Invariant approach to CP in unbroken $\Delta(27)$

The invariant approach is a powerful method for studying CP violation for specific Lagrangians. The method is particularly useful for dealing with discrete family symmetries. We focus on the CP properties of unbroken $\Delta(27)$ invariant Lagrangians with Yukawa-like terms, which proves to be a rich framework, with distinct aspects of CP, making it an ideal group to investigate with the invariant approach. We classify Lagrangians depending on the number of fields transforming as irreducible triplet representations of $\Delta(27)$. For each case, we construct CP-odd weak basis invariants and use them to discuss the respective CP properties. We find that CP violation is sensitive to the number and type of $\Delta(27)$ representations.


Introduction
The origin and nature of CP and its violation remains a mystery both within and beyond the Standard Model (SM). In addressing the question of CP it was observed some time ago that phases which appear in the Yukawa matrices for example are not robust indicators of CP violation since their appearance is dependent on the choice of basis. On the other hand, physical CP violating observables only depend on particular combinations of Yukawa matrices which are invariant under different choices of basis. Such weak basis invariants, which have the property that they are zero if CP is conserved and non-zero if CP is violated therefore provide unambiguous signals of CP violation which are closely related to experimentally measurable quantities. The use of such CP -odd weak-basis invariants (CPIs), rather than particular phases in a given basis, is generally referred to as the Invariant Approach (IA) to CP violation.
In the IA to CP violation [1], one starts by separating the full Lagrangian of the theory in two parts, one denoted L CP that is known to conserve CP , typically the kinetic terms and pure gauge interactions [2] 4 , and the remaining Lagrangian, denoted L rem. . The crucial point is that L CP allows for many different CP transformations and as a result, CP is violated if and only if none of these CP transformations leaves L rem. invariant. In the case of the SM, L CP includes the gauge interactions and the kinetic energy terms, while the relevant components of L rem. are the Yukawa interactions. Using the IA, one can readily derive [1] some specific conditions that the Yukawa couplings have to satisfy in order to have CP invariance. It is well known that the Yukawa couplings in the SM have a large redundancy which results from the freedom that one has to make redefinitions of the fermion fields which leave the gauge interactions invariant but change e.g. the quark Yukawa couplings Y u , Y d without changing the Physics. The great advantage of the IA is that it allows one to derive CPIs which, if non-vanishing, imply CP violation. In the SM, it has been shown [1] that the relevant CPI is Tr [H u , H d ] 3 , where we define the Hermitian For the 3 fermion generation case this CPI leads to the Jarlskog invariant [4]. The IA can be applied to any extension of the SM, in particular to extensions of the SM with Majorana neutrinos [5].
It should be emphasised that the IA not only enables one to verify whether a given Lagrangian violates CP , but also provides an idea of how suppressed CP violation might be. A notable example is the possibility of showing why CP in the SM is too small to generate the baryon asymmetry of the Universe (BAU). One simply observes that the dimensionless number 3 /v 12 is of order 10 −20 , where we used the Hermitian quark mass matrices and v = 246 GeV denotes the scale of electroweak symmetry breaking. This dimensionless number should be compared to the size of observed BAU, n B /n γ 10 −10 [6]. The IA, leading to basis invariant quantities, also identifies what combination of parameters are physical such that, e.g. there is no need to count how many phases can be eliminated through rephasing, which can be laborious in complicated Lagrangian, and specially in the presence of family symmetries.
Recently [7] the use of CPIs, valid for any choice of CP transformation, was advocated as a powerful approach to studying specific models of CP violation in the presence of discrete family symmetries. Examples based on A 4 and ∆ (27) family symmetries were discussed and it was shown how to obtain several known results in the literature. In addition, the IA was used to identify how explicit (rather than spontaneous) CP violation arises, which is geometrical in nature, i.e. persisting for arbitrary couplings in the Lagrangian.
Here we intend both to further highlight the usefulness of the IA in dealing with discrete family symmetries and also to systematically explore the CP properties of ∆ (27). By using the IA, we are able to construct CPIs independently of the specific group and need to consider the group details only to compute coupling matrices by using the respective Clebsch-Gordan coefficients in any particular basis. By combining the coupling matrices with the CPIs, basis-independent quantities are obtained which indicate if there is CP violation.
In this paper we explore in depth the CP properties of unbroken ∆ (27) invariant Lagrangians using the IA as outlined in [7] (see also the proceedings [8]). The method is based on [1]. We focus on ∆(27) since it involves many features which may be encountered in more general discrete groups, such as complex representations and multiple non-trivial singlet representations. It therefore constitutes a rich playground for exploring the IA in the case of discrete family symmetries. Although the cases discussed do not represent realistic models, since the ∆(27) is unbroken, the work here lays the foundation for future models based on spontaneously broken ∆ (27).
Following the IA we consider several cases that highlight how the CP properties depend both on the field content and on the type of contractions considered (which may be controlled e.g. by additional symmetries, even though here we don't always consider those explicitly). We focus on tri-linears terms, which we refer to as Yukawa-like couplings, keeping in mind that most (but not all) cases considered are meant as fermion-fermionscalar terms. We start by considering Lagrangians with just ∆ (27) singlets where the IA identifies the relative phases that are physical, and then concentrate on Yukawa-like terms involving a triplet, an anti-triplet, and a singlet.
