Nondecoupling in Multi-Higgs doublet models

We consider models with any number of Higgs doublets and study the conditions for decoupling. We show that, under very general circumstances, all the quadratic coefficients of the scalar potential must be present, except in special cases, which include terms related to directions of vanishing vacuum expectation values. We give a few examples. Moreover, we show that the decoupling of all charged scalars implies the decoupling of all extra neutral scalars and vanishing $\mathcal{CP}$ violation in scalar-pseudoscalar mixing.


Introduction
It is has been determined experimentally that there are four electroweak gauge bosons [1,2,3,4], as predicted by the SU(2) L × U(1) Y gauge group of the Standard Model (SM). The SM gauge structure does not predict the number of fermion families, which was established at LEP and SLD by the invisible width of the Z [5]. Neither does the SM structure predict the number of scalar doublet families. This is the most fundamental open question, and it is being actively pursued at LHC.
The Atlas and CMS experiments have already identified a 125 GeV Higgs particle (h 125 ) [6,7], as established in the (minimal N = 1 Higgs version of the) SM, and have checked that its couplings are consistent with those predicted, within errors of order 20% [8]. In N Higgs doublet models (NHDM) there are in general 2N − 1 neutral scalars S 0 α (α = 2, . . . 2N ) and N − 1 charged scalar pairs S ± a (a = 2, . . . N ). 1 States beyond a e-mail: francisco.faro@tecnico.ulisboa.pt b e-mail: jorge.romao@tecnico.ulisboa.pt c e-mail: jpsilva@cftp.ist.utl.pt 1 We keep the first entries S 0 1 = G 0 and S ± 1 = G ± for the would-be Goldstone bosons. the S 0 2 = h 125 scalar have not been found, nor its number limited. The reason is that the generic NHDM has a decoupling limit, where the extra scalars have very high masses and the remaining h 125 has basically the properties of the SM one.
There are, however, NHDMs in which the extra Higgses do not decouple. That is the case, for example, with the 2HDM with an exact Z 2 symmetry [9]. Nevertheless, Gunion and Haber [10] have shown that the decoupling limit is recovered by including in the potential a term of dimension two which breaks softly the Z 2 symmetry. Nondecoupling has also been analyzed by Nebot in the 2HDM with spontaneous CP violation, both with and without soft symmetry breaking terms [11]. One further example with 3HDM and a S 3 symmetry has been discussed by Bhattacharyya and Das [12]. In this article we discuss the situation in the NHDM. If these have an exact symmetry limiting the number of quadratic and quartic couplings, then one can expect that nondecoupling occurs. Conversely, as we show in this article, for the existence of a decoupling limit, all soft breaking terms must be included in the scalar potential, except those related to directions with vanishing vacuum expectation values. We introduce our notation in section 2, where we also present our first results. Section 3 is devoted to theorems valid when there are no vanishing vacuum expectation values (vev) and/or small mixing angles. We present a few 3HDM examples in section 4, both of the theorems and of what happens when the assumptions of the theorems are violated. We discuss briefly CP violation in section 5, and we draw our conclusions in section 6. Some detailed discussions have been relegated to the Appendices.

Notation and first results
Consider a SU(2) L × U(1) Y theory with N complex scalar doublets Φ i with hypercharge Y = 1/2.

The scalar potential
Following the notation in [13,14,15], we write the scalar potential as whose hermiticity implies, Requiring a massless photon implies that the global minimum corresponds to vacuum expectation values (vev) which preserve the charge symmetry, U(1) Q , generated by Q = T 3 + Y . Expanding the fields around those vevs ν i , we write where each v i is in general complex. The stationary conditions are given by Implicit summation of repeated indices is used throughout, except where noted otherwise. The mass matrix for the charged scalars is, in terms of which the stationarity conditions in (4) may be written as The mass matrix for the neutral scalars is, where, under the canonical definition of CP, M 2 ρρ is the mass matrix of the CP-even scalars, M 2 χχ of the CP-odd scalars, and M 2 ρχ = M 2 χρ T gives the mixing between the CP-even and CP-odd scalars. Substituting (5), we obtain

