Symmetries of the 2HDM: an invariant formulation and consequences

Symmetries of the Two-Higgs-Doublet Model (2HDM) potential that can be extended to the whole Lagrangian, i.e. the CP-symmetries CP1, CP2, CP3 and the Higgs-family symmetries Z2, U(1) and SO(3) are discussed. Sufficient and necessary conditions in terms of constraints on masses and physical couplings for the potential to respect each of these symmetries are found. Each symmetry can be realized through several alternative cases, each case being a set of relations among physical parameters. We will show that some of those relations are invariant under the renormalization group, but others are not. The cases corresponding to each symmetry group are illustrated by analyzing the interplay between the potential and the vacuum expectation values.


Introduction
The discovery of the Higgs boson by the LHC collaborations in 2012 [1,2] was a remarkable achievement. A series of precision measurements of its properties (see for instance [3,4]) revealed that the particle observed at the LHC has spin 0 and behaves, to a good degree of precision, as one would expect for the Higgs boson in the Standard Model (SM) [5][6][7][8][9][10]. Since then, a great deal of effort has been put into searches for Beyond the Standard Model (BSM) physics, but so far no significant deviation from SM phenomenology has been observed, no significant excess hinting at new particle resonances has been found.
The coming years will bring about a wealth of LHC results as we progress towards its high luminosity phase. This will provide us with the opportunity to further test BSM theories, some of which are already quite constrained by current data. The drive to extend the Standard Model is obvious when one considers the amount of observed facts that the model does not explain: the hierarchy in fermion masses, the astrophysical and cosmological data indicating the existence of Dark Matter, and the universe's matter-antimatter asymmetry, among other puzzles. There are many interesting proposals for BSM physics. One of the most popular consists in enlarging the scalar sector, and one of the simplest models of this kind is the Two-Higgs-Doublet Model (2HDM), proposed in 1973 by Lee [11] as a means to obtain an extra source of CP violation from spontaneous symmetry breaking (see [12,13]). In this model the SM field content is complemented with a second SU(2) doublet, which yields a larger scalar spectrum -a charged scalar field and three neutral ones (in versions of the model where CP is conserved two of those scalars are CP-even, the third being odd). The model has a rich phenomenology, and different versions of the 2HDM allow for dark matter candidates, spontaneous or explicit CP violation, tree-level flavour-changing neutral currents (FCNC) mediated by scalars, and many other interesting phenomena. In fact, these "versions" of the 2HDM correspond in many cases to different symmetries imposed on the model, which reduce the number of free parameters (thus increasing its predictive power) and change the phenomenology of the theory. The first such symmetry was introduced by Glashow, Weinberg and Paschos [14,15] -a discrete Z 2 symmetry corresponding to one of the doublets being odd under it, which, when extended to the whole Lagrangian, eliminated the tree-level FCNC mentioned above. Another symmetry, a continuous U(1), was proposed by Peccei and Quinn [16] in an attempt to solve the strong CP problem. Other symmetries were eventually proposed and thoroughly studied.
The study of 2HDM symmetries, however, is complicated by the fact that the model possesses a basis invariance. In fact, a general 2HDM can be formulated adopting different bases for the doublets, therefore, e.g., the scalar sector of the model is not uniquely defined. Different (while being physically equivalent) potentials could be related by a U(2) basis transformation which is not a symmetry of the model. Such (basis) transformations will in general make the parameters of the potential change, and a symmetry of the potential that is manifest in one basis will in general not be obvious in another. Therefore, the symmetries might be hidden and difficult to recognize. However, there exist physical parameters of the scalar sector of the 2HDM that are independent of the basis adopted to formulate the model and those could be utilized in the identification of the symmetries. The ultimate goal of this work is to provide a formulation of all possible symmetries of the 2HDM potential in terms of physical (observable) parameters, like masses and measurable coupling constants. Knowing the physical symmetry conditions would make the verification of invariance unambiguous, without any reference to a particular basis.
Symmetries are of fundamental relevance both for classical and for quantum field theories. Hereafter we limit ourselves to internal symmetries, even though space-time transformations also play a fundamental role in contemporary physics, the Lorentz invariance in special relativity and reparametrization invariance in general relativity being famous examples. The presence of continuous global symmetries implies, via the Noether theorem, the existence of conserved currents and charges. Conservation of electric charge, lepton or baryon numbers could serve as other examples of consequences of U(1) invariance. Even if a symmetry is broken explicitly by the presence of non-invariant terms in the Lagrangian, still, if the breaking is small, the notion of the symmetry might still be very useful. When global continuous symmetries are not respected by vacuum states the Goldstone theorem requires the existence of massless scalars that correspond to all the broken generators of the symmetry group. Here again the role of the symmetry is crucial while trying to understand the mass spectrum of particles, pions as the Goldstone bosons of spontaneously broken SU(2) L × SU(2) R serve here as a spectacular example. On the other hand, if a symmetry remains unbroken, its presence implies constraints on parameters, e.g. mass degeneracies appear and/or some couplings are related, while others might vanish. This is why symmetries are one of the main tools for model building. If continuous symmetries are local, their importance is even amplified, as in that case they lead to gauge theories such as QED or the SM itself. When symmetries are broken by terms of dimension < 4 ("soft" symmetry breaking terms) then, according to Symanzik [17,18], a theory remains renormalizable. That is yet another illustration of the power of symmetries.
Another class of symmetries is formed by discrete transformations that leave the action invariant. In particular, space (P) and time (T) reflections, and charge conjugation (C) have to be emphasized. In fact, composite symmetries such as CP and CPT play fundamental roles in quantum field theory. Discrete symmetries are also often adopted in theories of dark matter where e.g. the Z 2 symmetry mentioned above may be used to stabilize DM particles (taken to be odd under the symmetry). One may then wonder: how many different internal symmetries can one impose on the 2HDM scalar sector? The answer was found by Ivanov [19,20], using a bilinear field formalism to prove that there were only six different classes of symmetries that could be imposed. Three of these were socalled Higgs family symmetries, in which invariance of the scalar potential is required for doublet transformations of the form Φ i → Φ i = U ij Φ j , with U ij elements of a 2 × 2 unitary matrix U , they include the Z 2 symmetry, for which U = diag (1, −1), and the Peccei-Quinn U(1) symmetry, U = diag(1, e iθ ) for a generic real phase θ, mentioned above; and the SO(3) symmetry, for which one takes a general U(2) matrix. The remaining three symmetries arise from requiring invariance under generalised CP transformations of the where again X is a unitary 2 × 2 matrix. Different choices of X yield different CP symmetries, to wit CP1 (the "standard" CP symmetry, with X equal to the identity), a discrete CP2 symmetry [21][22][23] and a continuous CP3 one [24]. These, then, are the only six symmetries for an SU(2) × U(1) invariant 2HDM scalar potential. If one chooses to ignore hypercharge, then other symmetries arise, such as the custodial symmetry. The full classification of those possibilities, which we will not consider in the present work, may be found in [25,26].
We are going to find basis-independent conditions for invariance of the 2HDM potential under the field transformations which yield the six 2HDM symmetry classes mentioned above. Such conditions were expressed in a covariant way in terms of basis-dependent parameters in Ref. [27], whereas here we shall express these conditions in terms of basisindependent observables, and discuss spontaneous breaking of the symmetries. This work is a natural extension of the papers [28,29], where we have discussed the invariant formulation of the 2HDM under CP transformation. Here, our intention is to provide conditions (in terms of measurable parameters) for invariance of the 2HDM potential under all the remaining possible symmetries.
The symmetries in question satisfy the following hierarchy [27], These relations will be reflected by the physical constraints to be quoted in the following. The formulation of basis-independent conditions for global symmetries in the 2HDM has recently been addressed by Bento et al [30] in the framework of a rather mathematical formalism. While that approach is general and may have interesting applications also in other theories, we think that at least in the case of 2HDM our approach is more useful, directly expressing constraints in terms of physical quantities.
The paper is organized as follows. In section 2 we review the model, and discuss the choice of parameters. In section 3 we review the approach of Ref. [27], and outline the mapping to physical parameters. Then, in section 4 we present our results for the different cases in compact form, with the detailed analysis presented in section 5. In section 6 we address the issue of stability under the renormalization group equations (RGE), and in section 7 we provide a brief discussion, highlighting the RGE-stable cases. More technical material is collected in three appendices.

The model
We shall start out by parametrizing the scalar potential of the generic (CP-violating) 2HDM in the common fashion: 1 All parameters in (2.1) are real, except for m 2 12 , λ 5 , λ 6 and λ 7 , which in general could be complex.

