$\mathcal{CP}$ violation with an unbroken $\mathcal{CP}$ transformation

A $\mathcal{CP}$ conserving $\mathrm{SU}(3)$ gauge theory is spontaneously broken to $\mathrm{T}_7$ by the vacuum expectation value (VEV) of a $\boldsymbol{15}$-plet. Even though the $\mathrm{SU}(3)$-$\,\mathcal{CP}$ transformation is not broken by the VEV, the theory exhibits physical $\mathcal{CP}$ violation in the broken phase. This is because the $\mathrm{SU}(3)$-$\,\mathcal{CP}$ transformation corresponds to the unique order-two outer automorphism of $\mathrm{T}_7$, which is not a physical $\mathcal{CP}$ transformation for the $\mathrm{T}_7$ states, and there is no other possible $\mathcal{CP}$ transformation. We explicitly demonstrate that $\mathcal{CP}$ is violated by calculating a $\mathcal{CP}$ odd decay asymmetry in the broken phase. This scenario provides us with a natural protection for topological vacuum terms, ensuring that $\theta\,G_{\mu\nu}\widetilde{G}^{\mu\nu}$ is absent even though $\mathcal{CP}$ is violated for the physical states of the model.


Introduction
Violation of the combined transformation of charge conjugation and parity (CP) has been observed in decays and oscillations of K and B mesons [1,2]. Complementary to that, violation of the time reversal transformation (T ) is implied by the CPT theorem and has been verified independently [3]. Likewise, recent global fits of neutrino oscillation data point towards CP violation also in the lepton sector [4]. Given that the measurement of leptonic mixing angles has entered its precision phase, and anticipating the near future measurement of neutrino masses, the experimental efforts to pin down all parameters which constitute the SM flavor puzzle is close to being completed.
Even though the observed fermion masses, mixings, and CP violation can be consistently parametrized in the framework of 3 × 3 (CKM [5] or PMNS [6]) unitary mixing, the overall theoretical situation is in many ways unsatisfactory. First of all, CP violation beyond the standard model is necessary in order to explain the baryon asymmetry of the universe [7]. Furthermore, CP violation is observed only in flavor changing transitions mediated by the weak interaction, not in strong interactions, thereby giving rise to a severe fine-tuning problem of the θ QCD parameter. Ultimately, the sheer multitude of parameters in the flavor sector and their distinct pattern cries out for an underlying fundamental explanation. Therefore, more theoretical work is necessary in order to unveil the underlying structure of the flavor puzzle, including the origin of CP violation.
From a formal point of view, CP transformations are quite particular. While P or C transformations individually relate otherwise disconnected representations of the spacetime and gauge symmetries of a model, a CP transformation maps each irreducible representation (irrep) to its own complex conjugate representation. This implies that in theories with a real valued action, CP cannot be broken maximally by simply leaving out or adding fields, in vast contrast to, for example, the parity transformation in a chiral theory. From a group-theoretical point of view this is reflected by the fact that CP transformations are special outer 1 automorphisms of the continuous [8], discrete [9,10] and space-time [11] symmetries of a model, which map the irreps of each group to their own complex conjugate representations [12]. This should be contrasted to other outer automorphisms, such as parity or charge conjugation for example, which map irreps to other irreps that may not be present in a model to begin with.
This notion defines CP transformations as special automorphisms of symmetry groups. However, it is not guaranteed that such an automorphism exists for a given symmetry group [10]. While it is known that CP outer automorphisms exist for simple Lie groups [8] and the Poincaré group [11] it has been pointed out that there are certain discrete groups which violate CP by the intrinsic complexity of their Clebsch-Gordan coefficients [13]. In more detail, these so-called "type I" groups prohibit simultaneous r i ↔ r i * (for all i, labeling the irreps of the corresponding group) transformations [10]. That is, these type I groups do not allow for outer automorphisms which could be identified with physical CP transformations for all of their irreps. If a model features such a group as global symmetry and a sufficiently large number of irreps, then CP transformations are not possible in consistency with the symmetry group of the model. Explicitly it has been found that the necessarily complex Clebsch-Gordan coefficients of type I groups then enter CP odd basis invariants, thereby giving rise to particle-antiparticle asymmetries in oscillations and decays [10]. Due to the fact that the arising physical CP-odd complex phases are discrete and calculable, this phenomenon has been termed explicit geometrical CP violation [13,14].
In this context, an important fact is that type I groups can arise as subgroups of simple Lie groups. This gives rise to the puzzling situation in which a Lie group G allows for a perfectly well-defined CP transformation, i.e. a particular complex conjugation outer automorphism, whereas the type I subgroup, H ⊂ G, does not. Thus, CP is not conserved at the level of H, and the question arises when and how CP is broken in a possible breaking of G → H. Naïvely, one might expect that in a dynamical setting it should be the vacuum expectation value (VEV) which gives rise to CP violation. Rather surprisingly, it turns out that this is not necessarily the case. We will show that the VEV which spontaneously breaks G → H does not break the complex conjugation outer automorphism, i.e. the VEV is CP conserving. Nevertheless, physical CP is violated at the level of H, and we will substantiate this claim by an explicit calculation of non-vanishing CP odd basis invariants that give rise to a physical decay asymmetry. The way the conundrum is resolved is the following: The conserved outer automorphism which gives rise to CP conservation at the level of G is conserved by the VEV and, hence, also a conserved outer automorphism at the level of H. Nevertheless, at the level of H this outer automorphism is no complex conjugation automorphism. Thus, once the physical states of a theory are H states, the conserved outer automorphism can no longer be interpreted as a physical CP transformation. An anticipated distinct outer automorphism transformation, which would correspond to a physical CP transformation at the level of H, is prohibited by the group structure of H, and it would also not be a consistent automorphism of G to begin with.
The investigation in this paper is based on an economic toy example. We use an SU(3) gauge theory which is broken to the type I group T 7 by a the VEV of a complex scalar φ, transforming in the 15-plet representation of SU(3) with the Dynkin indices (2, 1). We assume physical CP conservation at the level of SU(3) and show that the VEV does not break the corresponding outer automorphism. Yet we will find CP violating decays of a physical scalar to gauge bosons in the broken phase.
The rest of the paper is organized as follows. In Section 2, we define our model. Section 3 details the spontaneous breaking of SU(3) to T 7 . In Section 4, we discuss how the T 7 states are related to the states of the SU(3) theory. In Section 5, we show explicitly that CP is broken in the T 7 phase. Section 6 contains a comment on the θ term of the SU(3) theory. Finally, Section 7 contains our conclusions. Some details are deferred to the appendices.
Re σ 0 , Im σ 0 1 0 Table 1. Physical fields before and after the symmetry breaking. The number of real degrees of freedom before and after SSB coincide as 16 + 30 = 24 + 22.