Basis freedom and symmetries
We may choose to describe the theory in terms of new fields Φ ′ i , obtained through a basis transformation which leaves the kinetic terms unchanged where U is an N ×N unitary matrix. Since the theory is invariant under a global U(1), we may take U in SU(N ). Under this basis change, the potential parameters and the vevs are transformed as, This means that not all parameters have physical significance; only basis invariant combinations can be observed experimentally [13]. Still, any model with more than one scalar has many free parameters. Most often, these are curtailed by invoking some specific symmetry also given by a U(N ) matrix, which imposes relations among the potential parameters, Recall that in a basis change the potential parameters do not remain the same, whereas under a symmetry these must remain invariant. 2 The symmetry may (or not) be spontaneously broken, according to whether (or not) Consider a theory in which V H , when written in terms of the fields Φ i , has the symmetry S. Now, perform the basis transformation in eq. (15). When written in terms of the new fields Φ ′ i , V H is no longer invariant under S; rather, it is now invariant under As an example, the Z 2 2HDM symmetry becomes in the basis Eq. (23) is a conjugacy relation within U(N ). In a suitable basis, S may be brought to the form [16] or, using the invariance under global hypercharge, This choice makes the presence of the symmetry in the Higgs potential more transparent. Strictly speaking, we have just described the situation where there is only one generator S. Imagine that there are two generators S 1 and S 2 . Then, it may be that one can bring both generators to diagonal form. In that case, the symmetry generated by both, S = {S 1 , S 2 }, is Abelian and easy to guess from the form of V H when it is written in the basis where both generators are diagonal. If S 1 and S 2 do not commute, then S = {S 1 , S 2 } is a non-Abelian subset of U(N ). 3 One can diagonalize S 1 , or S 2 , but not both. 3 We leave a subtlety for Appendix A.