Choice of basis and basis independence
The potential has been written out in terms of two doublets that we have named Φ 1 and Φ 2 . Since both doublets have identical quantum numbers and there is nothing a priori to distinguish them, we could equally well have expressed the potential in terms of linear combinations of these (initial) doublets, i.e. if we defineΦ i = U ij Φ j , where U is a U(2)matrix, we can instead choose to express the potential in terms ofΦ 1 andΦ 2 . This is referred to as a change of basis. Note that the parameters of the potential will in general change under a change of basis. How the parameters change under the most general change of basis is given explicitly in Eqs. (5)-(15) of [31]. This reparametrization freedom means that some of the parameters in (2.1) are superfluous and can be eliminated via a judicious basis choice. Thus, the number of free parameters in the most general 2HDM potential is not the 14 shown in (2.1), but rather 11 [21], as we will discuss later on. But basis changes also introduce complications when it comes to an attempt to recognize whether a given 2HDM potential is invariant under a particular symmetry.
Clearly, physics cannot depend on an arbitrary choice of basis for the Higgs doublets. All measurable quantities must be basis independent, thereby leading to the study of basis invariant quantities in multi-Higgs-Doublet Models (NHDMs). Of course, the scalar masses are basis invariant. The same holds for most of the physical couplings, the exception being the couplings f i that are defined in section 2.3. They occur in couplings involving charged fields, whose phases are arbitrary, and are thus pseudo-invariants [28].
In the present work we shall derive relations between the basis-invariant masses and couplings needed in order to respect certain symmetries imposed upon the potential. In order to do so we shall choose to derive these relations in a particularly simple basis, namely the Higgs basis [32,33]. The Higgs basis is a basis in which only one doublet has a nonvanishing, real and positive vacuum expectation value (VEV), whereas the other doublet has a vanishing VEV. If the original doublets have neutral (and in general complex) VEVs then the Higgs basis is obtained via the field redefinition 2 so that the new fields have VEVs given by We must make sure that the vacuum corresponds to a minimum of the potential, and by demanding that the derivatives of the potential with respect to the fields should vanish we end up with the stationary-point equations in the Higgs basis, Demanding that the vacuum should correspond to a stationary point does not guarantee that it is a minimum of the potential. One must also demand that the squared masses of the physical scalars are positive in order for the potential to have the curvature of a minimum point. In the present study we shall encounter situations where some physical scalar has a vanishing mass. Then we shall relax the requirement of positive squared masses by simply demanding that the physical scalars have non-negative squared masses. One should also add that within the 2HDM there may be coexisting minima for the same set of parameters [19,20,[34][35][36][37], so in fact one must also verify whether the minimum we are interested in is the global one. We will however not address this issue in the present work.

Scalar fields and mass eigenstates
Having chosen to work within the Higgs basis, we may parametrize the two doublets as The great advantage of working in the Higgs basis is that the massless Goldstone fields, which we represent here by G 0 and G ± , are immediately present in the VEV-carrying doublet. Then, H ± are the massive charged scalars. The neutral fields η i are not mass eigenstates, so we relate them to the mass eigenstate fields H i (whose CP properties are in general undefined) by an orthogonal rotation matrix R as As for the charged sector, the masses of the charged scalars H ± can be read directly off from the corresponding bilinear terms in the potential, and are given in the Higgs basis by As for the neutral sector, the bilinear terms can be written as where the mass-squared matrix is in the Higgs basis found to be Then, by using (2.7) we obtain the masses of the neutral scalars from the diagonalization of the mass-squared matrix, M 2 , We shall use indices i, j, k ∈ {1, 2, 3} to refer to these neutral mass eigenstates.

Physical couplings of scalar eigenstates
Having identified and diagonalized the mass terms of the potential, the remaining terms are trilinear and quadrilinear in the scalar fields, thereby representing trilinear and quadrilinear couplings among the scalars. Some of these couplings play an important role in the present work, namely the three trilinear neutral-charged H i H + H − couplings and the quartic charged self-interaction, that is H + H + H − H − . We denote these by q i and q, respectively. In the Higgs basis they are given by One can show explicitly that these couplings are all basis independent [38]. The LHC is already probing one of these couplings, q 1 , via the diphoton decay of the discovered Higgs boson, since in the 2HDM a scalar loop contributes to that amplitude. The scalar-gauge boson couplings will also be necessary for the present work. They originate from the kinetic term of the Lagrangian, which may be written as where we have adopted the usual definitions, Relevant couplings can now be read off from the kinetic terms, It is not a coincidence that different vertices are proportional to the same quantities e i and f i , but rather a consequence of the gauge invariance of the model. The factors e i and f i are given, in terms of Higgs basis parameters, by In a general basis, the factors e i = v 1 R i1 + v 2 R i2 are found to be explicitly invariant under a change of basis [38]. Unitarity of the rotation matrix in this multi-doublet model forces these factors to satisfy a sum rule, to wit The factors f i (and their conjugate partners f * i ) appear in couplings between scalars and gauge bosons whenever an H + W − pair (H − W + pair) is present at the vertex. In a general basis they are given by . These factors are not invariant under a change of basis, they transform as pseudo-invariants, meaning that their lengths are invariant, but their phases change, see [39]. The product f i f * j is, however, invariant under a change of basis. This can also be seen from the following identity

The physical parameter set
While the potential of the 2HDM has a total of 14 real parameters, the number of observable quantities arising from the potential is in fact less than 14. Through a series of basis changes one can reduce the number of potential parameters from 14 to 11, leaving us with a total of 11 physical independent quantities as stated in [21]. A simple way of seeing this is by considering once again the most general 2HDM potential of (2.1) -it is easy to imagine a doublet rotation such that m 2 12 is set to zero, thus eliminating two parameters from the potential (since this coefficient is in general complex). With this "diagonalization" of the quadratic part of the potential the quartic couplings will also change, of course. Then, with m 2 12 = 0 in the new basis we can still rephase one of the (new) doublets to absorb a complex phase from λ 5 , for example, thus eliminating a third parameter. Now, instead of working with 11 independent potential parameters, we will choose a set of 11 physical parameters, masses and couplings, that we denote by P [28,29,38,39]. For this purpose we pick the mass of the charged scalars as well as the masses of the three neutral scalars along with the scalar couplings H i H + H − and H + H + H − H − and the coefficients e i of the gauge couplings to get 3 which we denote as our physical parameter set, consisting of 11 independent invariant quantities. All the other purely scalar couplings of the model are expressible in terms of these 11 parameters [39] along with the auxiliary complex couplings f i and f * i (which do not appear separately in physical observables because of (2.22)). All physical properties of the scalar sector are thus expressible in terms of masses and couplings. For a 2HDM with some symmetry, then, some of the 11 parameters of the physical parameter set P will either be related or set to zero.

The bilinear formalism and symmetries
In this section we will briefly review the bilinear formalism, in which the scalar potential is expressed not in terms of the doublets themselves but rather using their gauge-invariant bilinear products. This formalism is rather useful when studying symmetries and possible vacua of NHDM models. An earlier application of this method appeared in [40] and was used to establish tree-level theorems about the stability of 2HDM minima [34,41,42]. A remarkable formulation of bilinears in a Minkowski space was developed in [19,20,[43][44][45]. The bilinear formulation used in this paper is that of [46][47][48][49][50]. The formalism was adopted to investigate the custodial symmetry of the 2HDM in [51]. Similar formalisms have also been used for other models, for instance the 3HDM [52,53], the complex singlet-doublet model [54] and the N2HDM [55,56].

Field bilinears
It is very convenient to express the potential in terms of four gauge-invariant bilinear products of the doublets. We will follow closely the conventions of [27], defining the bilinears as and the four-vectorK Then one can express the potential of the 2HDM as and in our notation 4 The authors of the paper [27] classified in their Table II all 5 possible symmetries of the SU(2) × U(1) 2HDM potential in terms of the two vectors, ξ and η, together with eigenvectors and eigenvalues of the three-by-three matrix E.

Translating to the physical parameter set
Our goal is to express the conditions for the different symmetries of the 2HDM in terms of constraints among the masses and couplings of the physical parameter set P. For this purpose it is convenient to introduce the following vectors 6 Using the results from Appendix A we find that in the Higgs basis In [27] some of the potential parameters are defined slightly differently than ours. 5 Only those symmetries of the scalar potential which could be extended to the whole Lagrangian of the model are discussed, so e.g., custodial symmetry has not been considered there as it would require no hypercharge coupling, g = 0. The custodial symmetry has been studied using the bilinear formalism in [51]. 6 Note that F a i , F b i and F c constitute a basis for R 3 in the case where fi = 0.
Likewise, one can translate the elements of the matrix E, yielding the following results, valid for the Higgs basis Thus we see that by working in the Higgs basis, we managed to express ξ, η and E in terms of the 11 parameters of P as well as the auxiliary quantities f i and f * i .