When the Yukawa-like couplings are between triplet, anti-triplet and singlet, we study cases with a single independent triplet, with independent triplet and anti-triplet, and with three or more independent 3-dimensional irreducible representations. Some of the cases considered are similar to adding ∆(27) to a type II 2 Higgs doublet model (2HDM) 5 or N Higgs doublet model (NHDM). For each of these frameworks the IA allows to identify if the Lagrangian has the possibility to explicitly violate CP , how this depends on how many different ∆ (27) singlets are coupled, and what CPIs are relevant and non-vanishing when there is CP violation. This serves to further illustrate the convenience and power of the IA for the study of CP properties of specific Lagrangians.
The layout of the remainder of this paper is as follows. In section 2 we briefly review the IA to CP in family symmetry models. We continue with the group theory of ∆ (27) in section 3. Section 4 considers just singlets. In section 5 the field content includes one triplet. Two triplets are considered in detail in section 6, where we differentiate also based on the number and type of singlets present. We generalise to three triplets in section 7 and to four and more triplets in appendix A. For comparison with the IA, we present some examples of specific CP matrices in appendix B. Finally we conclude in section 8. The CP transformation X is consistent with the group G, as following it with the transformation ρ(g) associated with element g of G, and then with X −1 , is equivalent to the transformation ρ(g ) associated with some other element g of G.
2 Invariant approach to CP in family symmetry models As mentioned in the introduction, the IA as outlined in [7,8] is based on [1], where to study the CP properties of a given Lagrangian one starts by splitting it where L CP denotes the part that is known to conserve CP (kinetic terms and gauge interactions, as pure gauge interactions conserve CP [2]). L rem. includes non-gauge interactions such as the Yukawa couplings. A review of how the IA is applied to the Standard Model (SM) lepton sector can be found in [7], which also includes its application to a model of spontaneously broken A 4 and to a model of ∆ (27) which features explicit geometrical CP violation.
In this paper we study many different Lagrangians invariant under unbroken ∆(27), relying on the IA. As pointed out in [7], the presence of a family symmetry does not change the most general CP transformation which leaves invariant L CP -these are the kinetic terms and the gauge terms which are flavour blind. Before continuing with the IA, a relevant question is what role is played by the consistency relations [10,11] in this type of analysis.
The consistency relations can be obtained by considering that a Lagrangian invariant under both a family symmetry and a CP symmetry should be the same whether one considers doing a consistent CP transformation before or after a family symmetry transformation. The concept is illustrated in Fig. 1 for a field φ. When this is considered rigorously one obtains a relationship between CP transformations X and family symmetry transformations ρ(g) Xρ(g) * X −1 = ρ(g ), g ∈ G . (2) If we find an explicit CP transformation that leaves the Lagrangian which respects some family symmetry G invariant (even one is enough) then we can be quite sure that the theory conserves CP as well as respecting the G. In this case the consistency relation in Eq.2 is automatically satisfied. This is clear since under a CP transformation, followed by a family symmetry transformation, followed by another CP transformation, etc., leaves the Lagrangian invariant, The consistency of CP and G is clear since the Lagrangian is left invariant at each stage. However we can be even more explicit than this in order to demonstrate the equivalence of the two approaches. Consider a mass term m in the Lagrangian, then define Suppose that the Lagrangian is invariant under some family symmetry transformation, ρ(g), then this implies that the mass term in the Lagrangian remains unchanged under a family symmetry transformation and hence The condition for the invariance of the Lagrangian under a CP transformation, X, requires that the mass term swaps with the H.c. mass term hence, Taking the complex conjugate of Eq.5 we find, using Eq.6 for the last equality. From Eq.7 and Eq.6 we obtain, hence where we have used Eq.5 but for a different group element g in the last equality. By comparing both sides of Eq.9 we identify, which is just the consistency condition in Eq.2.
As we further illustrate in the following sections, using the IA one need not specify a CP transformation. For a given Lagrangian it is sufficient to input the invariance conditions imposed by the symmetries. This makes the IA very useful to study CP violation in the presence of family symmetries.
Using the IA, [7] found that a specific ∆ (27) invariant Lagrangian features a different type of geometrical CP violation, where CP is explicitly violated (rather than spontaneously). In this paper we go beyond this result, considering many ∆ (27) invariant Lagrangians to study in depth how the interplay between ∆ (27) and CP changes depending on the representations and how they couple with one another.