The symmetry basis and the charged Higgs basis
The basis freedom in eq. (15) may be used to study a given theory in a specific basis. There is always a specially simple basis, the so-called Charged Higgs basis (CH basis), in which the U(N ) basis freedom is used in order to diagonalize the mass matrix of the charged Higgs in eq. (5) [15,17]. In this basis, v 1 = v, v i =1 = 0, and the fields may be parametrized as . . .
where S + 2 , . . . , S + N are the physical charged Higgs mass eigenstate fields, with corresponding masses m 2 ±,i , and S ± 1 = G ± is the would-be Goldstone boson with m 2 ±,1 = 0. This basis is especially adapted to study the decoupling limit. In this limit, all (massive) charged Higgs (S ± i =1 ) should acquire a very large mass. Since perturbativity and unitarity constrains the quartic parameters to lie below some upper bound, which one may take to be of order 4π ∼ O(10), we find Substituting eq. (30) in eqs. (12)- (14), one obtains [15] ( were no sum is implied. This leads to our first important results. First, as the charged Higgs masses become very large, all neutral particles also become very massive. This is easy to understand. As Y CH i i =1 becomes positive and very large, the whole doublet Φ CH i decouples from the rest. Thus, within the framework of effective field theory, M 2 i is the interesting measure of decoupling -see, for example [18]. Second, if all charged Higgs become very massive, then M 2 ρρ and M 2 χχ become very large and almost diagonal, while M 2 ρχ remains of (small) order v 2 . Thus, all CP violation in scalar-pseudoscalar mixing vanishes. This could be viewed as our Theorem 0.
At first sight, this second result may seem puzzling. Consider two very heavy Higgs doublets. Can't there be scalar-pseudoscalar mixing among those two heavy doublets? No! To understand the reason, we notice that eqs. (34)-(36) mean that, in order to have significant CP violation in scalar-pseudoscalar mixing, the charged Higgs masses m 2 ±,i =1 must be of order v 2 = |v 1 | 2 . . . |v N | 2 . Now, the vev of each doublet is bounded by |v j | ≤ v and, thus, it cannot be very large. With vevs of order v, one is left with two options. Charged Higgs masses of order v and possible large scalar-pseudoscalar mixing, or alternatively, charged Higgs masses much larger than v and necessarily small scalar-pseudoscalar mixing. Ref. [15] showed how crucial the CH basis was in interpreting unitarity bounds. The two results presented above provide another striking example of the usefulness of this basis. Indeed, these general results would be very difficult to guess in a generic basis.
For the most general NHDM, the Y CH and Z CH parameters are free to take any value (consistent with perturbativity and unitarity). However, such models suffer from several problems. On the one hand, they have too many free parameters and their study (bounds from current experiments and proposed new signals) is effectively very difficult. On the other hand, such models tend to lead to very large scalar flavour changing neutral couplings (sFCNC) with fermions, which are very tightly bound by flavour experiments. There are three solutions: make the new scalar masses large (precisely the decoupling limit); take the sFCNC small; or, make the sFCNC exactly zero by enforcing some symmetry. We now focus on models with some symmetry S. As seen in eqs. (20)-(21), such a symmetry imposes conditions on the parameters of V H . Eq. (23) implies that the specific constraint depends on which representation is chosen for the symmetry. We consider some specific representation for the symmetry and name the basis where the symmetry has that particular form as the Symmetry basis (S basis). And, because it constrains the available parameter space, such a symmetry will have an impact on the nature of the parameters in the CH basis.
The CH equations (30)-(33) and (37) can be written compactly as where Recall that as M 2 i increases, it drives the decoupling and defines the energy scale of the states within the Φ CH k =1 doublet. The CH basis and the S basis are related by Eqs. (38) and (42) can be combined into This equation shows explicitly how the quadratic parameters in the S basis evolve with the decoupling parameters M 2 k =1 . In order to have a decoupling limit with all M 2 k =1 ≫ v 2 it is certainly sufficient to include all Y S ij . However, most symmetries preclude some of these quadratic coefficients, possibly precluding a decoupling limit. Our aim is to study when this can and when it cannot happen.
Recall that the matrix U CH is the unitary transformation which diagonalizes the mass matrix of the charged scalars in eq. (5), when written in the S basis. The first line of U CH must have the form in eq. (41) because, through eq. (6), that guarantees that the first eigenvector has zero mass, corresponding to G + . Eq. (41) also guarantees that the vev of the first doublet in the CH basis is v, while all other doublets in the CH basis have zero vev. A generic N × N unitary matrix is defined by the N (N − 1)/2 angles in an orthogonal N × N matrix, and by N (N + 1)/2 phases. From the original N (N − 1)/2 angles, (N − 1) are determined by the vevs, which appear the first line of U CH , c.f. eq. (41). We denote such angles by β i and, in our notation, they cannot be multiples of π/2 if we wish to keep all vevs different from zero. There remain (N − 1)(N − 2)/2 angles, which we denote by ω i . In contrast with the β i , these ω i angles depend not only on the vevs, but also on the independent Y S ij and Z S mn,op . We may choose regions of parameter space such that some ω i , and, thus, some entries in the U CH matrix are zero.
Consider a real 3HDM, with real vevs. Then, the unitary transformation from the S basis to the CH basis can be written as, Parametrizing the vevs as one obtains with s 1 ≡ sin β 1 , c 1 ≡ cos β 1 , s 2 ≡ sin β 2 , c 2 ≡ cos β 2 , s ω ≡ sin ω, c ω ≡ cos ω. As mentioned, the angles β 1 and β 2 are determined solely by the vevs, while the angle ω will depend on the quadratic and quartic parameters in the S basis in some complicated fashion.
One final notational issue must be addressed. If there are two doublets with the same group charge, then any basis change among those two doublets is allowed. For example, the 3HDM with the Z 2 symmetry S = diag(1, −1, −1) does not have a well defined symmetry basis, since one can mix at will the last two scalars. We will concentrate on models which do have a well defined S basis, such as the Z 2 × Z 2 3HDM generated by In summary, the S basis is useful to identify the set of independent parameters, while the CH basis is useful to discuss decoupling. Going from the former to the latter involves diagonalizing N ×N (for the charged scalars) and 2N × 2N matrices (for the neutral scalars), and it is basically only manageable for N = 2 or other exceedingly simple cases. We will now show that the converse procedure of starting from quadratic parameters in the CH basis and looking for generic properties of the quadratic parameters in the S basis can lead to results valid for any N .

Decoupling or nondecoupling
Consider a NHDM with some symmetry. It may decouple to the SM in two ways:

