on angles). In contrast, our approach remains applicable albeit with an extended set of the α-parameters. It would provide further insight into the all-directions NAS-BFB conditions for example in the study of Left-Right symmetric models, unlike in [60] which relied on [41,59] where specific directions were considered. A distinctive feature in this case is the appearance of positivity conditions for full quartic polynomials as was found for the model with two singlets discussed in [42] and 54 in the present study with two triplets.

VI. CONCLUSION
We carried out in this paper a comprehensive study of tree-level necessary and sufficient First we note that ∆ being traceless implies the identity (see also Eq. (C2)), from which follows immediatly Since H † ∆ † ∆H is positive definite one then has and thus Furthermore ξ is trivially greater than zero since it is the ratio of two positive definite quantities. Finally the two values 0 and 1 are effectively reached respectively when H † ∆ = 0 and ∆H = 0, which is always possible for some given configurations of the H and ∆ field components provided that Det∆ = 0 when H = 0. Thus Then, using Det∆∆ † ≡ |Det∆| 2 ≥ 0 implies straightforwardly from Eq. (A6) that Note that the value 1 is indeed reached when ∆∆ † has one zero and one non-zero eigenvalues, which is always possible to find for some configurations of the ∆ field components.
Also, we trivially have ζ ≥ 0 since it is the ratio of two positive definite quantities. However, the value 0 cannot be trivially reached, since if the numerator of ζ vanishes then the denominator should vanish as well! In fact ζ cannot go below 1/2. To see this we rewrite ζ in terms of M 2 1 , M 2 2 the two (real and positive) eigenvalues of ∆∆ † , It is now easy to study the function ζ(x) = (1+ x 2 )/(1+x) 2 where x ≡ M 2 1 /M 2 2 ≥ 0, to show that it has a minimum of ζ = 1/2 at x = 1, that is when ∆∆ † has degenerate eigenvalues.
One also retrieves the fact that ζ(x) ≤ 1 and reaches 1 for x → 0 or x → ∞. Thus and of the parameters defined in Eqs. (3.19 -2.7). Since U(x) is unitary and ∆∆ † hermitian, we can always find, for any given field configuration ∆, a gauge transformation U ∆ (x) that diagonalizes ∆∆ † . Then ζ takes the form given in Eq.(A8) and ξ reads where the tilde denotes the components of the transformed doublet H = U ∆ (x)H. It is then natural to define with their obvious range of variation c 2 ∆ , c 2 H ∈ [0, 1]. Equations (2.12, 2.13) follow then straightforwardly from Eqs. (A8, A10 -A12): This allows to determine the lower envelope in the ξ, ζ plane, i.e. when saturating the inequality in Eq. (2.11) as discussed in [32]. We will however rely directly on Eqs. (A13, A14) when determining the BFB conditions in the next section. .
At this point a careful study is needed, as the dependence on c 2∆ is not trivially monotonic so that a priori one does not necessarily have, It is nonetheless noteworthy that this equivalence does hold in half of the parameter space region where λ 3 < 0 despite the non-monotonicity of F in c 2∆ , as we will see in a moment.
Irrespective of the sign of λ 3 the first and second derivatives of F read, In the sequel we will assume without further reference the conditions given in Eq. (B1). It then immediately follows from Eq. (B7) that F < 0, ∀c 2∆ ∈ [−1, 1], whenever λ 3 < 0. This implies that if F admits an extremum it will be necessarily a maximum so that Eq. (B5) is valid, since in this case the value of F at one of the two boundaries of [−1, 1] is necessarily the smallest value it can take. On the other hand, if F does not admit an extremum then Eq. (B5) is obviously valid as well, and one retrieves Eqs. (B15, B16).
We thus conclude that the BFB conditions Eqs. (B15, B16) initially found in [31] are necessary and sufficient, and thus complete, when λ 3 < 0.