Properties of the vectors ξ and η
Both the vectors ξ and η, as well as their cross product ξ × η, will be needed in the discussion to follow. Also, we will need to formulate conditions for the vanishing of either of these vectors. For that purpose it is convenient to write out expressions for the squared length of each vector. Using (3.10)-(3.11) along with (2.22), first we find that where χ i = ξ 3 + 2v 2 η 3 e i M 2 i + v 2 ξ 3 q i , and Im J 1 is a quantity encountered in the study of the CP properties of the 2HDM [28,29,31,57,58]. This quantity is part of a set of three invariant quantities, {J 1 , J 2 , J 2 }, such that if all J i = 0 the 2HDM vacuum preserves CP. 7 Im J 1 can be expressed in terms of the physical parameter set as In fact, J3 of the early papers [57,58] corresponds to the present Im J1.
For the squared lengths of the vectors we then find Let us at this point also introduce a shorthand notation for a quantity which we will encounter later in our study, This quantity is always non-negative, and vanishes iff (e 1 q 2 − e 2 q 1 ) = (e 1 q 3 − e 3 q 1 ) = (e 2 q 3 − e 3 q 2 ) = 0.

The eigenvalues and eigenvectors of the matrix E
In [27], many of the symmetries we are about to discuss are formulated in terms of properties of the eigenvalues Λ m of the three-by-three matrix E, and the corresponding eigenvectors, e m : E e m = Λ m e m , m = 1, 2, 3. (3.20) Note that m ∈ {1, 2, 3} labels the eigenvalues, it should not be confused with the set {i, j, k} used to label the three neutral scalars. Furthermore, the eigenvectors e m should not be confused with the couplings e i , e j or e k . The characteristic equation of the matrix E will be a cubic one, and in general we must express the roots of the characteristic equation using cube roots. In many of the physical configurations encountered, the characteristic equation factorizes and can be solved without the need for cube roots. The discussion of the eigenvalues and the eigenvectors of E is relegated to Appendix A.

The vanishing of Im J 1
Most symmetries we are about to discuss require that the cross product ξ × η vanishes. From (3.17) we see that this will require Im J 1 = 0. There are several ways for Im J 1 to vanish, we need to explore them all. We list six physical configurations which together cover all situations under which Im J 1 vanishes: and not Configuration 1.
We know from earlier work [28] that Configurations 1-3 imply CP conservation for both the potential and the vacuum, since then all Im J i = 0. Configurations 4-6 all imply Im J 1 = 0, but some other Im J i will be non-zero, thus CP is not conserved. The potential may still be CP invariant, in which case we will have spontaneous CP violation. In Configuration 6 it is implicitly understood that there is no mass degeneracy, and that all three gauge couplings e 1 , e 2 and e 3 are non-vanishing.

Results
In Table II of [27], the six symmetry classes of the 2HDM, and the corresponding constraints on the scalar potential parameters, are listed. At this point, we shall make note of the fact, that except for a single constraint for the CP1 symmetry that requires ξ × η to be an eigenvector of E (thus requiring ξ × η = 0), all other constraints require ξ × η = 0. Therefore we may split the analysis into two parts-first we analyze how to get CP1 conservation when ξ × η = 0. Next, when continuing the analysis for the situations where ξ × η = 0 (both the second and the third option for CP1, as well as all the other symmetries), we employ the fact that this also implies Im J 1 = 0, working systematically through the six configurations listed in section 3.5. The amount of configurations, subconfigurations and special configurations needed to be explored in order to arrive at the final results is substantial. We omit details of these calculations, however we believe that we have provided the reader with enough tools to explore and reproduce results listed hereafter on one's own, given the preliminary results in section 3 and Appendix A.

CP1 symmetry
If there exists a basis in which the potential of the 2HDM is invariant under the transformation where X = I 2 , then we say that the potential is invariant under CP1, i.e. there exists a basis in which the potential is invariant under complex conjugation, often referred to as "standard" CP symmetry. From Table II of [27] we see that the potential possesses the CP1 symmetry iff either of the following two conditions is met:
Performing the analysis, we recover four already known [28,29] cases when the 2HDM potential is CP conserving, where the auxiliary sum D is given by Cases A, B and C are identical to what we have referred to as Configurations 1, 2 and 3 in section 3.5. From earlier work we know that these are cases under which we not only have a CP-invariant potential, but also a CP-invariant vacuum [28]. Also from earlier work we know that the constraints of Case D only guarantee a CP-invariant potential, but the vacuum may or may not be CP-invariant, opening the possibility for having spontaneous CP violation [29]. The reason for putting a bar over A and B is because these two cases of CP1 symmetry are unstable under the renormalization group equations. We shall in fact always put a bar over RGE unstable cases encountered, whereas the cases encountered that are RGE stable will be written without the bar. We have devoted section 6 to a discussion of stability under RGE.
It is worth commenting on Case B, where mass degeneracy of the two fields H i and H j is accompanied by the constraint (e j q i − e i q j ) = 0 on the couplings. The mass degeneracy allows for an arbitrary angle 8 in the neutral-sector rotation matrix allowing one to construct linear orthogonal combinations H a and H b out of H i and H j . In general, by a suitable rotation, one can arrange to make one of the new H's even and the other odd under CP. In Figure 1 we make an attempt at visualizing the constraints of each of the four cases of CP1. Each circle with a letter i, j, k represents one of the three neutral scalars. This graphic representation is quite useful for a quick analysis of the several configurations of masses/couplings yielding a given symmetry. Below we list some details of this convention: • If a given circle is filled with a color (black or green), the corresponding scalar can have a non-vanishing mass. Scalars rendered massless by a symmetry will be represented as empty circles (see figure 3 below).
• Whenever there is a line connecting two circles, it means that the two connected neutral scalars are mass degenerate. We then see in the visualization that Case A corresponds to full mass degeneracy and Case B to partial mass degeneracy.
• Whenever there is a green cross on a line connecting two mass degenerate neutral scalars labeled i and j, this tells us that, in addition to the mass degeneracy, the constraint e j q i − e i q j = 0 applies to the couplings of those two scalars. This is seen in the visualization of Case B.
• Whenever a circle is filled with green color (rather than black) this means that the corresponding neutral scalar does not couple to ZZ, W + W − or H + H − pairs. This illustrates, for Case C, that e k = q k = 0.
• Whenever the circles representing the three neutral scalars are enclosed by a larger circle (as shown for Case D), this means that the two constraints characterizing Case D apply (one constraint on M 2 H ± and another on q).
As for the remaining symmetries, Z 2 , U(1), CP2, CP3 and SO (3), we know that if the potential respects any one of them, it will also be CP1 symmetric. The vacuum will be CP-invariant as well in any of those cases 9 , indicating that Configurations 1, 2 and 3 are the only configurations that need to be explored for the remaining symmetries. This also means that they will all have to satisfy at least the constraints of cases A, B or C. 9 Remember that we are only dealing with exact symmetries and are not considering soft breaking terms in the potential.

Z 2 symmetry
If there exists a basis in which the potential of the 2HDM is invariant under the transformation then we say that the potential is invariant under Z 2 . From Table II of [27] we see that the potential possesses the Z 2 symmetry iff the following condition is met We find a total of six cases when the 2HDM potential is Z 2 invariant 10 , Case AD: Case CD: In Figure 2 we visualize the constraints for each of the six cases of Z 2 . The namings of these six cases are related to the four individual cases of CP1. We note that each of the six cases of Z 2 is obtained by simultaneously imposing two or more of the conditions yielding CP1 (this is in agreement with theorem 1 of [27]): • Case AD is the combination of cases A and D.
• Case ABBB is the combination of three different cases B (with different sets of indices), also satisfying case A.
• Case BD is the combination of cases B and D.
• Case BC is the combination of cases B and C.
• Case CD is the combination of cases C and D.
• Case CC is the combination of two different cases C (with different indices).
From this labeling it is apparent that all cases of Z 2 satisfy at least the constraints of cases A, B or C as stated at the end of the previous subsection.
It follows from the discussion of Case B above that in the fully degenerate Case ABBB two linear combinations (call them H a and H b ) could be formed out of H i , H j and H k that both decouple from gauge bosons (e a = e b = 0) and from the charged scalars (q a = q b = 0), while the third one carries the full-strength couplings. The two states that decouple have opposite CP.