The model
We consider an SU(3) gauge theory with the Lagrangean with D µ = ∂ µ − ig A µ being the ordinary gauge covariant derivative and G a µν the field strength tensors. The complex scalar φ is charged under SU(3), transforming in the 15plet representation. The scalar potential is given by where we take µ 2 > 0, and the I These invariants are real, which allows us to take λ i ∈ R without loss of generality. Further details on the derivation of the invariants can be found in Appendix A. There is a region in the parameter space in which this potential has a global minimum which gives rise to a VEV φ that spontaneously breaks SU(3) → T 7 [15,16]. The linear symmetry of the vacuum is T 7 , a discrete group with 21 elements. The physical spectrum after spontaneous symmetry breaking (SSB) is given in Table 1. We will give further details of T 7 , including its embedding in SU(3) and branching rules in Section 3.1.
The action derived from (2.1) is automatically invariant under the simultaneous outer automorphisms of SU(3) and the Lorentz group under which the gauge and scalar fields transform as Here P = diag(1, −1, −1, −1) is the usual spatial reflection, while R and U are the representation matrices of the outer automorphism of SU(3) fulfilling Table 2. Character table of T 7 . The arrows illustrate the action of the unique Z 2 outer automorphism of T 7 . We use the definitions ω := e 2πi/3 and η = ρ + ρ 2 + ρ 4 with ρ := e 2πi/7 .
Here, t a and f abc , with the usual relation [t a , t b ] = i f abc t c , are the generators and structure constant of the Lie algebra, respectively. At the level of SU (3), this outer automorphism maps all irreps to their own complex conjugate and, therefore, is the most general possible physical CP transformation. The transformation is conserved by the action implying that the model is CP symmetric in the unbroken phase. In addition, the VEV fulfills U φ * = φ and, therefore, does not break the outer automorphism. Surprisingly, we will find that even though the SU(3)-CP transformation is not broken, physically, CP is no longer conserved once SU(3) gets broken to T 7 .
As will be detailed below, the unique Z 2 outer automorphism of SU(3) corresponds to the unique Z 2 outer automorphism of T 7 . Therefore, the actual symmetry breaking chain of the model is given by (2.5) Crucially, the Z 2 transformation on the left-hand side corresponds to a CP transformation, while the identical transformation on the right-hand side cannot be interpreted as CP. In more detail, at the level of T 7 , the unique Z 2 outer automorphism acts as (cf. also Table 2) Out(T 7 ) : This transformation does not map all T 7 irreps to their own complex conjugate representation. Therefore, it does not correspond to a physical CP transformation if triplet and non-trivial singlet representations of T 7 are present simultaneously. This is the case in the given model. That is, the model does not allow for a physical CP transformation at the level of T 7 . Hence, the setting exhibits CP violating processes once SU(3) gets broken to T 7 . For definiteness, we will explicitly show in Section 5 that there is a CP asymmetry in the decay of a heavy charged scalar to massive gauge bosons of the broken SU(3) symmetry.
Here it should also be noted that any supposed physical CP transformation r T 7 ↔ r * T 7 is inconsistent with the structure of T 7 . For this reason the group has been classified as a finite group of "type I" in [10]. In fact, a supposed transformation r T 7 ↔ r * T 7 is not a consistent automorphism at the level of SU(3) to begin with. Imposing this transformation as a symmetry nonetheless, enforces g = λ i = 0, i.e. forbids all interactions.