NHDM → SM: single scale
Regarding the first possibility, we present our main Theorem 1: Barring some zero vevs (v i = 0 for some i), and/or very small mixing angles in the matrix U CH , then all Y S ij must be present in order for a decoupling limit NHDM → SM with one single scale to exist.
In most cases, a symmetry S forces some Y S ij to vanish. Our claim is that, under the conditions of the theorem, such models will not have a decoupling limit. Conversely, in order to keep the decoupling limit, the way out is to break the symmetry softly by including all the quadratic terms. This applies to any N and any symmetry S.
The proof uses eq. (44). Perturbativity implies that all Z CH ij,kl must be smaller than about 4π. Thus, we conclude from eq. (39) that each Ω CH kl must be smaller that O(10)v 2 . Taking all doublets to share a decoupling scale eq. (44) leads to Thus, The right-hand side (RHS) of eq. (50) must be of order one. Similarly, the RHS of eq. (51) must be of order one, unless some vev is very small.
There are a few caveats. The theorem does not hold in directions with v i = 0; ie, in inert models. The theorem may also cease to hold if there are vevs of order v 2 /M, such that there is a cancellation between the two terms on the RHS of eq. (51). In fact, in such circumstances, it could even happen that some Y S ij is exactly zero. 4 Finally, one could consider decoupling in multiple scales.
Notice that we followed an uncommon strategy. Usually, one discusses the decoupling limit by starting from the restricted set of parameters in the S basis and then finding how to rotate from this basis into a new basis. For example, Gunion and Haber [10] start from the S basis of the Z 2 2HDM, construct the mass matrices in this basis, and then diagonalize them, transforming from this basis into the mass eigenstates directly. Later, the result was revisited by Bernon et. al. [19] by starting again in the symmetry basis and going into the Higgs basis. 5 Again, this requires that one minimizes the potential explicitly and finds the matrix going from the S basis into the Higgs basis, and then from this into the mass basis of the neutral scalars. This is easy for N = 2, but unmanageable for larger N . Here we follow the opposite strategy. We write the S basis in term of the CH basis, where decoupling is very easy to describe.
For our theorem, it was sufficient to use the approximate results in eqs. (49)-(51). To compare with exact results in the literature, incorporating in the quartic couplings the constraints from the symmetry S, one can apply the following Strategy: find the constraints that the symmetry imposes on the quartic couplings in the S basis, Z S mn,op ; -write the quartic parameters in the CH basis, Z CH ij,kl , using eq. (43). This guarantees that the quartic parameters in the CH basis already encode the constraints that the symmetry places on the quartic couplings; -use eq. (44) to see how the quadratic parameters in the S basis evolve with the decoupling parameters M 2 i . Taking as an example the softly broken CP conserving Z 2 2HDM, we find 4 See eq. (78) below. 5 There is no distinction between the Higgs basis and the CH basis when N = 2. See ref. [15] for details.

NHDM → SM: multiple scales
We now consider the possibility that there are multiple Y CH kk ≡ M 2 k ≫ v 2 (k = 1) scales in the decoupling NHDM → SM. For definiteness, one could imagine that M 3 ≫ M 2 ≫ v. We assume that the two scales are distinct, but also independent. For example, we exclude the possibility that M 3 = 100 M 2 , both growing coordinately towards very high values. We find the following Theorem 2: Barring some zero vevs (v i = 0 for some i), and/or very small mixing angles in the matrix U CH , in order for a multiple scale NHDM → SM decoupling to exist: a) all Y S ii must be present and large; b) moreover, if there are no judicious cancellations, then all Y S i j =i must be present and with large magnitudes.
The proof of a) is very similar to that in the previous section. Taking eq. (44) leads to For a fixed a, Y S aa can only be of order v 2 (or vanish) if U CH ka = 0 for all values of k = 2. But this is impossible. Indeed, if it were true, the unitarity condition would force |v 1 | = v and all v k =1 = 0, contradicting our hypothesis. Of course, the result a) is not very useful in cases in which all generators are simultaneously diagonalizable and chosen to be represented as in eq. (27). Indeed, in such cases, all Y S aa are allowed by the symmetry from the start.
To prove b) we start from We assume that there is no judicious cancellation among large terms, such as Such a cancellation could occur because one is looking at a particularly fine-tunned region of parameters. But it could also occur naturally due to the symmetry. Recall that, given some non-Abelian symmetry, one can diagonalize one generator, but not another. The latter will necessarily impose relations between parameters corresponding to different doublets and, under those circumstances, an equation such as (58) cannot be excluded a priori. The theorem b) applies when there is no such cancellation. In addition, we are also excluding situations in which there are small entries U CH * ka U CH kb , such that the RHS of eq. (57) has some term in M 2 k exactly canceled by some term of order v 2 . Excluding such judicious cancellations, for given a and b = a, Y S ab in eq. (57) can only vanish if all combinations U CH * ka U CH kb = 0 for all values of k = 2. But this is impossible. Indeed, if it were true, the unitarity condition would force some v i =1 = 0, contradicting our hypothesis.