U(1) symmetry
If there exists a basis in which the potential of the 2HDM is invariant under the transformation for an arbitrary angle θ, then we say that the potential is invariant under U(1). From Table II of [27] we see that the potential possesses the U(1) symmetry iff either of the following two conditions is met: • ∆ = 0; and ξ × e = η × e = 0 where e is an eigenvector from a one-dimensional eigenspace of E, (4.8) • ∆ = 0; ∆ 0 = 0; and ξ × η = 0, (4.9) where ∆ and ∆ 0 are defined in appendix A. The condition ∆ = 0 implies that the E matrix has two degenerate eigenvalues, and requiring ∆ 0 = 0 as well, E will have three degenerate eigenvalues.
We find a total of four cases when the 2HDM potential is U(1) invariant, In Figure 3 we visualize the constraints for each of the four cases of U(1). We follow the same pattern as before when giving names to these cases, with the additional subscript "0" whenever there are neutral scalars with vanishing masses. All U(1) invariant potentials are also Z 2 invariant, as is seen when comparing Figures 2 and 3.

CP2 symmetry
If there exists a basis in which the potential of the 2HDM is invariant under the transformation then we say that the potential is invariant under CP2 11 . From Table II of [27] we see that the potential possesses the CP2 symmetry iff the following condition is met: We find a total of three cases when the 2HDM potential is CP2 invariant,  11 The CP2 symmetry cannot be extended to the fermion sector in an acceptable manner, as it always implies at least one massless family [24,48,49].
In Figure 4 we visualize the constraints for each of the three cases of CP2. All CP2 invariant potentials are also Z 2 invariant, as is seen when comparing Figures 2 and 4.

CP3 symmetry
If there exists a basis in which the potential of the 2HDM is invariant under the transformation for any 0 < θ < π/2, then we say that the potential is invariant under CP3 12 . From Table II of [27] we see that the potential possesses the CP3 symmetry iff the following condition is met We find a total of five cases when the 2HDM potential is CP3 invariant, In Figure 5 we visualize the constraints for each of the five cases of CP3. All CP3 invariant potentials are also both U(1) invariant and CP2 invariant, as is seen when comparing Figures 3, 4 and 5. 12 The only viable extension of CP3 to the fermion sector occurs for θ = π/3, any other choice of angle implies a massless family [24].

SO(3) symmetry
If there exists a basis in which the potential of the 2HDM is invariant under the transformation where U is any U(2) matrix, then we say that the potential is invariant under SO(3) 13 . From Table II of [27] we see that the potential possesses the SO(3) symmetry iff the following conditions are met: We find a total of two cases when the 2HDM potential is SO(3) invariant, In Figure 6 we visualize the constraints of each of the two cases of SO(3). All SO(3) invariant potentials are also CP3 invariant, as is seen when comparing Figures 5 and 6.

Analysis
We shall now demonstrate explicitly how the different cases presented in the previous section can be realized for specific choices of the potential, together with a suitable basis. That is, we find the explicit contraints on the potential parameters and the vacuum parameters that correspond to each of the cases presented. Thus, we will see that each of the cases presented as contraints on masses and couplings is in fact realizable by explicit construction of potential and vacuum.
For each of the symmetry classes we write out the most general potential for which the symmetry is manifest, along with a vacuum written out for a general basis. Next, we write 13 It might seem that the symmetry group involved in these transformations would be the full U(2), but as argued in [19,20], taking into account the U(1) hypercharge symmetry underlying the theory, the largest Higgs-family symmetry is indeed SO(3). This is particularly evident in the bilinear formalism. out the resulting stationary-point conditions, which often can be solved in more than one way. Some solutions imply one of the doublets having a VEV equal to zero, for others both doublets have non-zero VEVs. Different ways of solving the stationary-point equations often lead to the manifestation of different cases within each symmetry class. The cases in which the symmetry is manifest in a Higgs basis versus the cases in which the symmetry is manifest in a non-Higgs basis often (but not always) depends upon whether the symmetry under consideration is spontaneously broken or not.
We will also stress that these are tree-level classifications. As we will discuss in the next section, some of the constraints corresponding to the cases listed in the previous section are not stable under the RGE. However, since the symmetry under consideration is preserved also by the higher-order effects, we must necessarily remain within the same symmetry class. We return to this issue in section 6.

CP1 symmetry
If the potential is invariant under CP1, there exists a basis in which all the parameters of the potential are real. The VEV can either be real (CP conservation) or complex (spontaneous CP violation). Cases A, B and C correspond to complete CP conservation (potential and vacuum) while case D allows for spontaneous CP violation. These are known results [28,29] so we do not repeat the details. Anticipating results of section 6, we will find that cases A and B, involving mass degeneracies, are not RGE stable, these will "migrate" into case C which is RGE stable at the one-loop level.
Since hereafter we are going to discuss vacuum structure and the possibility of spontaneous symmetry breaking, we switch from the Higgs basis to a generic one, as this is more convenient for the discussion.

Z 2 symmetry
If the potential is invariant under Z 2 , there exists a basis in which m 2 12 = 0 and λ 6 = λ 7 = 0. Writing out the potential in such a basis, it is given by Since we do not know the form of the vacuum, we shall assume the most general chargeconserving form, and parametrize the Higgs doublets as Here v j are real numbers, so that v 2 1 + v 2 2 = v 2 . The fields η j and χ j are real, whereas ϕ + j are complex fields. Then the most general form of the vacuum reads Note that the phases ξ j are extracted from the whole doublet, not from the VEVs only. Next, let us define orthogonal states and so that G 0 and G ± become the massless Goldstone fields and H ± are the charged scalars. The neutral fields η i are related to the mass-eigenstate fields H i by (2.7). The masses of the physical scalars are read off from the bilinear terms in the potential and, using (2.11), the mass-squared matrix (which now has a different form than in the Higgs basis) is diagonalized. Without loss of generality we can rephase Φ 2 in order to make λ 5 real. Next, utilizing a simultaneous rephasing (by the weak hypercharge) of both Φ 1 and Φ 2 we can make Φ 1 real and non-negative.
This leaves us with the following potential and the vacuum with v i ≥ 0. Minimizing the potential with respect to the fields yields the stationary-point equations We shall assume that Re λ 5 = 0, or otherwise we would have a U(1)-symmetric potential, which we shall treat later. Allowing for solutions with one vanishing VEV, the stationarypoint equations can be solved by simply putting This solution corresponds to a Z 2 -symmetric vacuum. There is no need to also consider v 1 = 0, since this is related to v 2 = 0 simply by an interchange of the two doublets.
Another solution of the stationary-point equations can be found whenever both v i = 0. Then the stationary-point equations are solved by This solution corresponds to a spontaneously broken Z 2 -symmetry. We see that these two ways of satisfying the stationary-point equations are topologically different, meaning one cannot get from the spontaneously broken vacuum solution to the Z 2 -symmetric vacuum solution simply by letting v 2 → 0 in a continuous way. These two situations will therefore lead to different physics, as we will now see.

Z 2 symmetric potential and vacuum
With the potential of (5.6) and a vacuum of the form we have obtained the potential and the vacuum of the Inert Doublet Model (IDM) [59][60][61][62].
In this model the Z 2 symmetry is preserved by the vacuum, and the lightest neutral scalar from the second doublet becomes a viable dark matter candidate. With a vanishing VEV, the phase ξ can be absorbed into Φ 2 and the parametrization of the doublets becomes This is equivalent to simply putting ξ = 0. The mass-squared matrix becomes 13) and the charged mass is given in (2.8), (5.14)