To do so, some understanding of the group properties is required. ∆(27) has three Z 3 generators [15,16] but we only need to use two, which we refer to as c (for cyclic, with c 3 = 1 ) and d (for diagonal, with d 3 = 1 ). This notation refers to their respective 3-dimensional representation matrices in the basis we use. We define ω ≡ e i2π/3 . Starting with the 9 distinct singlets which we conveniently label as 1 ij , the generators are represented by c 1 ij = ω i and d 1 ij = ω j for that particular singlet. A field transforming as a 1 00 (trivial singlet) is explicitly invariant under ∆ (27) transformations, and the other 8 singlets simply get multiplied by the respective powers of ω when acted upon by c or d. The other irreducible representations of ∆ (27) are triplets, two distinct ones which we take as 3 01 and 3 02 . The generator c is represented equally for both d is represented as a diagonal matrix with entries that are powers of ω, with the exponents denoted by the indices of the triplet representation The determinant of the matrices is 1 (∆(27) is a subgroup of SU (3)) and the two indices identify d 3 01 = diag(1, ω, ω 2 ), d 3 02 = diag(1, ω 2 , ω). The representations 3 01 and 3 02 behave as a triplet and anti-triplet, so in analogy with SU (3) we refer to them mostly as the 3 and3 representations. The subscript notation is useful to remember the powers of ω that each component transforms with under d 3 ij so we refer to it occasionally throughout the paper, e.g. if we take A = (a 1 , a 2 , a 3 ) 01 transforming as triplet 3 = 3 01 andB = (b 1 ,b 2 ,b 3 ) 02 transforming as (anti-)triplet3 = 3 02 , the explicit construction of the trivial singlet is (AB) 00 = (a 1b1 + a 2b2 + a 3b3 ) 00 . This can be verified by acting on A and B with generators c and d and checking that the prescribed (AB) 00 remains invariant. Indeed, 3 ⊗3 = i,j 1 ij and rules for constructing the non-trivial singlets from triplet and anti-triplet follow (a 2b1 + a 3b2 + a 1b3 ) 01 , 6 Spontaneous geometrical CP violation has also been found in other groups [25][26][27].
(ω 2 a 1b2 + ωa 2b3 + a 3b1 ) 12 , All these can be verified by acting on the triplets with the generators and tracking how each product transforms.

Just singlets
To illustrate how the IA would proceed, we start by considering Yukawa-like terms without ∆(27) triplets. Throughout we refer to fields transforming as singlets under ∆ (27) as h ij where the subscript refers to the field being assigned as a 1 ij under ∆ (27).
A simple example where the field content is h 00 , h 01 , h 10 would have Yukawa-like terms L III =z 00 h 00 h 00 h 00 + z 01 h 01 h 01 h 01 + z 10 h 10 h 10 h 10 It is clear there are further ∆(27) invariant terms such as h 00 h 00 , but for the sake of illustrating the IA we consider the CP properties of the Yukawa-like terms in L III by itself as the only part of the full Lagrangian that can violate CP (i.e. as the L rem. for this case). 7 The next step in the IA is to consider the most general CP transformation for each field consistent with the kinetic terms etc., in this case this means each singlet transforms with its own phase which we denote as p ij When we apply these transformations on L III and demand it remains invariant, we obtain a set of necessary and sufficient conditions for CP conservation restricting the parameters in L III z 00 e i3p 00 = z * 00 , y 00 e ip 00 = y * 00 (26) y 10 e ip 00 = y * 10 .
We note that the CP conservation conditions on couplings y 01 and y 10 are independent of p 01 , p 10 . To build CPIs we combine conditions that cancel dependence on the CP transformations, which involving y 01 and y 10 requires only the cancellation of all dependence on p 00 . A simple and useful CPI is then Im[y 01 y * 10 ], as where the y ij are complex numbers so y † ij = y * ij . The CPI vanishing is a necessary (but not necessarily sufficient) condition for CP conservation and it constrains the relative phase between the two couplings.
There are also CPIs of this type constraining the relative phases between y 00 and the other two y ij couplings. CPIs involving z 00 can also be built noting that the CPI needs to cube the other couplings e.g. Im[z * 00 y 3 ij ], as If any of these CPI are non-zero, then CP is violated. Other CPIs like Im[z ij z * ij ] (or Im[y ij y * ij ]) do not provide useful constraints as they automatically vanish. As a first generalisation of L III we add a field h 02 . This allows a mixed invariant whose coupling constant will be sensitive to the CP transformation of the non-trivial singlets. We continue to consider only Yukawa-like terms in L rem. so we write The new CP conditions are y 02 e ip 00 = y * 02 , y 1 e i(p 00 +p 01 +p 02 ) = y * 1 .
With the added couplings, we have from the direct generalisations of Eq. 29 CPIs involving y 02 like which constrain the relative phases of the y ij couplings. It is however not possible to obtain such simple, useful constraints on the relative phase of the mixed coupling y 1 , because it involves p 01 and p 02 . The simplest CPI, Im[y 1 y 1 * ] automatically vanishes and is therefore not useful. The conclusion from this CPI alone would be that the phase of y 1 is unconstrained by requiring L IV to conserve CP . However, we note that by using a more complicated invariant involving z 01 , z 02 and either z 00 or 3 insertions of any y ij , we can build non-trivial CPIs involving y 1 , such as The situation involving the coupling of mixed terms of the type y 1 qualitatively changes if there are sufficient mixed terms. Consider now a field content with all 9 ∆(27) singlets h ij . To reduce the number of allowed terms we may impose a Z 3 symmetry where each h ij transforms equally. Such a Z 3 symmetry forces y ij = 0 (in addition it forbids a multitude of other Yukawa-like terms like h 01 h 10 h † 11 , where h † 11 would play the role of h 22 ). There are 9 Yukawa-like terms like z 00 h 00 h 00 h 00 (one for each singlet). CPIs involving the z ij will be like Eq. 38. Focusing solely on the mixed terms like y 1 h 00 h 01 h 02 , there are 12 combinations of singlets whose indices add up to a mixed invariant The CP conservation condition for each coupling depends on the 3 phases of the respective singlets, e.g.