The situation is quite different when λ 3 > 0. In this case F is non-negative over the full with the consistency condition 0 ≤ (c (±) 2∆ ) 2 ≤ 1. The latter condition reads Note that the second of these two inequalities always implies the first due to the case assumption λ 3 > 0 and the validity of Eq. (B1). Moreover this second inequality can be rewritten equivalently as where we again relied on the case assumption λ 3 > 0. Thus Eq. (B10) is necessary and sufficient for the existence of minima within the domain [−1, 1]. In this case one of the two functions F (c 2∆ , +), F (c 2∆ , −) will have a minimum at c (+) 2∆ and the other at c (−) 2∆ . Moreover, the values of the two F functions at these minima turn out to be the same, given by, It follows that even though the two functions F (c 2∆ , +) and F (c 2∆ , −) do not reach their minimum for the same value of c 2∆ , requiring when Eq. (B10) is satisfied, will be equivalent to Eq. (B4). Note in particular that Eq. (B12) should imply F (±1, ±) > 0 and F (∓1, ±) > 0, that is, which is indeed the case. 18 Finally, when Eq. (B10) is not satisfied, but still λ 3 > 0, then either c (±) 2∆ are not real-valued or they lie outside of the [−1, 1] domain. In both cases the two functions F (c 2∆ , ±) are monotonic on [−1, 1] and Eq. (B5) applies, which is similar to the previously discussed case of λ 3 < 0. Putting everything together one can summarize the conditions that are equivalent to λ 1 + ξλ 4 + λ(λ 2 + ζλ 3 ) > 0 (or Eq. (B4)), as follows: Adding Eq. (B1) to these conditions, we obtain the Boolean form of the necessary and sufficient BFB conditions as given by Eqs. (2.14, 2.17). 18 This is due to the inequality λ( and Eq. (B1) valid and thus consistently also 2λλ 3 − λ 2 4 > 0.

b. The old conditions
We recall here for further reference the sufficient and almost necessary BFB conditions [31]: thus trivializing Eqs. (C6, C7).
Similarly, Eq. (C2) leads to, Note that products of two dim-2 traced operators should also be added. Thus a systematic strategy would be to reduce in the above list the traces of the product of four matrices to products of two traces, whenever possible. This is done using the same tricks as illustrated for dim-3. E.g., T rB 4 = T r( 1 2 1T rB 2 )B 2 = 1 2 (T rB 2 ) 2 as a consequence of Eq. (C11), or . Note that T rAA † AA † can be transformed similarly but we chose not to do so in Thanks to gauge invariance and to the fact that B is self-adjoint one can always find, for each given value of α AB , an SU (2) L transformation U L that diagonalizes B leading to Then all dependence on B drops out from α AB , and one is then left with Using the fact that −|z| ≤ Re(z) ≤ |z| for any complex number z, one immediately finds where the upper (lower) bound is effectively reached in the field directions where arg(ã ++ (φ + * ) 2 ) = 0(π), arg(ã 0 (φ 0 * ) 2 ) = π(0). Moreover, since |ã 0 | 2 + |ã + | 2 + |ã ++ | 2 ≥ |ã 0 | 2 + |ã ++ | 2 , α ABH scans a larger domain in the directionã + = 0, namely Defining x = |φ + |/|φ 0 | and y = |ã ++ |/|ã 0 |, the above domain is rewritten as  To prove that Eq. (3.52) gives the lower boundary we show hereafter that δ ≡ α 2 ABH − 2 α AH is either negative or vanishing. This combination is of the form with x ≡ sin θ and a, b easily read from Eqs. (3.49, 3.48), a = − cos 2ϕ cos 2ψ + 2(c 1 cos ϕ cos 2 ψ − c 2 sin ϕ sin 2 ψ) 2 , and c i ≡ cos θ i . The study of the δ(x) function shows that it reaches only one stationary point where we took into account the fact that 0 ≤ x ≤ 1, cf. Eq. (3.44), and defined r = a/b.
One also finds This in turn is equivalent to since a ≤ 1, Eq. (D15) being valid independently of the sign of b. Expressing sin ψ and cos ψ in terms of T ≡ tan ψ, the above inequality is equivalently rewritten as with T ∈ [0, +∞) and We can now examine the NAS positivity conditions for the biquadratic polynomial in T , namely The first two are trivially satisfied. To prove the third we should take into account the correlation θ 1 = θ 2 + θ 3 + θ 4 (modulo multiples of 2π), see Eq. (3.50). Moreover, since sin θ 1 and sin θ 2 can take either signs, we can include the two cases by simply using the inequality (1 − c 2 1 )(1 − c 2 2 ) ≥ sin θ 1 sin θ 2 to write: This ends the proof that α 2 ABH − 2 α AH ≤ 0 holds for all field directions and that Eq. (3.52) gives the lower boundary in the (α ABH , α AH ) plane.
with the obvious notations, x = sin ϕ, y = sin ψ, c 1 = √ 2 cos θ 1 sin θ, c 2 = √ 2 cos θ 2 sin θ. We seek the conditions on α ABH , c 1 and c 2 that ensure the existence of at least one value for x ∈ [0, 1] for each value of y 2 ∈ [0, 1] and vice versa. This can be worked out by solving for y 2 and considering the (relative) signs of c 1 and c 2 . One finds: • When c 1 × c 2 ≥ 0, α ABH can be of any sign, with x and α ABH satisfying • When c 1 × c 2 ≤ 0, α ABH and c 2 should have the same sign, with x and α ABH satisfying We can study now the allowed domain in the plane (α ABH , α A ). We first determine the allowed (α ABH , α A ) sub-domain corresponding to sin θ = 1, then show that all sub-domains that correspond to sin θ < 1 are necessarily within that sub-domain, which thus turns out to be the full (α ABH , α A ) domain.