No mass degeneracy (RGE stable)
Provided there is no mass degeneracy, the rotation matrix for the neutral sector is simply the three-by-three unit matrix, so that the relevant couplings become This is then seen to be a a realization of Case CC.
Partial mass degeneracy M 1 = M 2 (RGE unstable) We will not yet allow for mass degeneracy between M 2 and M 3 or full mass degeneracy, since this would require λ 5 = 0 which yields a U(1) symmetric model (which will be treated later). Allowing for mass degeneracy 14 between M 1 and M 2 requires m 2 22 = (−2λ 1 + λ 3 + λ 4 + Re λ 5 )v 2 . The mass-squared matrix now becomes (5.16) and the charged mass One may now argue that the neutral-sector mass matrix is already diagonal, and the rotation matrix will simply be the unit matrix. While this argument is notably correct, one can also (because of the mass degeneracy present) use the rotation matrix where α is a completely arbitrary angle to preserve the diagonal structure of the mass matrix, but the mass eigenstate fields will have a different admixture of the fields H 1 and H 2 for each value of α, affecting the couplings involving the neutral scalars. In particular, using (5.18), we find the following couplings some of which depend on the arbitrary angle α. One can easily see that for any value of α, e 1 q 2 − e 2 q 1 = 0. Thus, this is a realization of Case BC. It should be emphasized that the arbitrariness of the angle α has nothing to do with basis freedom, it is only an artifact resulting from the mass degeneracy. Physics cannot depend upon the value of α, all physical observables must in this mass-degenerate case be independent of α. Since the couplings e 1 , e 2 , q 1 and q 2 all depend on α, none of these couplings are physical. Neither can they be made physical simply by picking a particular value of the unphysical α. However, combinations of these couplings like e 2 1 + e 2 2 = v 2 or q 2 1 + q 2 2 = v 2 λ 2 3 or e 1 q 1 + e 2 q 2 = v 2 λ 3 are independent of α and are physical. Thus, in processes with external H 1 and H 2 one should sum corresponding squares of amplitudes (no interference) 15 , while in the case of virtual H 1,2 , summation should be made at the level of amplitudes 16 . In the end, any dependence on the mixing angle α must vanish. One possible construction is to define one field H a carrying the full-strength interaction, e a = v and q a = vλ 3 , and another field H b , that decouples both from the vector bosons and from the charged Higgs bosons. Note that interactions of H a and H b are easily reproduced from the interactions of H 1 and H 2 by picking the specific value of α = 0, yielding (unphysical) couplings e a , e b , q a and q b : e a = v, e b = e 3 = 0, q a = vλ 3 , q b = q 3 = 0, (5.20) 15 Consider H + H − → H1,2 (external). The sum of squared amplitudes becomes proportional to q 2 1 + q 2 2 , which is physical since it is independent of α. 16 Consider W + W − → H1,2,3 (virtual) → H + H − . The amplitude becomes proportional to e1q1 + e2q2, which is physical since it is independent of α.
coinciding with Eq. (5.15), implying that Case BC is physically indistinguishable from Case CC (IDM) with the additional constraint of mass degeneracy between one inert and one non-inert neutral scalar. A similar interpretation will be applicable in other cases of mass degeneracy. This is an example of a mass degeneracy which is not preserved by radiative corrections, which we will discuss in more detail in section 6.

Spontaneously broken Z 2 symmetry
With the fields expanded as in (5.2), the potential of (5.6) and a vacuum of the form of (5.7), setting 17 ξ = 0, the mass-squared matrix becomes and the charged mass

No mass degeneracy (RGE stable)
Provided there is no mass degeneracy, the rotation matrix for the neutral sector is simply where α is fixed from the diagonalization procedure. It should be emphasised that, in distinction from that used in eq. (5.18), this angle is not arbitrary or unphysical. We find the couplings e 1 = v 1 cos α + v 2 sin α, e 2 = −v 1 sin α + v 2 cos α, e 3 = 0, (5.24) and We find the following two identities to be satisfied in this case for i, j = 1, 2: Thus, this is a realization of Case CD. 17 Other values of ξ satisfying s 2ξ = 0 yield similar results.

Partial mass degeneracy M 2 = M 3 (RGE unstable)
If we allow for mass degeneracy 18 between M 2 and M 3 , the rotation matrix for the neutral sector becomes where α is fixed by the diagonalization of the mass-squared matrix, and θ is an arbitrary angle, again an artifact resulting from the mass degeneracy. This yields the couplings We also find We find that for any value of θ, e 2 q 3 − e 3 q 2 = 0, and the following two identities are satisfied for i, j, k = 2, 3, 1: Thus, this is a realization of Case BD. The situation is analogous to Case BC already discussed, except that now it is H 2 and H 3 that are mass degenerate, and θ is the arbitrary unphysical angle. We adopt the same approach, defining states H b and H c (equivalent to setting θ = 0) resulting in couplings together with 18 Mass degeneracy between M1 and M3 yields a similar result.
Comparing to (5.24) and (5.25), we see that Case BD is in fact physically equivalent to Case CD with the additional constraint of mass degeneracy between one CP-odd and one CP-even scalar. However, the condition for mass degeneracy, is not RGE stable.

Full mass degeneracy (RGE unstable)
If we allow for full mass degeneracy, M 1 = M 2 = M 3 , this requires λ 3 + λ 4 + Re λ 5 = 0, v 2 1 λ 1 = v 2 2 λ 2 and v 2 Re λ 5 = −v 2 1 λ 1 . The mass-squared matrix becomes The rotation matrix for the neutral sector takes the generic form where α 1 , α 2 and α 3 are all arbitrary. In particular, we find the couplings We also find that the following two identities are satisfied for any combination of α 1 , α 2 and α 3 . Thus, this is a realization of Case AD. We may define states H a , H b and H c (equivalent to setting α 2 = α 3 = 0) to regain the couplings e a = v 1 cos α 1 + v 2 sin α 1 , e b = −v 1 sin α 1 + v 2 cos α 1 , e c = 0 (5.38) which we recognize as the couplings of Case CD with the additional constraint of full mass degeneracy imposed 19 . However, the conditions needed for full mass degeneracy are not preserved by the RGE.
With the exception of Case ABBB, which will be discussed in the U(1) section, we have demonstrated that all the cases of Z 2 symmetry are in fact realizable in terms of explicit construction of the potential and vacuum as intended. Some of those symmetry conditions, however, are not preserved by radiative corrections. The only radiatively stable symmetry conditions, then, correspond to cases CC: the IDM, with one "active" neutral scalar and two "dark" ones which do not couple to gauge bosons, and CD: a vacuum with spontaneous breaking of Z 2 , with two CP-even and one CP-odd neutral scalar.

U(1) symmetry
If the potential is invariant under U(1), there exists a basis in which m 2 12 = 0 and λ 5 = λ 6 = λ 7 = 0. In such a basis, it is given by Again, we parametrize the Higgs doublets in the most general way, following the steps in Eqs. (5.2) through (5.5). Without loss of generality we can independently rephase Φ 1 and Φ 2 in order to make both Φ 1 and Φ 2 real and non-negative. Minimizing the potential with respect to the fields yields the stationary-point equations The v 1 = 0 case is related to the v 2 = 0 case by simply interchanging the two doublets. If both v i = 0, then the stationary-point equations are solved by We see that these two ways of satisfying the stationary-point equations are topologically different, meaning one cannot get from the second solution to the first one by simply letting v 2 → 0 in a continuous way. We would therefore expect these two situations to lead to different physics.

U(1) symmetric potential and vacuum
With the potential of (5.40) and a vacuum of the form the mass-squared matrix becomes 45) and the charged mass Note that this vacuum does not break the U(1) symmetry defined by (4.7) -when the Φ 1 doublet receives a phase as a result of the U(1), that phase could be absorbed via simultaneous (global) rephasing of both doublets, which is always allowed due to the hypercharge gauge symmetry. Therefore effectively the vacuum would remain invariant.
Partial mass degeneracy M 2 = M 3 (RGE stable) We see from the mass matrix that the U(1) symmetry dictates mass degeneracy between two of the neutral scalars. Provided there is no full mass degeneracy, we obtain the physical couplings of the neutral scalars to electroweak gauge bosons and charged scalars 20 , This is a realization of Case BCC, which is a version of the IDM with a Peccei-Quinn symmetry instead of the Z 2 one. This model predicts degenerate dark matter candidates, and is disfavoured by astronomical observations [62]. The mass degeneracy is here of a different kind than what we encountered when discussing Z 2 -symmetric cases, since now both states are inert. While the cases of mass degenerate states discussed for Z 2 are radiatively unstable, the mass degenerate states encountered here are radiatively stable under the RGE, as will be discussed in section 6.

Full mass degeneracy (RGE unstable)
Allowing for full mass degeneracy requires m 2 22 = (−2λ 1 + λ 3 + λ 4 )v 2 . The mass-squared matrix then becomes 48) and the charged mass Due to the full mass degeneracy, the rotation matrix for the neutral sector is now given by the most general form (5.35), yielding couplings We find that Q 2 = 0 for any combination of α 1 , α 2 and α 3 . Thus, this is a realization of Case ABBB. Like before, we may define states H a , H b and H c (equivalent to setting α 1 = α 2 = α 3 = 0) to regain the couplings e 1 = v, e 2 = e 3 = 0, q 1 = vλ 3 , q 2 = q 3 = 0.
(5.51) which we recognize as the couplings of Case BCC with the additional constraint of full mass degeneracy imposed. Thus, we see that Case ABBB is equivalent to Case BCC with the additional constraint of full mass degeneracy imposed. The full mass-degeneracy constraint is unstable under RGE, see section 6.

