Discriminating between $CP$ and family transformations in the bilinear space of NHDM

The scalar potential of the N-Higgs-doublet model (NHDM) is best analyzed not in the space of N complex doublets $\phi_a$ but in the $N^2$-dimensional space of real-valued bilinears constructed of $\phi_a^\dagger \phi_b$. In particular, many insights have been gained into CP violation in the 2HDM and 3HDM by studying how generalized CP transformations (GCPs) act in this bilinear space. These insights relied on the fact that GCPs, which involved an odd number of mirror reflection, could be clearly distinguished from Higgs family transformations by the sign of the determinant of the transformation matrix. It was recently pointed out that this criterion fails starting from 4HDM, where the reflection/rotation dichotomy does not exist anymore. In this paper, we restore intuition by finding a different quantity which faithfully discriminates between GCPs and Higgs family transformations in the bilinear space for any number of Higgs doublets. We also establish the necessary and sufficient conditions for an orthogonal transformation in the bilinear space to represent a viable transformation back in the space of N doublets, which is helpful if one prefers to build an NHDM directly in the bilinear space.


I. INTRODUCTION
N -Higgs-doublet models (NHDM) is a popular conservative framework of building models beyond the Standard Model (bSM). Driven by the idea that the Higgs fields can also come in generations, just like fundamental fermions, and by the numerous opportunities to impose new global symmetries, such models allow one to resolve or alleviate many of the problems faced by the SM. Sparked by the original proposals of the 2HDM by T. D. Lee in 1973 [1] and the 3HDM by S. Weinberg in 1976 [2], the field has evolved into a vibrant bSM playground, see for example book [3] and recent reviews [4,5].
The rich phenomenology of NHDMs comes at the price of having to face technical challenges. The scalar potential of the NHDM involves very many free parameters, which is further complicated by the notorious issue of basis dependence. Namely, when building models with several fields having identical quantum numbers such as the NHDM, one is allowed to perform an arbitrary unitary transformation mixing these fields without changing the physical content of the model. Although the Lagrangian and intermediate calculations may look vastly different in different bases, the phenomenological consequences must be the same.
Therefore, in order to efficiently explore all the phenomenological situations offered by N Higgs doublets, one must be able to tell which potentials are linked by a mere basis change and which represent truly distinct models. In short, one needs tools which allow one to explore the NHDMs in a basis independent way.
A particularly intriguing topic in the NHDM physics is the issue of CP violation coming from the scalar sector [3]. A tricky issue here is that the CP transformation itself can be defined in a variety of ways. The standard definition in the space of complex scalar fields φ a , a = 1, . . . , N , is φ a ( r, t) → φ * a (− r, t). However, this definition is basis dependent: the same physical consequences will follow from imposing provided the CP transformation is of order 2 (denoted CP2), that is, applying it twice produces the identity transformation. This extra condition restricts the matrix X by the relation XX * = 1 N . However, one can also impose (1) without requiring it to be of order 2 [6][7][8]. If, as a result of imposing this higher-order CP symmetry, one does not acquire an accidental CP2, one obtains a new CP -conserving model which is truly different from the one based on CP2. Examples of NHDMs have been constructed based on the CP symmetry of order 4 (CP4) [9] and of higher orders [10]. The vast experience gained in the 2HDM shows that it is much more convenient to discuss its CP properties not in the space of Higgs doublets φ a , a = 1, 2, but in the 1 + 3-dimensional space of their gauge-invariant bilinear combinations φ † a φ b grouped into the following real-valued variables: where σ i are the familiar Pauli matrices. The map (2) from doublets φ a to the (r 0 , r i ) covers not the entire 1 + 3-dimensional space but only the forward cone defined by the inequalities A pure family transformation φ a → φ ′ a = U ab φ b with U ∈ U (2) leaves r 0 invariant and induces an SO(3) rotation of the vector r i : The standard CP transformation φ a → φ * a induces the following planar reflection: A GCP transformation (1) induces a rotary reflection, which can always be presented as a product of a pure rotation and the standard CP transformation: C = RC (s) . Clearly, Since the map SU The geometric picture emerging from this dichotomy was discussed at length in [11][12][13][14]. It gave a more direct proof of the basis-invariant conditions for the explicit CP conservation in the 2HDM scalar potential which had been previously established in [15][16][17] with the aid of extensive computer algebra scans. 1 A detailed analysis of various paths leading to explicitly CP conserving 2HDMs was presented in [13,14].