When sin θ = 1 the dependence on cos ρ drops out from Eq. (3.42) and one can easily solve for x(= sin ϕ) as a function of α A , The two ± solutions should be kept in the discussion as their union scans the full , 1] is allowed in the (α ABH , α A ) plane.
Note also that since |α ABH | > 1 the min and max in Eqs. (D28) become uniquely defined, equaling respectively 1 − gives the lower boundary for α A when α ABH > 1.
This completes the proof that when sin θ = 1, the allowed domain in the (α ABH , α A ) plane is as defined by Eqs. (3.53) and illustrated in Fig. 4.
Since θ 1 and θ 2 appear only in α ABH , they can be safely chosen without biasing the correlations between α A and α ABH , as long as they maximize the allowed domain of the latter. The angle θ is however common to α A and α ABH . One should then be careful that the value sin θ = 1 does not miss points in the allowed domain. A necessary condition for this not to happen is that sin θ = 1 still allows α A and α ABH to take any value in their 69 respective domains as given by Eqs. (3.26, 3.29). This is indeed the case as one can check from Eqs. (3.42, 3.49) by varying all the other angles at fixed sin θ = 1.
However this is not sufficient. One should still check that for sin θ strictly smaller than one there exists no set of values for the remaining angle variables giving a point in the (α ABH , α A ) plane that is outside the domain defined by Eqs. (3.53). To show this it suffices to prove (cf. Eq. (D31)) that whenever Rewriting Eq. (3.49) as where Y = sin ϕ sin 2 ψ cos θ 2 − cos ϕ cos 2 ψ cos θ 1 (D35) and Y ∈ [−1, 1], condition (D33) implies since none of |Y | and sin θ can exceed one. We can thus replace Eq. (D33) by On the other hand, as seen from Eqs. in which case (D32) can be recast in the form where we defined τ = tan θ and dropped out a positive denominator. The coefficients of τ 2 and τ 4 in Eq. (D40) both satisfy as a consequence of the lower bound in Eq. (D37). The first is immediate to establish. The positivity of a 4 is less obvious. Rewriting |Y | ≥ 1/ √ 2 and a 4 respectively as and noting that Y is linear in cos θ 1 and cos θ 2 , cf. Eq. (D35), one can easily study the sign of a 4 when Eq. (D45) is satisfied, in terms of a bundle of six parallel straight lines with slope cot ϕ cot 2 ψ in the (cos θ 1 , cos θ 2 ) plane; the sign alternates each time one of these lines is crossed. Moreover, since they are all parallel it suffices to study the change of sign along a given axis in the (cos θ 1 , cos θ 2 ) plane, say the axis defined by cos θ 2 = 0. On this axis the inequality Eq. (D45) is satisfied if and only if This implies Moreover, the quadratic function in τ 2 is a monotonically increasing function as can be seen from its derivative and Eqs. (D43, D44). Its minimum is thus reached for τ 2 min = 1 2Y 2 −1 and is given by where we defined x ≡ cos 2ψ, and a, b are readily obtained from Eqs. (3.45,3.48), It is easy to study the structure of maxima and minima of α AH (x) at fixed a, b. One finds that it always has only one stationary point, at x = a sgn b √ a 2 + b 2 ∈ [−1, +1], given by Moreover, this stationary point is found to be a minimum (resp. maximum) when b < 0 (resp. b > 0), and thus with a corresponding maximum (resp. minimum) of α AH given by max{α AH (±1)} (resp. min{α AH (±1)}). This leads to: The parameter b as defined by Eq. (D56) can take either signs when all the angles are varied (since cos θ 3 , cos θ 4 ∈ [−1, 1] and ϕ ∈ [0, π 2 ], cf. Eq. (3.44)). It is thus more relevant to combine the α AH domains given above, reducing them for fixed a and |b| to or equivalently to Given Eq. (D56), the domain in Eq. (D61) is obviously maximized for cos θ 3 = cos θ 4 = ±1.