Decoupling of one doublet
It is interesting that a strong result is possible, even under the simple situation in which only one doublet decouples. Theorem 3: Barring some zero vevs (v i = 0 for some i), for every doublet, Φ CH k =1 , that decouples from the low energy theory, at least three quadratic parameter in the Symmetry basis, Y S ab =a Y S aa and Y S bb , will depend on M 2 k =1 . The proof uses eq. (44), eq. (41), and the fact that the matrix U CH is unitary. The latter implies that the sum of the squares of the entries in a given row (column) must be unity. Imagine that for a given k = 1, M 2 k =1 ≫ v 2 , and consider the k-th row of U CH . Since the sums of squares along the row must add to one, there must be at least a column a such that |U ka | = 0. However, we know that also |U ka | = 1; otherwise, considering now the sum of all squares in column a, we would find |U 1a | = 0. That, according to eq. (41), would imply v a = 0, violating our hypothesis. So, there must be at least one other column b such that |U kb | = 0. But, looking back at eq. (44) we see that, as a result, Y S ab =a , Y S aa , and Y S bb grow with M 2 k =1 .
It may help to visualize the argument made here to look back at eq. (47). Imagine that we are taking M 2 2 ≫ v 2 . One can make U CH 23 = 0 by setting ω = 0; the matrix simplifies into Then, according to eq. (44), none of Y S i3 and Y S 3i depend on M 2 2 . Nevertheless, both U CH 21 and U CH 22 must be nonvanishing, or there would be one zero vev. As a result, even taking ω = 0, Y S 12 , Y S 11 , and Y S 22 will grow with M 2 2 .

Simple examples
To illustrate both the application of our theorems (decoupling) and some relevant violations of the assumptions (thus, nondecoupling or decoupling using fine tuned regions with vevs of order v 2 /M), we concentrate on 3HDM models with Abelian symmetries. These have been classified in [16] and [20]. The symmetries U(1) × U(1), U(1) × Z 2 , Z 2 × Z 2 , Z 3 , and Z 4 , yield a well defined symmetry basis. The corresponding potentials are shown in Appendix B. All these symmetries allow for quadratic diagonal coefficients Y S ii and preclude quadratic off-diagonal coefficients Y S ij =i . When off-diagonal quadratic couplings Y S ij =i are needed for decoupling, then one must either give up decoupling or else break the symmetry softly.
This model was suggested by Weinberg [21] and explored for spontaneous CP violation with three quark families by Branco [22]. The potential is written in eqs. (B.8) and (B.10). We now follow the strategy outlined at the end of section 3.1.
Using eqs. (38), (39), and (43), we find Substituting in eq. (42), we get the leading order terms We start by noticing that, as proved in section 3.3, Y S 12 , Y S 11 , and Y S 22 grow with M 2 2 . Let us now assume, as in section 3.2, that there are two very different scales M 3 ≫ M 2 ≫ v in the decoupling 3HDM → SM. Then, all Y S ij must be present. However, as we see in eq. (B.8), the Z 2 × Z 2 symmetry precludes the Y S ij =i terms. Thus, one either keeps the symmetry and looses the decoupling limit or, else, one must break the symmetry softly with all possible terms. It is true that one could (for example) make Y S 13 in eq. (69) parametrically small by choosing fine tuned regions with β 2 ∼ v 2 /M 2 3 . Small but not zero. Again, for a 3HDM → SM with several scales one must include all Y S ij =i terms to have a decoupling limit. Notice that this holds despite the fact that in this particular case we are even allowing small vevs. Now, we notice that the leading term for Y S 12 in eq. (68) is proportional to M 2 3 − M 2 2 . So, we consider the situation where there is only one scale M = M 2 3 = M 2 2 ≫ v 2 in the Z 2 × Z 2 3HDM → SM decoupling, as discussed in section 3.1. One finds and, taking Y S 12 to higher order in β's, Now it is possible to make Y S 12 exactly zero, by choosing fine tuned regions of λ 13 − λ 23 , M (very large) and β 2 (correspondingly very small), such that the RHS of eq. (78) vanishes. Notice that, according to eq. (46), β 2 → 0 implies v 1 → 0 and v 2 → 0, in contradiction with the conditions of Theorem 1. For the general NHDM, we do not know how many Y S ij =i need to be included in order to attain this parametric decoupling with M k =1 = M ≫ v. Nevertheless, we suspect that it may be possible to exclude Y S ab =a when both v a → 0 and v b → 0; granted, using a very fined tuned choice of parameters.
Although simplified by taking ω = 0 and small β 1 and β 2 , eqs. ≫ v 2 we see that all quadratic couplings must be present in the S basis. Moreover, taking only M 2 2 ≫ v 2 we see that some quadratic couplings may be absent, but that there are at least three which must be present. 6 This illustrates Theorem 3.
This section also illustrates the caveats in the theorems. For example, by taking small angles ω = 0 and very fine tuned regions of parameter space where the vevs decrease with the decoupling scale (eg, β 2 ∼ v/M), eq. (78) shows that we can set Y S 12 = 0 in the 3HDM → SM decoupling with one single scale. Eqs. (67)-(72) show that that is not possible with multiple independent scales M 2 3 ≫ M 2 2 ≫ v 2 . 7 In all cases illustrated here, the Z 2 × Z 2 3HDM can have decoupling if and only if there is at least some off-diagonal term breaking the symmetry softly. It is easy to see that, indeed, this is a general feature of the Z 2 × Z 2 3HDM by writing the stationarity equations in the symmetry basis: We see that, in the exact Z 2 ×Z 2 3HDM (that is, with no Y S ij =i terms), all diagonal quadratic couplings Y S ii = m 2 ii are or order v 2 , and thus there is no decoupling limit. This is not, however, a general feature of NHDM, as we will see next.