With these conditions it is now possible to combine several of the mixed couplings to form a CPI. An example is Im[y 1 y * 2 y * 6 y * 10 y 11 y 12 ] , meaning this particular combination of couplings is constrained by CP conservation to be real. Other combinations of this type can be built from the couplings in L IX .

One triplet
We will now consider Lagrangians involving Yukawa-like couplings with just one ∆(27) triplet [8]. In order to make invariants, the terms will necessarily involve the conjugate of that triplet. In this case it is not possible to construct Yukawa couplings involving a We construct the Lagrangian with one scalar triplet φ ∼ 3 and 2 scalar singlets h 01 , h 10 , such that the Yukawa-like terms are The most general CP transformations are where U is a general unitary matrix. Assuming CP invariance of L 2s and using matrices Y ij corresponding to the couplings y ij we have We can build a useful CPI for this Lagrangian [8] The CPI applies for any matrices Y 01 and Y 10 . Imposing ∆(27) invariance we have from Eq. 13,15 which we input into I 2s and obtain Finding a non-vanishing CPI means that CP is violated, which clearly happens for any non-zero values of y 01 and y 10 . Given that CP is explicitly violated by a phase only originating from the group structure and not from arbitrary Lagrangian parameters, this is a minimal case with explicit geometrical CP violation [7,8].
In [22] it was pointed out that ∆ (27) provides an example of a group where not all Clebsch-Gordan coefficients can be made real by a change of basis, when several of the singlets are used. Indeed, this fact was already referred in the earlier ∆ (27) works [19][20][21] where CP is violated spontaneously and therefore only a few singlets were used. The change of basis analysis presented explicitly in [8] further clarifies the connection between the inevitability of complex Clebsch-Gordan coefficients (which are basisdependent) and the presence of multiple singlets. The physical consequences are of course basis-independent as illustrated elegantly in the invariant approach, and depend crucially on the field content, not just of singlets but also of triplets as shown in the following sections.

Two triplets
We continue our exploration of unbroken ∆ (27) invariant Lagrangians with Yukawa-like terms by considering in detail the class with two distinct 3-dimensional representations. We take these to be explicitly a triplet Q ∼ 3 and an anti-triplet d c ∼3. 8 The notation we are following is suggestive of identifying the 3 and3 as fermions, as considered in [7]. Nevertheless, it is also possible to consider just scalars [8], or that one of the triplets is a scalar and Yukawa couplings are formed by having ∆(27) singlet fermions, as considered in [17][18][19][20][21]. The conclusions we derive with the IA apply to all cases.
As a further point of notation, if e.g. Q does refer to the SM quark fields, the most general CP transformation consists in Here we need not specify if Q is a fermion or scalar, and in particular we are more concerned with identifying which matrix corresponds to each field (in Eq. 58, U corresponds to Q). For these reasons we use a simplified notation along the lines of which is exact for scalars, and is more convenient to identify which general CP transformation corresponds to each field (in Eq. 59, U T Q corresponds to Q). We assume the L rem. part of the Lagrangian consists of Yukawa-like terms between triplet Q ∼ 3, anti-triplet d c ∼3, and singlets h ij ∼ 1 ij . 9 This class of Lagrangians is a good framework to illustrate several interesting points, so we go into some detail of what happens when varying the number of coupled singlets. The first model we consider of this type is with Q, d c and 2 singlets h 10 and h 01 [8] L 3 = y 10 (Qd c ) 20 h 10 + y 01 (Qd c ) 02 h 01 + H.c. (60)

Adding a specific CP symmetry
Before applying the IA to Lagrangians of this type, lets consider what happens when applying a specific CP transformation.
Arguably the simplest CP transformation is the trivial CP transformation, which we refer to as CP 1 . This corresponds to U Q = 1 in Eq. 59, i.e.
where we used the subscript to denote that Q * transforms as a3 = 3 02 , given that under action by generator d, Q 2 → ωQ 2 and therefore we must have Q * 2 → ω 2 Q * 2 under d (as expected from complex conjugation). Similarly, For the two ∆(27) singlets h 10 and h 01 in L 3 where the conjugated versions transform under ∆(27) as 1 20 and 1 02 respectively. This means that under the trivial CP transformation all four fields go into their respective conjugate ∆ (27) representations.