We note that these functions do not depend on the sign of y. Starting from Eq. (E22) it is straightforward to determine the configurations whereζ 0 reaches its absolute minimum value 1 3 . One finds,ζ The lower boundary for the portionω 0 ∈ [− 1 4 , − 1 6 ] is thus to be found within theζ − 0 branch. A straightforward analytical study shows thatζ − 0 (y,ω 0 ) is a strictly decreasing function of y 2 for anyω 0 ∈ [− 1 4 , − 1 6 ]. 19 It follows that the lower boundaryζ 0 min (ω 0 ) is given byζ − 0 (y,ω 0 ) at y 2 = 1 (strictly speaking at y = −1 sinceω 0 < 0), The functions given in Eqs. (E21, E25, E27) provide the full boundary in the (ω 0 ,ζ 0 ) domain. Given that χ ++ and Im χ + are put to zero by a gauge choice, i.e. Eq. (3.99), we have proven under the working assumption Re χ + ≡ u = 0 in Eq. (3.99), that this boundary is obtained when y = ±1 and sin 2 θ + = 1, that is for Im χ 0 = ξ + = 0, cf. Eqs. (E1, 3.105). This agrees with [33] where the domain was determined by a numerical scan. There is however more to the proofs we provided: sin 2 θ + = 1 is not only sufficient but also necessary; indeed as one can see from the various steps of the proofs given above, all the inequalities and monotonicity are strict.
It is important to stress that there is a priori no simple reason to believe that the domain (ω 0 ,ζ 0 ) will be identical to the full domain of (ω,ζ), i.e. when relaxing the working assumption u = 0. The necessity of sin 2 θ + = 1 proved instrumental while completing the determination of the domain when u = 0, see Sec.III F. 19 More specifically, we find that the derivative ∂ that are necessary and sufficient to ensure where P (ξ) is a quartic polynomial: Our derivation does not rely on the known form of the four roots of P (ξ) = 0, and will actually allow to cast the conditions in a simpler and more compact form than the ones usually relied upon in the literature, [42,61]. To achieve this we take a different path than just writing down the well-known expressions of the four roots of P (ξ).
We are interested in determining the exact {a i } space region for which P (ξ) is positive valued for any ξ in (−∞, +∞). Recalling a classic theorem on positive definiteness of even degree polynomials defined on R and having all their coefficients real-valued, if P (ξ) satisfies Eq. (G1) then it can be written in the form where the x i , y i and z i denote real numbers. 20 . 20 Note that taking Q and R as in Eq. (G4) is more general than actually needed. Indeed, P (ξ) will satisfy The exact {a i } space is then defined by the NAS conditions on the a i coefficients such that there exist real numbers x i , y i and z i satisfying eq. (G3). To determine these conditions we find useful to geometrize this statement. Introducing the vectors, x = (x 1 , x 2 ), y = (y 1 , y 2 ), z = (z 1 , z 2 ), the identification of the coefficients of each ξ monomial in Eq .(G3) leads to 2x.y = a 1 , 2y.z = a 3 , (G9) so that the problem is equivalent to determining three vectors knowing some of their moduli and scalar products and relations among them. The NAS conditions on the a i will thus be determined by requiring consistent moduli of and angles between the three vectors x, y, z. Eq. (G1) if and only if its four roots are non-real complex-valued, that is P (ξ) of the form P (ξ) = r(ξ − s)(ξ −s)(ξ − t)(ξ −t) = r|(ξ − s)(ξ − t)| 2 , with Im(s), Im(t) = 0, s, t and their complex conjugatess, t being the four roots, and r a positive real number. Expanding this form as the squared modulus of a complex number, leads to Eq. (G4) but with one of the two polynomials Q and R being only linear in ξ.
The symmetric choice made in Eq. (G4) lends itself however to a more convenient geometric discussion.
Its equivalence with the more specific case above, results from the invariance of Eq. (G3) under any rigid rotation of the three vectors x, y and z defined in Eq. (G5). and using the boundedness of the cosine one finds the necessary condition for the existence of the modulus of y: It should be stressed that while this condition is necessary to ensure the existence of at least one choice of the angle (x, z), not knowing the sign of a 2 , for which the modulus of y exists, Eqs. (G11, G13) are in general not sufficient to guarantee the existence of the vectors themselves (apart from the special case a 1 = a 3 = 0); one has still to check for the consistency of the three scalar products: Eqs. (G8, G6, G10) lead to and Eqs. (G9, G7, G10) to a 3 = 2 √ a 4 a 2 − 2 √ a 0 a 4 cos (x, z) cos (y, z).