Embedding
The discrete group T 7 can be generated by two elements with the presentation For the triplet representation we choose a basis in which a and b are represented by respectively, with ρ := e 2πi/7 .
The group elements of SU (3) for an arbitrary representation r are given by denote the generators of the representation r and α a the parameters. We emphasize that there are two generally different spaces which are relevant here. While both, [X (r) ] ij and [t (r) a ] ij , live in the "ij-space" there is also the "a" or adjoint space in which the parameters α as well as the gauge bosons live. This is important, because it is possible to choose bases independently for each of these spaces. 2 For practical reasons, we do not work in the standard Gell-Mann basis of the adjoint space but follow the basis choice of Fonseca's Mathematica package SusyNo [17]. The generators of the fundamental representation r = 3 used in this work are specified in Appendix B.
The discrete subgroup T 7 is embedded into SU(3) via the irreducible triplet representation. In the given basis we find that A and B are obtained from (3.3) by the choice of parameters The branching rules of representations under SU(3) → T 7 can be calculated with the help of [18]. Branchings relevant to this work are

Outer automorphism
Fonseca's basis choice has the virtue of being a CP basis [8], meaning that U = 1 in (2.4) irrespective of the specific representation. At the same time, the outer automorphism transformation in the adjoint space in this basis is given by which can easily be computed from (2.4). We stress that in contrast to the U 's, it is not possible to chose a basis in which R is trivial, for a non-trivial automorphism.
Since we ultimately will break SU(3) to T 7 , it is convenient to rotate the ij-space of the scalar representation to a T 7 -diagonal basis, in which (3.5b) is explicitly realized for the T 7 generators parametrized by (3.4). The corresponding matrix for the basis change is given in Appendix B. Most importantly, the transformation matrix of the outer automorphism, U (cf. Equation (2.4)), is not invariant under such a basis change. 3 In particular, we find that where 1 n denotes the n-dimensional unit matrix. This shows that the SU(3) outer automorphism acts like (2.6) on the T 7 representations.
Analogously, it is possible to choose a basis for the gauge bosons in the a-space such that (3.5a) is explicitly realized. The corresponding basis change is also given in Appendix B. We will see later that this basis also corresponds to the physical basis for the gauge bosons.
Using the basis changes in reverse, one can also show that the naïve CP transformation at the level of T 7 (which would have to take r i → r i * ∀ T 7 irreps i) does not fulfill (2.4) and, hence, does not correspond to a valid SU(3) automorphism.