In 3HDM, the necessary and sufficient basis-invariant conditions for existence of a CP2 symmetry were first derived in [12] and recently put in a wider context in [19,20]. The meticulously elaborate analysis of the order-2 condition of the very recent study [21] reconfirmed that CP2 can always be brought by a basis change to the standard form. In addition, three Higgs doublets allow one to implement a novel type of CP symmetry of order 4 [9,22], and a basis-independent algorithm for detection of CP4 in 3HDM was given in [23].
All these results were obtained only with the bilinear space formalism, stressing its superior role in analyzing the scalar sector of NHDMs. Despite its complexity in the 3HDM as compared to the 2HDM, one still observes the discriminating role of the determinant: all Higgs family transformations are described in the 1 + 8-dimensional bilinear space by pure rotations R ∈ SO(8) with det R = +1, while all GCPs induce transformations C, which involve an odd number of reflections, so that det C = −1.
However when moving to 4HDMs, a peculiarity was recently pointed out in [21]. It turns out that in this case, both the family transformations and the GCPs induce transformations O in the bilinear space with det O = +1. Thus, the intuition linking Higgs family transformations with pure rotations and GCPs with the orientation-flipping reflections fails starting from 4HDM.
In this paper, we resolve this disturbing loss of intuition. Recalling that the bilinear space inherits the SU (N ) invariant tensors, we replace the determinant with a different indicator which serves as a faithful discriminator between GCPs and family transformations in the bilinear space for any number of Higgs doublets. As a byproduct, we establish the necessary and sufficient conditions on an orthogonal transformation O in the bilinear space of NHDM to represent a viable transformation back in the space of N doublets, either a basis change or a GCP. These conditions are especially helpful if one prefers to build an NHDM scalar sector directly in the bilinear space.

A. The problem
Let us first remind the reader how the bilinear formalism is extended to 3HDM [12,24,25]. We define 1 + 8 gauge-invariant bilinear combinations (r 0 , r i ): If λ i are the standard Gell-Mann matrices, then (3) structure constants f ijk and the fully symmetric SU (3) invariant tensor d ijk . These bilinears satisfy not only (3) but also an additional constraint [24]: (3) induces an SO(8) rotation R of the vector r i , with det R = +1. With the usual notation for the Gell-Mann matrices, the standard CP transformation acts on r i by flipping components 2, 5, 7: so that det C (s) = −1 still holds. By the same logic as before, any GCP transformation (1) can be written as a product of the standard CP and a rotation, which again leads to det C = −1. The geometric interpretation of GCP transformations becomes more involved [12,19,21], but the determinant still offers a faithful distinction between GCP transformations and pure basis changes. The construction can be repeated for any number of Higgs doublets N . The N 2 − 1-dimensional real-valued vector r i is constructed in a similar way based on traceless hermitean matrices λ i , [12,19,21]. Out of them, there are N − 1 diagonal matrices and N (N − 1)/2 pairs of the off-diagonal ones, one real symmetric and one imaginary antisymmetric in each pair, for an explicit numbering scheme see, for example, [21]. The components of r i do not fill the entire space R N 2 −1 but lie in an algebraic manifold called the orbit space and defined, in addition to Eqs. (3) and (8), by a series of algebraic equalities of orders up to N with coefficients constructed from SU (N )-invariant tensors [24].
All Higgs family transformations and GCPs of the NHDM are represented by those orthogonal transformations from O(N 2 − 1) which leave this orbit space invariant. The standard CP transformation φ a → φ * a acts on the components of r i by ±1 factors, with a −1 factor corresponding to each imaginary antisymmetric generator λ i which produces r i = Imφ † a φ b for an suitable pair of doublets. As a result, the determinant of the standard CP transformation depends on the number n a of such antisymmetric generators [12]: This determinant is −1 for N = 2, 3 but changes to +1 for N = 4, 5. This pattern "two −1's followed by two +1's" repeats itself as N grows further [21]. Clearly, the same applies to all GCPs.