Again, using −1 ≤ cos ≤ 1, one retrieves two necessary conditions from these two equations that can be summarized as These conditions are stronger than condition (G13). There is however a further constraint that correlates Eqs. (G14, G15), namely (y, z) = (y, x) + (x, z). This transforms Eqs. (G14, where η ≡ cos (x, z), and yx (resp. yz ) indicates the relative sign between sin (y, z) and sin (z, x) (resp. between sin (y, x) and sin (x, z)). Note also that Eqs. is the critical value above which at least one of the square roots in Eqs. (G17, G18) turns complex and thus becomes invalid. 21 To study further the conditions for the existence of η we square both sides of Eq. (G17). This leads to a cubic equation in η: where we define for later use I(η) ≡ 2 √ a 0 a 4 (2 √ a 0 a 4 η − a 2 ) (η + 1) + a 1 a 3 (η − 1), (G22) It is important to note that Eq. (G21) would equally result from squaring Eq. (G18) due to the invariance under the permutation (a 1 ↔ a 3 , a 0 ↔ a 4 ). It follows that (G21) encodes by itself the information contained in (G17) as well as that contained in (G18), except for the one that is lost by squaring, namely the signs yx , yz . This loss of information is however not problematic, as the signs can be retrieved by plugging back in Eqs. (G17, G18) whatever valid solutions for η are found by solving (G21). Moreover, the constraint that only the solutions satisfying Eqs. (G19, G20) are valid, is implicitly embedded in Eq. (G21): Whenever a solution is found satisfying η ∈ [−1, +1], it automatically satisfies (G19, G20).
The reason is that squaring both sides of Eq. (G17) enforces the positivity of the term under the square-root. We thus conclude that the sought-after NAS conditions are those which guarantee the existence of (at least one) real-valued η satisfying simultaneously (G21) and η ∈ [−1, +1], together with Eq.(G11). The function I(η) being a cubic polynomial in η, one can in principle solve I(η) = I which has at least one, and up to three, real-valued solutions.
One could of course proceed numerically, but this is not our aim. On the other hand, 21 Note that a necessary condition for the existence of η is obviously η * ≥ −1, leading back to Eq. (G16).
extracting an information from the analytical expressions of the three roots of this cubic equation is not particularly tractable. In fact I(η) has some interesting properties listed below, that are straightforward to establish and will allow us to determine analytically the NAS conditions without solving the equation. A straightforward calculation shows that one always has: where we define ∆ 0 = a 2 2 + 12a 0 a 4 − 3a 1 a 3 . We can now write down the full NAS conditions. As clear from Eq. (G21), they correspond to ensuring all possible configurations in the (η, I) plane for which I(η) crosses (at least once) the positive horizontal line I = I within the η domain given by Eq. (G19). We will refer to these configurations as existence configurations (EC). To proceed, we begin by identifying a set of four necessary conditions for the EC, then show that they form together with Eq. (G16) a set of sufficient conditions as well.
We note first that, when it exists, η + is always the position of the local minimum of I(η). Properties (b) and (d) then imply that this minimum is necessarily negative whenever We therefore conclude that adding the necessary condition Eq. (G16) to Eq. (G11) and (G26 -G29), one obtains a set of necessary and sufficient conditions. There is however 87 more to it. One can show that (G28) actually implies Eq. (G16). The latter can hence be discarded without loss of generality. 22 Putting everything together, the NAS conditions read finally: where we defined ∆ 1 = 2a 3 2 + 27(a 0 a 2 3 + a 4 a 2 1 ) − 72a 0 a 2 a 4 − 9a 1 a 2 a 3 .
Note that we have switched all the inequalities over to strict. The non generic equality cases can lead to different conditions. However, as argued at the beginning of Section III C, only strict positivity will be relevant. We have performed a numerical check of the above NAS conditions by scanning randomly over a 0 , a 4 ∈ [0, 100] and a 1 , a 2 , a 3 ∈ [−100, 100] for 10 5 points, then solved numerically P (ξ) = 0 for each point and checked that whenever Eqs. (G30a -G30d) are satisfied, P (ξ) has no real roots, and whenever one of the conditions is violated P (ξ) has at least one real root. We also performed another non-trivial check based on the obvious fact that a translation of P (ξ) to P (ξ + ξ 0 ) for any ξ 0 ∈ R * should not affect the positivity. It follows that the NAS conditions obtained after the translation, where the modified coefficientsã 0,1,2,3 depend explicitly on ξ 0 whileã 4 = a 4 , should be equivalent to the initial ones. Incidentally we find that ξ 0 cancels out in the modified ∆ 0 and ∆ 1 , which 22 The proof consists in showing that (G28), more explicitly Eq.