The Yukawa-like terms in L 3 are explicitly invariant under ∆ (27) and y 10 , y 01 are arbitrary complex numbers. We now impose additionally that L 3 is invariant under CP 1 . For the y 10 coupling, expanding from Eq. 18 and using Eq. 61, 62, 63 In identifying how (Qd c ) 20 has transformed under CP 1 , note the CP 1 -transformed product (Qd c ) 20 still transforms as a 1 20 under ∆ (27), as it picks up a phase of ω 2 when acted by c. For the y 01 coupling, expanding from Eq. 14 and using Eq. 64 In contrast, because of the action in Eq. 62, d c * 2 picks up a phase ω when acted by d, we identify that the CP 1 -transformed product (Qd c ) 02 transforms as a 1 01 under ∆ (27). What are then the physical consequences of imposing CP 1 on the ∆(27)-invariant L 3 ? We need to compare Eq. 65 and Eq. 66 to the H.c. part of L 3 . In the case of Eq. 66 this reveals exactly the same expression, except that y * 01 appears, therefore the conclusion is clear -imposing CP 1 on L 3 forces y * 01 = y 01 . However, when we compare what we obtained in Eq. 65 to the only way to make the expressions match (to have L 3 be invariant underCP 1 ) is to require y * 10 = y 10 = 0. That there is some incompatibility with y 10 was already hinted at by the fact that Eq. 65 is explicitly not invariant under ∆ (27), which we denoted through the subscripts -(...) 20 transforms as a 1 20 , as does h * 10 . Our interpretation of these results is not that CP 1 becomes incompatible with ∆(27) when the theory includes field h 10 together with Q and d c . Rather, that it is always possible to add CP 1 to a ∆ (27) invariant Lagrangian regardless of the field content, but we interpret this result as indicating that there will be physical consequences on the couplings that make the theory consistent by making the Lagrangian invariant under the full symmetry imposed. In other words, it is more correct to state the incompatibility is not with the field content but rather with the couplings.
That CP 1 restricts L 3 does not mean that CP is violated though, and in fact a CP transformation can be defined that leaves the Lagrangian invariant. A good way to check L 3 is CP conserving is to use the IA. Before doing so, lets examine the effects of another possible CP transformation CP 2 , keeping the same transformations as CP 1 for the singlets but where the triplet transforms as where some components swapped positions. Similarly, Note that unlike what happens with CP 1 , given that under action from generator d we have Q 3 → ω 2 Q 3 and therefore Q * 3 → ωQ * 3 , this Q * transforms as a 3 = 3 01 , and similarly this d c * transform as a3 = 3 02 .
Checking how the L 3 terms transform under CP 1 we have In identifying how (Qd c ) 20 has transformed under CP 2 , it now picks up a phase of ω when acted by c. The other combination and one can verify that the (Qd c ) 20 has transformed under CP 2 into an expression that picks up a phase of ω 2 when acted by d. By comparing Eq. 70, 71 we conclude that by imposing CP 2 invariance on Eq. 60 we are forcing y 10 to be real and y 01 = 0 (contrast with CP 1 ). While we used ∆ (27) and CP 1 (and CP 2 ) in this explicit example, our interpretation is general -we have the freedom to impose any family symmetry (discrete or not) together with any CP symmetry. Eventually what may happen in extreme cases, is that it will not be possible to form non-trivial combinations that are invariant under both symmetries. We feel it is important to stress that in this interpretation, the CP symmetry is consistently treated equally to other symmetries -the transformation is defined and it has consequences for the Lagrangian.
It is important to stress again that even if imposing a specific CP symmetry on a theory restricts the couplings of the Lagrangian, this does not mean that the Lagrangian violates CP . L 3 is an example of that as we now show using the IA. It is convenient to first rewrite L 3 in terms of coupling matrices Y 01 and Y 10 and to specify the general CP transformation properties One can verify that CP 1 and CP 2 are particular cases of this general CP transformation. Imposing invariance under the general CP transformation requires We again wish to build combinations of the Yukawa-like couplings that eliminate the dependence on the general transformation. Unlike what we considered with CP 1 and CP 2 , Q and d c transform in general with distinct unrelated matrices. This forces CPIs to alternate between Y ij and Y † kl . It is therefore convenient to define the Hermitian involving for each only the matrix associated with d c and Q, respectively. The CPIs are of the type (with integers n i ) When referring to CPIs of this type we are including also CPIs like The first one alternates between G 01 and H 10 by having a single additional Y † 01 in the middle, and eventually goes back to G 01 due to the lone Y 01 that is required to cancel the dependence on p 01 . The second one is similar by having Y † 10 Y 01 between G 01 , then requiring Y † 01 Y 10 somewhere else in order to cancel the dependence on p 01 , p 10 -note the ordering of the inserted Y ij and Y † kl is not arbitrary. It is also possible to mix and match odd and even insertions of Y ij and Y † kl We refer to the more complicated CPIs as being of the type Eq. 78, 79 because they are obtained by iteratively inserting some G ij or H ij inside an existing G kl or H kl , thus separating the constituent Y kl and Y † kl . 10 When considering the Yukawa-like matrices that comply with ∆(27) invariance that we have seen before, we have and it is possible to check that all these CPIs vanish automatically, for any complex y 10 , y 01 . Therefore L 3 conserves CP for any y 10 , y 01 , even though it is not in general invariant under some of the particular CP transformations like CP 1 and CP 2 .