Spontaneously broken U(1) symmetry
With the potential of (5.40) and a vacuum of the form the mass-squared matrix becomes with M 3 = 0, and the charged mass

No mass degeneracy (RGE stable)
Provided there is no mass degeneracy, the rotation matrix for the neutral sector is simply where α is fixed by the diagonalization of the mass matrix. The resulting couplings are e 1 = v 1 cos α + v 2 sin α, e 2 = −v 1 sin α + v 2 cos α, e 3 = 0, (5.56) and We find the following two identities to be satisfied in this case for i, j = 1, 2: Thus, this is a realization of Case C 0 D. This is the Peccei-Quinn model [16], in which a massless axion appears as a result of spontaneous breaking of the continuous U(1) symmetry. The introduction of a soft breaking term in the potential prevents the masslessness, however we will not discuss soft breaking of symmetries here. Another possibility, however, is to promote the U(1) symmetry to a local gauge symmetry, thus introducing a new gauge boson, Z (see, for instance, [63] and references therein). The massless scalar resulting from the spontaneous U(1) breaking is then responsible for giving Z its mass, and we are left with a scalar sector including a charged scalar and two CP-even scalars, which is contained in the mass spectrum predicted for Case C 0 D. This scalar sector can therefore be of phenomenological interest, even without a soft symmetry breaking parameter.
Partial mass degeneracy M 2 = M 3 = 0 (RGE unstable) Allowing for mass degeneracy 21 M 2 = M 3 = 0, the rotation matrix is again given as in (5.27) where α is fixed by the diagonalization of the mass matrix, and θ is an arbitrary angle. Working out the couplings, we find that e 2 q 3 − e 3 q 2 = 0, and the following two identities are satisfied for i, j, k = 2, 3, 1: for all values of θ. Thus, this is a realization of Case B 0 D. Like before, we may define states H b and H c (equivalent to setting θ = 0) and regain the couplings of (5.56) and (5.57). Thus, we conclude that Case B 0 D is physically equivalent to Case C 0 D with the additional mass degeneracy M 2 = M 3 = 0. The RGE unstable condition leading to the mass degeneracy is in this case given by λ 1 λ 2 = (λ 3 + λ 4 ) 2 . We will not discuss the full mass degeneracy case where all three masses vanish here, since this implies an SO(3) symmetric model which will be treated later.

CP2 symmetry
If the potential is invariant under CP2, there exists a basis in which m 2 12 = 0, m 2 22 = m 2 11 , λ 2 = λ 1 and λ 7 = −λ 6 . In a basis in which the CP2 symmetry is manifest, the potential is given by Davidson and Haber [21] have demonstrated that for this potential one can change basis in order to get a similar potential, but with Im λ 5 = λ 6 = 0. We shall employ this change of basis in order to simplify the analysis. Again, we parametrize the Higgs doublets in the 21 Mass degeneracy between M1 and M3 yields a similar result. most general way, following the steps in Eqs. (5.2) through (5.5). Minimizing the potential with respect to the fields yields the stationary-point equations We will not allow for Re λ 5 = 0, since this will lead to the CP3 symmetry, which we will study later. Allowing for solutions with one vanishing VEV, the stationary-point equations are solved by putting With a vanishing VEV, the phase ξ can now be absorbed into the field Φ 2 .
If the two VEVs are identical, the stationary-point equations are solved by If both v i = 0 and v 2 = v 1 , then the stationary-point equations are solved by This latter option leads to a CP3 conserving model, and will be studied in section 5.5.

(5.69)
We also find This is a realization of Case CCD, the default implementation of the CP2 model.
Partial mass degeneracy M 1 = M 2 (RGE unstable) Allowing for mass degeneracy 22 between M 1 and M 2 requires Re λ 5 = (3λ 1 − λ 3 − λ 4 ). The mass-squared matrix then becomes The rotation matrix is given as in (5.18) where α is an arbitrary angle due to the mass degeneracy. Working out the couplings, we find that e 1 q 2 − e 2 q 1 = 0, and that for any value of α. Therefore this is a realization of Case BCD. Again we may define states H a and H b (equivalent to letting α = 0), with couplings given by that is, we obtain two inert states, one of which is CP-odd. Thus, we interpret Case BCD as physically equivalent to Case CCD with the additional mass degeneracy between one inert and one non-inert neutral scalar. The constraint yielding mass degeneracy given above is RGE unstable, as will be discussed in section 6. We will not discuss mass degeneracy between M 2 and M 3 or full mass degeneracy yet, since this will result in a CP3 symmetric model which we will study later.

CP2 symmetric potential with
Analyzing the solution of the stationary-point equations with v 2 = v 1 again leads to Case CCD which we already encountered, implying that this is simply a description of the same physical model in another basis. We omit the details.
Again, we parametrize the Higgs doublets in the most general way, following the steps in Eqs. (5.2) through (5.5). Minimizing the potential with respect to the fields yields the stationary-point equations We will not allow for λ 1 − λ 3 − λ 4 = 0, since this will be studied in the section on SO ( which leads to an SO(3)-symmetric potential, which will be discussed in section 5.6. The CP3 model of reference [24] had non-zero VEVS v 1 = v 2 , but it included a soft breaking term, thus it is outside of the scope of the present work.

CP3 symmetric potential with v 2 = 0
With the potential of (5.74) and the vacuum the mass-squared matrix becomes 23 and the charged mass 23 CP3 is broken spontaneously in this case, so a massless Goldstone boson appears.

No mass degeneracy (RGE stable)
The mass-squared matrix is diagonal, so the rotation matrix is simply the identity matrix. The couplings are found to be We also find This is a realization of Case C 0 CD.
Partial mass degeneracy M 1 = M 2 = 0 (RGE unstable) Allowing for mass degeneracy between M 1 and M 2 requires λ 1 = 0. The mass-squared matrix then becomes and the charged mass The rotation matrix is given as in (5.18) where α is an arbitrary angle due to the mass degeneracy. Working out the couplings, we find that e 1 q 2 − e 2 q 1 = 0, 2M 2 H ± = e 1 q 1 + e 2 q 2 and q = 0 for any value of α. Therefore, this is a realization of Case B 0 CD.
Like before, we may define states H a and H b such that e a = v, e b = e 3 = 0, q a = vλ 3 , q b = q 3 = 0. (5.87a) Thus, we interpret Case B 0 CD as physically equivalent to Case C 0 CD with the mass degeneracy M 1 = M 2 = 0 imposed in addition. Again, the condition responsible for the mass degeneracy is RGE unstable, as will be seen in section 6.
Partial mass degeneracy M 1 = M 3 (RGE unstable) Allowing for mass degeneracy between M 1 and M 3 yields λ 4 = 2λ 1 − λ 3 . The mass-squared matrix then becomes and the charged mass We write out the most general rotation matrix 24 compatible with the mass degeneracy and work out the couplings to find e 1 q 3 − e 3 q 1 = 0 and 2M 2 H ± = e 1 q 1 + e 3 q 3 − M 2 1 and 1 for any value of α. This is a realization of Case BC 0 D, in which one would have a degenerate pair, one of which is CP odd, together with a massless CP-even scalar.
Defining states H a and H c , equivalent to putting α = 0, we get e a = v, e 2 = e c = 0, q a = vλ 3 , q 2 = q c = 0. (5.90) Thus, we interpret Case BC 0 D as physically equivalent to Case C 0 CD with the mass degeneracy M 1 = M 3 imposed in addition. The condition responsible for the mass degeneracy is however RGE unstable. We will not yet discuss mass degeneracy between M 2 and M 3 or full mass degeneracy, since this will lead to an SO(3) symmetric model which will be treated in section 5.6.
5.5.2 CP3 symmetric potential with v 2 = v 1 and s ξ = 0 Analyzing the solution of the stationary-point equations with v 2 = v 1 again leads to Case C 0 CD which we already encountered, implying that this is simply a description of the same physical model in another basis. We omit the details.

CP3 symmetric potential with
With the potential of (5.74) and the vacuum putting 25 ξ = π/2, the squared mass matrix becomes and the charged mass Partial mass degeneracy M 2 = M 3 (RGE stable) We have two mass degenerate scalars, and the mass matrix is diagonalized by where α is arbitrary due to the mass degeneracy. However, none of the masses or couplings depend on α. We find the couplings This is a realization of Case BCCD. 25 Putting ξ = −π/2 yields a similar result.

Full mass degeneracy (RGE unstable)
Allowing for full mass degeneracy requires λ 4 = 1 2 (λ 1 − 2λ 3 ), The neutral-sector masssquared matrix becomes and the charged mass The rotation matrix for the neutral sector is given by (5.35) with arbitrary α 1 , α 2 and α 3 . Now we find We also find that Q 2 = 0 and for any values of α 1 , α 2 and α 3 . Thus, this is a realization of Case ABBBD. Like before, we may define states H a , H b and H c (equivalent to setting α 1 = π/4 and α 2 = 0) to regain the couplings which we recognize as the couplings of Case BCCD with the additional constraint of full mass degeneracy imposed. Thus, we se that Case ABBBD is Case BCCD with the additional constraint of full mass degeneracy imposed. Once more, the condition responsible for the full mass degeneracy is RGE unstable.