The Z 3 3HDM
As we've just seen, the stationarity conditions of the exact Z 2 × Z 2 3HDM in the symmetry basis are enough to see that one can only make the diagonal Y S ii = m 2 ii large by including some off diagonal Y S ij =i term. This is no longer the case for the Z 3 3HDM, whose potential is written in eqs. (B.8) and (B.11). Indeed, the stationar-7 Notice that, even with multiple scales, we could parametrically make Y S 33 = 0 in eq. (72) by a judicious cancellation involving M 3 , β 2 , and λ 33 . However, this is of no importance since the symmetry allows for this coupling anyway.
ity conditions for the real Z 3 3HDM, contain ratios of vevs. To be specific, take the expression for m 2 22 . Now one can have m 2 22 ≫ v 2 by taking v 2 → 0 and v 1 , v 3 → 0.
To study the decoupling properties, we follow the strategy outlined at the end of section 3.1. To simplify the expressions, we take again the vevs real, use eq. (47) with ω = 0, and expand in β 1 and β 2 . These conditions ensure that v 3 → v and that the contributions for the Y S ij =i due to M 2 k =1 ≫ v 2 are suppressed. Using eqs. (38), (39), and (43), we find Y CH ij , which we then substitute eq. (42). The leading order terms are Here one can see that one can set Y S 12 = Y S 23 = 0, by choosing a fine tuned region of parameter space with decoupling energy scales given to order β i v 2 by One concludes that when v 1 → 0 and v 2 , v 3 ≫ v 1 , the decoupling energy scales can still be larger than the electroweak scale without including all quadratic parameters. 8 Through this procedure, it is possible to decouple a 3HDM → SM with two scales which become larger and farther apart as β 1 → 0. It is interesting to interpret the difference between the stationarity conditions in this case, eqs. (80), and in the previous case, eqs. (79), in the following way. The decoupling limit is easy to interpret in the CH basis. It says that two parameters must become very large. And, using eq. (42) and the fact that all |U CH ij | < 1, we conclude that some entries of Y S must be large. Now, the stationarity conditions of the general NHDM in the S basis are Eq. (89) shows that there are two ways to make the diagonal elements Y S ii large. First, one can make the off-diagonal elements in the first term on the RHS of eq. (89) large Or, one can use the ratios of vevs in the second term on the RHS of eq. (89) and make those large The resulting difference between the Z 2 × Z 2 3HDM and the Z 3 3HDM is the following. Decoupling in the Z 2 ×Z 2 3HDM with two different scales M 3 ≫ M 2 ≫ v requires all off-diagonal soft symmetry breaking terms Y S ij =i . In contrast, decoupling in the Z 3 3HDM with two different scales M 3 ≫ M 2 ≫ v is possible with only some Y S ij =i . 8 Notice that the two scales cannot be made equal or even close, since M 2 2 − M 2 3 also grows as β 1 → 0.