Two singlets
The conclusion is a bit more general and applies beyond the pair of singlets h 01 and h 10 .
Using the IA it is relatively easy to verify that a Lagrangian of type L 3 automatically 10 E.g. start with Eq. 78 and split one of the G 01 in the middle with H n2 10 and redefine the integers to obtain Eq. 80. Or in reverse, remove G n2 10 from Eq. 82, which allows an H 01 to be created and included into H n3 01 by redefining the integers and arrive back at Eq. 80 -which can then be related with Eq. 78.
conserves CP for any two singlets ∆(27) (even if a specific CP like CP 1 imposes restrictions on these Lagrangians). The field content is then Q, d c and any two singlets h ij , h kl . There are only two Yukawa-like terms and associated matrices The CPIs are trivial generalisations of the previous cases, e.g.
and the other types discussed above. Due to invariance under ∆ (27) we have regardless of the singlets used. Using these relations allows one to conclude that all of these CPIs automatically vanish for any couplings y ij , y kl , meaning CP is automatically conserved in Lagrangians of L 3 type -Yukawa-like couplings of any 2 ∆(27) singlets to 2 triplets. However, the possibility for explicit CP violation exists for 3 singlets and beyond -see also [22,24], but note that the conclusion depends on the number of triplets as well (e.g. 2 singlets are sufficient to violate CP in the presence of only one triplet [8]).

Three singlets
Applying the IA to Lagrangians of L 3 type with 3 singlets allows us to identify explicit geometrical CP violation. In this case we change notation from triplet Q and anti-triplet d c to the notation used in the model of [7], to allow an easier comparison. We introduce the SM fermions L ∼ 3 and also ν c ∼3 and singlet scalars h 00 , h 01 , h 10 . This model is a model of leptons, with a charged lepton Lagrangian that gives in this basis a diagonal mass matrix, with a ∆(27) triplet scalar φ ∼3 [7] − y e (Lφ) 00 e c 00 − y µ (Lφ) 01 µ c 02 − y τ (Lφ) 02 τ c 01 + H.c. .
This is similar to the L 3 Lagrangian, with Yukawa-like terms between the triplets and the singlets. The general CP transformations are Assuming CP invariance of L 3s and expressing the couplings in terms of Yukawa matrices Y ij , the CP conservation requirements are ∆ (27) imposes Because we have 3 Yukawa matrices, we can build a CPI that does not involve G ij or H ij combinations This CPI is qualitatively different from the ones that could be built with only 2 matrices. Indeed if we calculate it for this particular choice of 3 singlets I 3s = Im(3ω 2 |y 00 | 2 |y 01 | 2 |y 10 | 2 ) .
This means that in general, for arbitrary (non-zero) couplings, this Lagrangian violates CP as the condition I 3s = 0, necessary for CP conservation, is not fulfilled. This is the case originally identified to have explicit geometrical CP violation in [7], with the phase appearing in I 3s independent of the arbitrary phases of couplings. We can again check what are the consequences of adding specific CP transformations, this time to a Lagrangian where explicit CP violation is possible. From the analysis of the rather similar L 3 Lagrangian, we can conclude that imposing CP 1 to L 3s leads to y 10 = 0, and that imposing CP 2 leads to y 01 = 0. As may have been expected, both make I 3s vanish. This is a clear demonstration of the meaning of adding different CP transformations -both lead to CP conservation but not necessarily the same consequences. 11 The type of CPI exemplified in I 3s is very useful in the study of Yukawa-like terms between triplet, anti-triplet and singlets. For CP to be conserved it is necessary that it vanishes. In cases with with 3 ∆(27) singlets, it is also sufficient that the respective CPI vanishes for this type of Lagrangian to conserve CP . The combinations of 3 singlets that 11 Consider Yukawa-like couplings involving Q, d c and singlets h 01 , h 10 and h 20 in the basis where these singlets correspond to 1 01 , 1 10 and 1 20 of ∆(27) respectively, then impose CP 1 in that basis. The two couplings y 10 and y 20 are forced to vanish. Conversely, CP 2 forces only one coupling to vanish, y 01 . If one changes the basis it may be that e.g. CP 1 is no longer associated with the 1 matrix in the new basis, but it will nevertheless force two couplings to vanish whereas CP 2 forces only one coupling to vanish. See [8] for more specific considerations regarding basis changes.
automatically conserve CP can be found [8] to be 12 out of the total 84 combinations of 3 singlets. Curiously, these combinations are the 12 combinations of 3 singlets appearing in each of the 12 terms of L IX in Eq. 40 of Section 4.
then use Yukawa-like matrices Y ij and build the CPI similar to I 3s Then we repeat these steps for the y 2 h 00 h 10 h 20 term in Eq. 40 and so on. When inserting the conditions imposed by ∆(27) on the Yukawa-like matrices Y ij , these 12 CPIs vanish automatically for any values of the Yukawa-like couplings One can explicitly build CP transformations to prove that indeed the 12 respective Lagrangians L 3s 1 , L 3s 2 , (...), L 3s 12 automatically conserve CP (for any values of the 3 Yukawa-like couplings of each respective Lagrangian).