SO(3) symmetry
If the potential is invariant under SO (3), there exists a basis in which m 2 12 = 0, λ 5 = λ 6 = λ 7 = 0, m 2 22 = m 2 11 , λ 2 = λ 1 and λ 4 = λ 1 − λ 3 [13,27]. Writing out the potential in such a basis, it is given by Minimizing the potential with respect to the fields yields the stationary-point equations solved whenever m 2 11 = v 2 λ 1 . The parameters of the SO(3)-invariant potential are insensitive to basis changes, and therefore we choose to do the analysis by working in the Higgs basis. With the potential of (5.102) and the vacuum The vacuum breaks U(2) so that there are 2 Goldstone bosons.
Partial mass degeneracy M 2 = M 3 = 0 (RGE stable) Again, there is arbitrariness in the rotation matrix 26 due to the two mass degenerate states. The couplings are found to be We also find This is a realization of Case B 0 C 0 C 0 D.

Full mass degeneracy (RGE unstable)
Allowing for full mass degeneracy requires λ 1 = 0. The neutral-sector mass-squared matrix becomes the three-by-three zero matrix, and the charged mass The rotation matrix for the neutral sector is given by (5.35) with arbitrary α 1 , α 2 and α 3 . Now we find We also find that Q 2 = 0, and In the previous section we found several cases of 2HDM symmetries which required specific conditions imposed on physical parameters specifying the potential. Whenever conditions defining the cases are satisfied, the scalar sector (i.e., the scalar potential) is invariant under a given symmetry, e.g. CP1, Z 2 , etc. For each symmetry the corresponding list of cases is complete in the sense that there exists no other case compatible with invariance of the potential under the considered symmetry. That suggests a natural classification of the cases: • cases which are stable under the RGE, • cases which are unstable under the RGE, where stability is defined by the vanishing of the perturbative beta functions corresponding to all conditions specifying a given case 27 . Importantly, the perturbative expansion can not violate the symmetry 28 , therefore after including radiative corrections the potential is still symmetric and, due to the completeness, we must remain within one of the cases available to the considered symmetry. If the case is preserved by radiative corrections, i.e., the relation between couplings specific to that case is RGE invariant, the case is stable. Otherwise, radiative corrections force us to move to another case, that happens when the beta function of a given condition is non-zero. The lesson is that it may happen (in unstable cases) that conditions crucial for the presence of a symmetry may be violated by radiative corrections, even though, still, the symmetry is preserved. The starting case would be replaced by some other case. This is an unfamiliar consequence of a symmetry. In other words, the cases unstable under loop corrections would correspond to treelevel fine tunings that are violated when radiative corrections are taken into account. The remaining, stable, cases constitute constraints on the parameters of the model which are preserved under renormalization, even when spontaneous (or soft) symmetry breaking is involved.
• If the potential has a Z 2 symmetry, a basis exists for which m 2 12 = λ 6 = λ 7 = 0. Notice how these conditions then imply that is, the m 2 12 = λ 6 = λ 7 = 0 "point" in parameter space is a fixed point for the RGE evolution of the parameters -if the potential obeys those conditions at some scale, it will obey them at any scale 30 .
• With λ 6 = λ 7 = 0, notice that the β-function for λ 5 becomes which possesses a fixed point at λ 5 = 0 -the conclusion is that the condition λ 5 = λ 6 = λ 7 = 0 is also RGE invariant, and indeed we know it corresponds to the quartic coupling conditions of the U(1) Peccei-Quinn model.
Notice how this last example leads to one of the mass degeneracies required for the symmetry conditions in the previous section. In fact, we had identified, in section 5.2.1 a possibility for Z 2 invariance which required two neutral scalars to be degenerate in mass, M 2 = M 3 . Looking at the mass matrix of Eq. (5.13), this then implies λ 5 = 0, which leads us to a potential with a symmetry larger than Z 2 , namely the Peccei-Quinn model, with a continuous U(1) symmetry unbroken by the vacuum. We indeed found it, Case BCC in section 5.3.1. This, then, is an example whereupon mass degeneracy between scalars is preserved to all orders in perturbation theory. One may follow the above procedure for all the parameter conditions presented in Table 5 of reference [13], and verify that all of those conditions are RGE invariant, at least to one-loop order. We have therefore a powerful tool to investigate the conditions we obtained for each case studied in the previous sections, and verify whether they are RGE stable. We provide several examples below where that does not happen: • In section 5.2.1, we analysed Case BC, wherein degeneracy of two neutral scalars implied the following relation among couplings: Invoking the minimisation conditions relating v to the parameters of the potential, Eq. (5.9), we can rewrite this as λ 1 m 2 22 = (−2λ 1 + λ 3 + λ 4 + Re λ 5 ) m 2 11 . (6.5b) At this point, using the β-functions from Eqs. (6.1g) and (6.2c), it is very easy to confirm that β(λ 1 m 2 22 ) = β{(−2λ 1 + λ 3 + λ 4 + Re λ 5 ) m 2 11 } (6.6) which implies that one-loop corrections would destroy the equality (6.5a). Thus, already at the one-loop level, the parameter relation (6.5a) corresponding to Case BC turns out to be merely an unstable tree-level fine-tuning of parameters that does not hold in the perturbative expansion.
• In section 5.3.1 we considered the Case ABBB which requires a condition similar to Eq. (6.5a), but with Re λ 5 = 0. This is also unstable, in analogy to the case above.
• In section 5.4 we studied the CP2 symmetry, which entails, in a given basis, λ 1 = λ 2 and λ 6 = λ 7 = Im λ 5 = 0. Case BCD further required (all couplings real) With the CP2 relations among the quartic couplings, we find (ignoring the gauge contributions for the moment) and it is clear that the RGE running of the two sides of equation (6.7) are different, the relation is not preserved by radiative corrections. Including the gauge contributions would not change this fact.
• As a final example, in the SO(3) symmetry cases discussed in section 5.6, for which λ 5 = λ 6 = λ 7 = 0 and λ 4 = λ 1 − λ 3 , mass degeneracies required λ 1 = 0. But for this symmetry class, the β-function for λ 1 is given by (ignoring the gauge contributions for the moment) β λ 1 = 2λ 2 3 , (6.9) which shows that λ 1 = 0 is not a fixed point in the RGE running. Again, including the gauge contributions would not change this fact. These are some of the examples of RGE instability found in the previous section. We leave the remainder as an exercise for the reader.

Discussion
We have seen that conditions for all six symmetries can be formulated in terms of constraints on physical quantities, masses and couplings. Each symmetry can be satisfied in a number of different ways, referred to as "cases". The recent results of Bento et al [30] translate into exactly the same set of "cases" as presented here, and are thus in full agreement with ours. Some of the cases encountered, involving mass degeneracies, are unstable under radiative corrections. This result is to some extent surprising, as it turns out that sometimes, i.e. for some "cases", even though the Lagrangian is symmetric under a given transformation, conditions that guarantee the invariance are not stable with respect to RGE. The following cases remain stable at the one-loop level: • CP1 [two constraints] Case C: e k = q k = 0, Case C realizes an unbroken CP1 symmetry, whereas case D realizes a spontaneously broken CP1 symmetry.