A note on CP violation
The examples above focused on models with real vevs in a real symmetry basis. Then CP is conserved, and only the magnitudes of the quadratic parameters are relevant for decoupling. The same type of arguments can be used to study cases where there is CP violation. This is beyond the scope of this article, but we make a few simple comments here. Let us consider cases in which the potential is invariant under the canonical definition of CP: Invariance of the potential (1) under this definition of CP implies that and all coefficients are real. Then one may have spontaneous CP violation if some vevs have a relative phase. Under this definition of CP, the fields ρ k and χ k in eq. (3) are CP-even and CP-odd, respectively. In that case, CP violation in scalar-pseudoscalar mixing is described by M 2 ρχ ij in eq. (14), which, since it is at most of order v 2 , becomes irrelevant as all charged Higgs become very massive.
There are several issues that complicate a general analysis. First, the definition of CP changes with basis transformations. For example, it is true that under the CP transformation (92), ρ k and χ k in eq. (3) are CP-even and CP-odd. However, the simple rephasing Φ k → Φ ′ k = iΦ k means that, under the same definition of CP, it is now χ ′ k which is CP-even and ρ ′ k which is CP-odd. 9 Second, when searching for CP violation one must study all possible definitions of CP -for an introduction to these problems, see for example ref. [24]. Finally, one may even have the odd situation that there is CP conservation even though there are irremovable complex couplings in the scalar potential. The first and simplest example is the so-called CP4 3HDM [25]. For these reasons, a complete analysis of the relation between decoupling and CP violation (explicit or spontaneous) lies beyond the scope of this article.
Still, the result discussed below eq. (37) is completely general. Indeed, if all charged Higgs become very massive, then eqs. (34)-(36) imply that M 2 ρρ and M 2 χχ become very large and almost diagonal, while M 2 ρχ remains of (small) order v 2 . Thus, all CP violation in scalar-pseudoscalar mixing vanishes, regardless of the precise details of CP.
We can also recover a recent result on the 2HDM [11], because the model is very simple. Consider a 2HDM with a softly broken Z 2 symmetry and complex vevs. As noted, the symmetry basis is defined up to rephasing of its doublets. Then one can without loss of generality choose λ 5 to be real. In this basis m 2 12 can either be real or complex. If it is real, there can be [26] (or not) spontaneous CP violation, and if it is complex, there is explicit CP violation. By parameterising the vevs as v 1 = vc β and v 2 = vs β e iθ , the transformation matrix from the S basis with real λ 5 , to the Higgs basis is given by, To obtain the decoupling limit conditions we repeat our procedure of setting Y CH ≡ M 2 2 and writing Y CH 11 , Y CH 12 as a function of the S basis quartic parameters. Then we write the S basis quadratic parameters as a function of the CH basis quadratic parameters. Note that both Y CH 12 and U CH are now complex. Then, and the soft breaking term will be given by Here, one can see that it is not possible to have a decoupling limit and spontaneous CP violation in a 2HDM with a Z 2 symmetry softly broken (by a real parameter). Y S 12 must be complex for a decoupling to exist, which in turn explicitly breaks the CP symmetry.

Conclusions
We have investigated the situations under which a generic NHDM might have a decoupling limit. This is important for model building, since one might wish to gain intuition on a complicated model by studying it analytically when it decouples into a simpler theory. It is also very convenient when simulating numerically a complicated model, by debugging with simulations of numerical limits when it reduces effectively to a simpler model.
We have found that, under the assumptions that the vevs and the mixing angles are not parametrically small, all quadratic couplings must be included in order for decoupling to occur. For the most part, this means that either one has nondecoupling or, else, one must break the symmetry softly. In addition, we showed that there is no CP violation in scalar-pseudoscalar mixing as the charged Higgs masses approach decoupling.
Our theorems were illustrated with a number of special examples. These were also used to explore violations of the assumptions and to discuss what form of decoupling can occur in the absence of some Y S ij =i , as long as one goes to a very fine tuned region of parameter space, where some vevs decrease as v 2 /M. and Z 4 , found in [16,20], and the Z 2 ×Z 2 put forward by Weinberg [21]. Such symmetries are realisable through the following representations, Here one can explicitly see that every doublet has a different group charge, so that these symmetry groups have a well-defined Symmetry basis. By writing the general U(3) matrix in a suitable diagonal basis, we obtain the generator of the U(1)×U(1) symmetry in (B.3). Then the parameters that are invariant under (B.3) will also be invariant under abelian symmetries whose generator is written in a diagonal basis. The U(1) × U(1) symmetric 3HDM can be parameterised as [20],