We are now ready to formulate the problem. Suppose we work in the bilinear space of NHDM and we are given an orthogonal transformation O ∈ O(N 2 − 1) that leaves the orbit space invariant. Is there a simple way to tell whether it corresponds to a GCP or a Higgs family transformation? In 2HDM and 3HDM, we developed the geometric intuition: all family transformations have det O = +1, all GCPs have det O = −1. Now we see that this geometric intuition is lost already for 4HDM, where all allowed transformations of Higgs doublets are represented by orientation-preserving SO(15) rotations in the bilinear space. What is then the true faithful discriminator between GCPs and family transformations within the bilinear space?

B. A resolution
Let us first mention that working with traces of (powers of) matrices instead of the determinant does not help much. Although TrR is a basis-invariant quantity, traces of different basis changes are also different: TrR 1 = TrR 2 . The same applies to traces of GCP transformations. Moreover, since TrAB = TrA · TrB, we cannot factor the trace of GCP into the traces of a rotation and the standard CP . Finally, it may happen that TrR = TrC, and one would need to compute traces of their higher powers to distinguish the two. In short, traces give too much detail of the transformation and are not suitable for a quick distinction between GCPs and basis-change. 2 Now, let us recall that the bilinear space is not the structureless R N 2 −1 but inherits the SU (N ) structure. Therefore, if we are given an orthogonal transformation matrix O which leaves the orbit space invariant, not only are we allowed to evaluate traces or determinants but we can also contract it with SU (N ) invariant tensors f ijk or d ijk . For example, CP odd invariants are easily constructed with the use of f ijk [12]. We propose the following invariant quantity as a discriminator between family transformations and GCPs: If O = R is a pure family transformation, then, due to the property of f ijk being invariant under SU (N ) basis The last equality comes from the fact that the tensor f ijk is non-zero only when its indices i, j, k involve an odd number of sign-flipping components. This can be seen, for example, directly from the definition of the structure constants f ijk , which link three generators through the imaginary unit. Therefore, each term in the implicit summation of (13) comes with C (s) These results hold for any N . Thus, J defined in (11) serves as a faithful discriminator between GCPs and family transformations.
It is interesting to note that for N = 2, the 3D orbit space recovers the full spherical symmetry. As a result, all O(3) transformations in the r i space can be induced either by a basis change or a GCP transformation. On the other hand, the SU (2) structure constants are f ijk = ǫ ijk , so that the invariant J coincides with the well-known definition of the determinant. This is why the intuition based on rotation/reflection worked in the 2HDM.

C. Which transformations can be used in the bilinear space?
Suppose one wants to build an NHDM and to implement a symmetry directly in the bilinear space. Which orthogonal transformations O can one use? The quantity J in Eq. (11) already gives a strong constraint: only those O which produce J = ±1 may be used as symmetries of the model. If one wants to implement an O with det O = +1 but |J| < 1, then it cannot be induced by any Higgs family or GCP symmetry in the space of doublets. Thus, |J| = 1 and | det O| = 1 represent a necessary condition for the transformation O to be a valid choice of a symmetry of the model. But is it sufficient?
The answer is no, and it can be illustrated with the following 5HDM example. Consider the C (s) transformation in the 5HDM. In the bilinear space, which is 24-dimensional, it flips 10 directions, thus det C (s) = +1 but J = −1. Now notice that the orthogonal transformation O = −1 24 satisfies the same conditions: det O = +1 and J = −1. However, O = −1 24 cannot be induced by any basis change or GCP back in the Higgs doublet space. The simplest way to see it is to observe that the overall sign flip r i → −r i does not leave (8) invariant, hence it does not preserve the orbit space. In a similar way, one can consider the orthogonal transformation O = −C (s) in 5HDM, which again cannot be induced by any basis change or GCP but which satisfies det O = +1 and J = +1, the conditions one normally expects from regular basis changes.