• Z 2 [four constraints]
Case CC: e j = q j = e k = q k = 0, Case CC realizes an unbroken Z 2 symmetry, whereas case CD realizes a spontaneously broken Z 2 symmetry.  Figure 7. Migration of unstable cases under radiative corrections. Each of the three small circles labeled i, j and k, represents a neutral scalar, with symbols as defined in section 4. The cases ABBB and ABBBD, which both occur for two symmetries, are shown only under the higher symmetry. Cases enclosed by a larger circle have a constraint on M H ± and the quartic coupling q, whereas those without the "overline" are RGE stable at the one-loop order. Symbols in white refer to massless states.
The different cases listed above all require a certain number of constraints. These numbers are collected in For example, a CP2-symmetric model has five independent physical parameters: e 1 = ±v, q 1 , M 1 , M 2 and M 3 . The rest are either zero or given in terms of these: e 2 = e 3 = 0, q 2 = q 3 = 0, M 2 H ± = 1 2 (e 1 q 1 − M 2 1 ) and q = M 2 1 /(2v 2 ). As another example, an SO(3)-symmetric model has three independent physical pa-rameters: e 1 = ±v, q 1 and M 1 . All other parameters of the model are either zero or given in terms of these three.
However, this does not mean that a model given in terms of any three parameters has SO(3) symmetry. It could also be a model with lower symmetry, but with parameters set to zero or related in a way which is unstable under radiative corrections.
We have performed a complete translation of conditions for 2HDM symmetries in terms of physical, measurable, basis-invariant parameters. Our parameter set P, as detailed in eq. (2.23), includes the four scalar masses, the couplings of the three neutral scalars to electroweak gauge bosons and to charged scalar pairs, and the quartic vertex interaction among four charged fields, in a total of 11 physical parameters. All 2HDM symmetries leave less than 11 free parameters, thus specific values for elements of P, or relations between them, were found for each symmetry class. We followed closely the work of ref. [27], wherein conditions for each of the symmetries were found in terms of a bilinear formalism, which we then translated in terms of physical parameters. We found that several of the symmetry conditions of [27] implied mass degeneracies among two or more scalars, which were then found to be unstable under renormalization. Those specific symmetry cases corresponded to zero-measure regions of parameter space: tree-level fine tunings corresponding to a very specific relation among couplings, not preserved by radiative corrections. The "migration map" just discussed shows how radiative corrections lift mass degeneracies and lead those RGE unstable cases to symmetry cases stable under renormalization.
The remarkable benefit of expressing each 2HDM symmetry in terms of physical parameters is the possibility of giving, in a simple and physically intuitive way, a description of each possible model. For instance, at its simplest, the IDM, a 2HDM with a Z 2 symmetry left intact by spontaneous symmetry breaking, is described in the simplest of fashions: a neutral scalar with SM-like couplings to gauge bosons, two neutral scalars with no couplings to either electroweak gauge bosons nor to charged scalars. Other, more elaborate conditions between physical parameters are found, and those arising from Case D (see section 4.1 for the general conditions) are particularly interesting. (See also Ref. [29].) Consider in fact the conditions obtained for the model with a spontaneously broken Z 2 symmetry, eqs. (5.24) to (5.26): the first specifies that one of the neutral states (H 3 ) does not couple to electroweak gauge bosons or charged scalars -ergo, it is the pseudoscalar A (with mass M A ). The formalism we used throughout this paper was necessary for a full discussion of many different symmetry cases, but let us now use the "usual" 2HDM notation and define tan β = v 2 /v 1 , let us assume H 1 is the lightest CP-even scalar, observed at the LHC (with mass M h ), and therefore H 2 will be the heavier CP-even particle (with mass M H ). In order to obtain the "usual" gauge couplings, we put H 1 = h, H 2 = −H, H 3 = A and introduce α =α + π/2 to get whereα is the "usual" rotation angle of the CP-conserving 2HDM. Current LHC measurements indicate sin(β −α) 1, also known as the alignment limit. Much more fascinating measurable quantities, which can be put to experimental test and easily communicated. To say that a potential has a spontaneously broken Z 2 symmetry, one might state: in the basis where all couplings of the potential are real and m 2 12 = λ 6 = λ 7 = 0 both doublets acquire non-zero vevs. But the physics of that case is much more interestingly described as: a pseudoscalar exists, and two CP-even scalars have couplings such that their relation to the charged mass is given by equation (7.2). With the current work, such elegant and concise statements become possible, and 2HDM symmetries are brought to light with newfound clarity.
In this work we only studied exact symmetries at the potential level, not considering the possibility of soft breaking terms. Those certainly yield interesting phenomenologies, and enlarge the range of 2HDM possibilities. For the Z 2 symmetry, for instance, the inclusion of a soft breaking term can make a CP-breaking vacuum possible, or an explicitly CPbreaking potential to start with (the so-called Complex 2HDM). The soft breaking terms will obviously change the relation between the physical parameter set P and the parameters of the potential, though it will not enlarge the number of physical parameters necessary to fully describe 2HDM symmetries. We will study softly broken 2HDM symmetries in a forthcoming work. It is worth noting here, that the zero vector is by definition not an eigenvector of a matrix. We will encounter physical configurations where the above expression for e m reduces to 0. For those configurations we will need to work out the eigenvectors anew. In particular, this is necessary for Configurations 1-3 listed in section 3.5.

A.1 Configuration 1
The three eigenvalues are in this physical configuration (see section 3.5) given as (A.14) 31 In case ∆0 = 0, we may need to choose a different sign in front of the square root, i.e. if our expression yields C = 0, then we need to replace our expression for C with C =  ± [3M 2 1 − 4M 2 H ± + 2(e 1 q 1 + e 2 q 2 + e 3 q 3 ) − 2v 2 q] 2 + 16Q 2 .
For the eigenvalue Λ 1 , our general expression (A.13) for the eigenvectors reduces to 0, so for this eigenvalue we need to calculate the associated eigenvector anew. For the two remaining eigenvalues, we can use the general expression, and after some simplifications we arrive at provided Q 2 = 0, in which case they all reduce to 0. Thus, the situation when Q 2 = 0 needs separate treatment. The eigenvalues are then given by Thus, it is convenient to divide Configuration 1 into two different sub-configurations, and refer to these sub-configurations when discussing the different symmetries.

A.2 Configuration 2
The three eigenvalues of the E-matrix are in this physical configuration given as subject to the constraint e j q i − e i q j = 0. The eigenvalue Λ 1 inserted into our general expression (A.13) for the eigenvectors make e 1 reduces to 0, so for this eigenvalue we need to calculate the associated eigenvector anew. We find provided v 2 (e i q k − e k q i ) + e i e k (M 2 i − M 2 k ) 2 + v 2 (e j q k − e k q j ) + e j e k (M 2 i − M 2 k ) 2 = 0 and e 2 i + e 2 j = 0. If v 2 (e i q k − e k q i ) + e i e k (M 2 i − M 2 k ) = 0 and v 2 (e j q k − e k q j ) + e j e k (M 2 i − M 2 k ) = 0, then e 2 or e 3 reduces to 0, and that eigenvector must be calculated anew. The eigenvalues simplify to (A.28) Λ 3 = 2v 4 q + (e 2 i + e 2 j )M 2 i + e 2 k M 2 k − 2v 2 (e i q i + e j q j + e k q k ) 8v 4 , (A. 29) and the eigenvectors are then given as provided e 2 i + e 2 j = 0. Whenever e i = e j = 0, all eigenvectors of (A.25)-(A.26) reduce to 0 and must be calculated anew. In this situation the eigenvectors are given by provided q 2 i + q 2 j = 0. Finally, if e i = e j = q i = q j = 0, also these eigenvectors reduce to 0, and must be calculated anew. The eigenvalues are then given by Then, it will be convenient to divide all these different situations into four different subconfigurations, and refer to these sub-configurations when discussing the different symmetries: • Configuration 2α: M i = M j , e j q i − e i q j = 0, e 2 i + e 2 j = 0, v 2 (e i q k − e k q i ) + e i e k (M 2 i − M 2 k ) 2 + v 2 (e j q k − e k q j ) + e j e k (M 2 i − M 2 k ) 2 = 0.
• Configuration 2β: M i = M j , e j q i − e i q j = 0, e 2 i + e 2 j = 0, v 2 (e i q k − e k q i ) + e i e k (M 2 i − M 2 k ) = 0, v 2 (e j q k − e k q j ) + e j e k (M 2 i − M 2 k ) = 0. • Configuration 2 : M i = M j , e i = e j = 0, q 2 i + q 2 j = 0. • Configuration 2ζ: M i = M j , e i = e j = q i = q j = 0.

A.3 Configuration 3
The three eigenvalues are in this physical configuration given as For Configuration 3, the eigenvalue Λ 1 inserted into our general expression (A.13) for the eigenvectors make e 1 reduce to 0, so for this eigenvalue we need to calculate the associated eigenvector anew. For the two remaining eigenvalues, we can use the general expression, and after some simplifications we arrive at provided v 2 (e i q j − e j q i ) + e i e j (M 2 i − M 2 j ) = 0. Whenever v 2 (e i q j − e j q i ) + e i e j (M 2 i − M 2 j ) = 0, then e 2 or e 3 reduces to 0, and that eigenvector must be calculated anew. The eigenvalues simplify to , (A. 46) and the eigenvectors are then given as Then, it will be convenient to divide these different situations into two different subconfigurations, and refer to these sub-configurations when discussing the different symmetries.
• Configuration 3α: e k = q k = 0, v 2 (e i q j − e j q i ) + e i e j (M 2 i − M 2 j ) = 0. • Configuration 3β: e k = q k = 0, v 2 (e i q j − e j q i ) + e i e j (M 2 i − M 2 j ) = 0.