As illustrated by I 3s itself, the CPIs for the other 3 singlet combinations do not vanish automatically and appear with the respective |y ij | 2 |y kl | 2 |y mn | 2 , multiplied by a factor of 3ω or 3ω 2 . They are further examples of explicit geometrical CP violation.

Four or more singlets
Any choice of 4 or more singlets will necessary include combinations of 3 that would allow CP violation. For example, by adding any other singlet to the set h 00 , h 01 , h 02 in L 3s 1 , we have a singlet with h ij with i = 0 and any of the I 3s -type CPIs involving Y ij with Y 00 , Y 01 , Y 02 We continue investigating Lagrangians with Yukawa-like terms between triplets and singlets, in the presence of 3 triplets of ∆ (27). In contrast to the situation studied in [17][18][19][20][21] where there is a scalar ∆(27) triplet coupling to fermions, we consider a situation where we generalise the L 3 Lagrangian with additional anti-triplet u c . The triplet Q ∼ 3 now contains the SM quark SU (2) doublets, and two anti-triplets u c , d c ∼3 contain the up and down quark SU (2) singlets. The scalars are Higgs doublets h u ∼ 1 10 and h d ∼ 1 01 (we deviate here slightly from the notation used for scalars in other sections). With a Z 2 symmetry it is possible to have the u c couple only to h u and d c couple only to h d , leading to a type II 2HDM (see e.g. [9]) where the actual Yukawa terms are constrained by ∆ (27). The Lagrangian is We express the Lagrangian in terms of Yukawa matrices Y u , Y d and apply the IA to this Lagrangian 12 The conditions on Y u , Y d from imposing general CP invariance on L 2HDM are We see that we can without loss of generality redefine U u to absorb e ipu and U d to absorb e ip d . We choose then e ipu = e ip d = 1 and redefine U u , U d accordingly. 13 In effect what this means is that we have, for Y u and Y d , the same type of CP conservation requirements that they would have in the SM. So we extrapolate from [1], rely on the Hermitian which happens to correspond to the usual basis where H u is diagonal and 12 For simplicity, even though Q and d c are explicitly fermions, we continue using the abridged notation of Eq. 59 rather than the more rigorous one of Eq. 58. 13 Note that we could not do this for h 10 and h 01 in L 3 as there was only one anti-triplet d c .
In the limit of unbroken ∆ (27) in fact H u = H d = 1 so it is clear that CP is automatically conserved for any y u , y d . Indeed, by using the IA on the Lagrangian we conclude that the only way to build CPIs is to keep Yukawa structures that couple to d c together, and also to keep Yukawa structures that couple to u c together, as that is the only way to cancel the respective U u and U d dependence. In addition to combining H u and H d , one can use just G u or just G d , but not mix both.

Type II NHDM with ∆(27) triplet fermions
Continuing from the Type II 2HDM Yukawa Lagrangian, using the IA it is relatively straightforward to generalise the conclusions for an increasing number of scalars. It is useful to classify each scalar and their respective Yukawa matrix according to their sector, i.e. an h d ij couples to d c or an h u kl couples to u c . It is in general no longer possible to absorb the respective phases into U u,d . The respective CP conservation requirements include which cancel the dependence on phases p u ij , p d ij , but as we have seen already CPIs with the Hermitian combinations are automatically verified in unbroken ∆ (27). However, it is now possible to build CPIs of the type of I 3s by using either 3 different Y u ij or 3 different Y d ij -without mixing the two sectors. Using the conclusions derived for the Lagrangians with 2 triplets (1 triplet and 1 anti-triplet) in Section 6, we can extrapolate to this 3 triplet case (1 triplet and 2 anti-triplets). Doing so, we conclude that each sector remains automatically CP conserving when coupled to up to any 2 ∆(27) singlets, and can remain automatically CP conserving when coupled to 3 ∆(27) singlets (if the 3 singlets are one of the 12 special combinations, as discussed in Section 6). If a specific representation is repeated in the up and down sector this still represents 2 fields, as they are distinguished by the type II Z 2 symmetry that distinguishes the sectors. On the other hand, the 3 singlet representations chosen for the up sector and for the down sector need not be the same, so one can couple up to 6 different singlet representations without enabling CP violation. Conversely, coupling the triplets to 7 or more distinct singlet representations will not allow automatic CP conservation, and the minimal singlet content that enables CP violation is 3 singlet representations all in the same sector.
Recall also that even without additional symmetries, for ∆ (27) triplet Weyl fermions one cannot construct QQ † h ij , d c d c † h kl or u c u c † h mn due to Lorentz invariance (even though the combinations would be ∆(27) invariant).
If one continues to generalise this class of Lagrangians to 4 or more triplets the conclusion is indeed that we can separately treat each Yukawa-like sector (a distinct triplet to anti-triplet pairing). This is because the general CP matrices appearing for each sector are unrelated, and that constrains the types of CPIs that can be built in the IA. In ∆ (27), each sector can couple to as many as 3 different singlets before CP violation arises as a possibility. This extends the conclusion derived in Section 6 for Lagrangians with a single Yukawa-like sector, L 3 , L 3s and the 12 special combinations L 3s 1 , (...).