We prove in the appendix that it is sufficient to complement the |J| = 1 and | det O| = 1 conditions with the following extra requirement: Thus, if one wants to implement an orthogonal transformation O in the NHDM bilinear space, one has only the following two options: These criteria exclude, for instance, the case of O = −1 mentioned above.

III. CONCLUSIONS
In order to exhaustively explore all phenomenological options offered by N Higgs doublets, one must perform an efficient scan in the immense parameter space, distinguishing truly different models from mere basis changes.
The 2HDM and 3HDM experience shows that it can be efficiently done in the bilinear space. In particular, it provides an intuitive geometric picture that generalized CP transformations induce an odd number of mirror reflections while the family symmetries always produce pure rotations. The two types of transformations can be immediately distinguished by computing the determinant and checking whether it is −1 or +1.
It was recently pointed out that this intuition fails already for 4HDM, where all allowed transformations are represented in the bilinear space with orthogonal matrices O such that det O = +1. We cure this disturbing loss of intuition by replacing the determinant with the quantity J given in Eq. (11) and proving that J = +1 for Higgs family symmetries and J = −1 for all GCPs. Also, to remove the invalid transformations which can also lead to the same values of J, we also require that another quantity J d defined in (14) is equal to 1. This opens up the possibility to imposes symmetries on NHDM scalar sector directly in the bilinear space: only those orthogonal transformations O which lead to J d = 1 and J = +1 (basis changes) or J = −1 (GCPs) can be used. These results hold for any number of the Higgs doublets N .
By construction, F 2 = 1 due to (12) and F, G = J in (11). Since O T O = 1, it also follows that G 2 = F 2 = 1. We have seen in the main text that if O = R is generated by a basis change in the space of doublets, that is, O is a member of adj(SU (N )) (adjoint representation of SU (N ) in the R N 2 −1 space), then Similarly, we have shown that if O = R · C (s) corresponds to a GCP transformation on the doublets, then Now we want to prove the converse: if J = F, G = 1, then O ∈ adj(SU (N )) and therefore it can be represented as a basis change in the space of doublets. Analogously, if J = −1, then O corresponds to a GCP transformation. As we explained in the main text, just requiring J = ±1 is not sufficient, and we want to find an additional condition to be imposed to make the converse statement valid.
First, since For J = −1, the replacement O → −O is understood and from here on O denotes a matrix that satisfies the invariance (A5). The transformations in (4) or (5), which we can generically denote by O, can be understood as a map ψ on the Lie algebra su(N ) of SU (N ): The invariance (A5) implies that and ψ represents an automorphism of su(N ). Now, all automorphisms of su(N ) are classified (see e.g. Ref. [8]). They form a group denoted as Aut(su(N )). The invariant subgroup Inn(su(N )) consists of inner automorphisms induced by conjugation of the group itself as where S ∈ su(N ). In other words, where R ji (S) ∈ adj(SU (N )). Automorphisms that are not inner are outer and for su(N ), N ≥ 3, they can all be characterized from Aut(su(N ))/ Inn(su(N )) ≃ Z 2 .
A nontrivial representative ψ ∆ that generates Z 2 can be chosen exactly as the transformation (9) where R ∈ adj(SU (N )). For both cases J = 1, but only the first option corresponds to basis transformations on φ a . For the second option, it implies that −O, the one which produces J = −1, is a viable option as it corresponds to a GCP. Finally, to resolve the ±1 ambiguity, we can use the transformations properties for d ijk : To show the case for C (s) , just use the definition of d ijk based on t i and of the automorphism ψ ∆ . We then define J d in (14) and require it to be +1 for the valid orthogonal transformations. We also remark that the fact that valid GCPs must leave d ijk invariant as in (A13) was also stated and proved in [21]. The proof was given only for 3HDM and with the aid of computer algebra program by considering special cases for r i . In our procedure, it is required only at the last step, when the structure of O is already established up to the ±1 ambiguity and only the property of the standard CP transformation is necessary.