It is interesting to note that with 3 sectors, which arises as a possibility with 4 or more triplets, the full representation content of ∆ (27) with all 9 singlets can be present while CP is still automatically conserved. Given that cases with 4 or more triplets no longer have a counterpart with the SM quarks, we relegate a more detailed analysis to appendix A. In addition we present some examples of specific CP transformations in appendix B, which includes an existence proof of CP transformations for a Lagrangian with all the irreducible representations of ∆(27).

Conclusions
The group ∆(27) is very interesting from the point of view of CP properties. In this work we considered several ∆ (27) invariant Lagrangians with Yukawa-like terms (tri-linears) and studied them with the invariant approach.
Our dual purpose was to demonstrate the usefulness of the invariant approach in Lagrangians invariant under discrete family symmetries, and simultaneously to explore the CP properties of ∆ (27). The method is independent of the group when the CPodd invariants are constructed, and the group details are needed only to obtain coupling matrices in some convenient basis, which can then be used in the CP -odd invariants to obtain basis-independent quantities signalling CP violation.
Starting with simple cases where the field content includes only 1-dimensional representations of ∆(27) (singlets), the invariant approach reveals what are the relevant physical phases, which turn out to be specific relative phases of the complex couplings.
We then turned to consider Yukawa-like terms involving ∆(27) triplet and anti-triplet, starting with a single 3-dimensional representation (triplet) and progressing to two and more triplets, where it becomes helpful to refer to sectors of distinct pairs of triplet and anti-triplet.
The conclusions derived for the two triplet case with one sector are that CP is automatically conserved for Yukawa-like terms involving up to any 2 ∆(27) singlets and for 12 special combinations out the total 84 combinations of 3 singlets. The other cases are examples of explicit geometrical CP violation.
Based on these results, the invariant approach allows us to extrapolate for cases with three or more triplets. The same type of conclusion holds independently for each sector, and therefore with 3 sectors it is even possible to have all 9 ∆(27) singlets present while automatically conserving CP .
We have therefore completed a fairly exhaustive analysis of unbroken ∆(27) Lagrangians. The analysis here should provide a useful guide for formulating future realistic models in which ∆(27) is spontaneously broken. However the main motivation was to highlight the utility and power of the invariant approach for Lagrangians with discrete family symmetries.
with the usual general transformations for the singlets each with their own phase and In this case there are 3 sectors like in L 4Q , because some additional symmetry distinguishes triplets Q and L such that L pairs only with e c , h emn . Consider instead a situation where Q and L couple to the same anti-triplets and singlets, like in the Lagrangian which has 6 sectors. Note though that while e.g. Qd c and Ld c appear to be distinct sectors, if Qd c has automatic CP conservation having up to 3 h d ij singlets, this applies also to Ld c , because the singlets h d ij are the same.

B Specific CP matrices
It is interesting to compare the IA to the construction of specific CP transformations for Lagrangians of L 3 type with any of the singlets L = QY ij d c h ij + (...) + H.c. , The matrices associated to each singlet h ij are, due to ∆ (27) Y 00 = y 00 For each Yukawa-like term QY ij d c h ij we have a CP conservation requirement on the therefore in general the U Q and U d matrices that respect this requirement are different for each Y ij . Within this point of view, the possibility of CP violation arises when it is impossible to have even a single set of U Q , U d and p ij transformations that simultaneously fulfil the distinct requirements of all Y ij that are present in the Lagrangian. Checking this can be quite laborious as it should be done with complete generality and indeed one of the advantages of the IA is that usually one need not check for the existence of such transformations. For illustration purposes we fix for simplicity the respective p ij = e −2i arg[y ij ] , U Q = 1 , and U d to be diagonal. For these choices we present for each Y ij the respective U d corresponds to the requirement of the h 00 , h 01 and h 02 Yukawa-like coupling; corresponds to the requirement of the h 10 , h 11 and h 12 Yukawa-like coupling; corresponds to the requirement of the h 20 , h 21 and h 22 Yukawa-like coupling. These CP transformations are with loss of generality (e.g. U d need not be diagonal), but they are still an existence proof of a valid CP transformation for each singlet by itself, and for the specific groups of 3 singlets shown. It naturally agrees with what was obtained through the IA for the more general case. For example, we knew already that with a single sector Qd c , these 3 combinations of three singlets belong to the 12 that automatically conserve CP . Furthermore, with 3 sectors Qd c , Qu c and Qx c we have in addition to U d also U u and U x , enabling the possibility to have all 9 ∆(27) singlets in Yukawa-like terms with triplets while automatically conserving CP for any arbitrary (non-zero) complex value of the nine y ij . In this 3 sector case with L 4Q in Eq. 123, an existence proof for h d 00 , h d 01 , h d 02 , h u 10 , h u 11 , h u 12 , h x 20 , h x 21 , h x 22 follows by keeping p ij = e −2i arg[y ij ] , U Q = 1 , U d = 1 and taking U u and U x respectively as the diagonal matrices appearing in Eq. 141,142.