Physical states in the T 7 phase
In the T 7 -diagonal basis of SU(3), and using unitary gauge, the scalar φ can be written as featuring 22 = 30 − 8 real degrees of freedom. In this basis, it is straightforward to identify the T 7 representations of the components as The VEV in this basis is simply given by Minimizing the potential one finds that In what follows, we will choose v to be real and positive without loss of generality. The physical T 7 states of the gauge bosons are complex linear combinations of the A µ a 's. The massive and T 7 charged gauge bosons are given by They obtain masses The physical scalars arising from φ are mixtures of the fields listed in (4.2). For the one-dimensional representations one finds with masses m 2 Re σ 0 = 2 µ 2 , m 2 Im σ 0 = 0 , (4.8a) The massless mode can be understood noting that Im σ 0 is the Goldstone boson of an additional global U(1) symmetry of the potential (2.2) which is spontaneously broken by φ . This symmetry prohibits a possible cubic coupling term for φ. The massless mode can be avoided by softly breaking the U(1) via a reintroduction of the cubic term. Alternatively one could also gauge the additional U(1) upon which the would-be Goldstone boson Im σ 0 gets eaten by the U(1) gauge boson. Either way, this mode does not play any role in our discussion.
In contrast to the one dimensional representations, the triplet representations appear in identical copies. Therefore, the physical states are mixtures of T 1 , T 2 , and T * 3 , with mixing parameters depending on the potential parameters λ i . The physical states are given by The mixing matrix V is an orthogonal matrix with entries depending on the potential parameters V ij = V ij (λ k ) which are too complicated to be displayed here. 4 Instead of discussing the most general case, we will settle to a specific set of potential parameters, which is sufficient to prove our point. We choose the parameter values 5 as well as The hierarchies appearing here originate from the mild hierarchies in λ 1,2 /λ 5 .

Physical CP violation in the T 7 phase
Given the physical fields, the stage is set for an explicit proof of physical CP violation in the broken phase. The conserved outer automorphism, which corresponds to the CP transformation at the level of SU(3), acts on the physical states as Clearly, this does not correspond to a physical CP transformation. 4 While the mixing of physical states is dependent on the potential parameters, we note that the composition of the unphysical Goldstone bosons is independent of the potential parameters. This allows for the general choice (4.1). 5 One should not be bothered by the fact that λ3,4 = 0, as there are also global T7 minima for parameter choices λi = 0 ∀i. This is easy to see for the quantum theory, where 2) and the transformation (5.1) corresponds to a map In contrast, a physical CP transformation of the complex scalar field σ 1 (x) would be a map σ 1 (x) → σ * 1 (x), corresponding to (see e.g. [19,20]) It is straightforward to confirm that this naïve physical CP transformation of the T 7 states, or any generalization thereof, is not a symmetry of the action. In fact, this directly follows from the fact that no class-inverting automorphism exists for T 7 . This already shows that physical CP is violated in the T 7 phase.
In order see this more explicitly, we construct CP-odd basis invariants and show that they give rise to an observable CP asymmetry in the decay of a heavy charged scalar σ 1 to a pair of mutually conjugate heavy gauge boson triplets. Here M (i → f ) denotes the corresponding matrix element.
In a perturbative expansion, CP violation arises from interference terms of diagrams that feature physical CP-odd phases (cf. e.g. [19]). Physical observables must be independent of basis choices for all internal spaces and, therefore, can only depend on basis invariant quantities. An alternative way to a diagrammatic expansion, thus, is to construct basis invariants directly. The basis invariant approach is eminently useful and widely used in the context of CP violation for example in the standard model [21,22], but also in extensions with multiple families [23], additional scalars [24][25][26][27][28][29] or for theories with discrete symmetries [14,30]. 6 The reason is that CP-odd invariants can often be constructed without the need of performing involved calculations, even if CP violation is arising only at higher loop order. 7 In addition, it has been argued that the appearance of a single CP odd basis invariant is enough in order to show that a model is CP violating. To the best of our knowledge, however, it is not known how basis invariants are related to physical observables in general. This means that even if CP odd invariants arise in a given model, it is still a logical possibility that the invariants delicately cancel against one another in all possible processes. Therefore, in addition to constructing specific CP-odd basis invariants we also give a specific process for which we have checked that the invariants do not (all) cancel against one another. For the choice of parameters given in equation (4.10) the decay σ 1 → W W * is kinematically allowed if g 0.822. The relevant couplings can be obtained by deriving the Lagrangean in the broken phase after the physical fields. The tree-level coupling of σ 1 to the charged gauge bosons is given by Loop corrections to this vertex are possible, for example with triplets τ 2 in the loop (cf. Figure 1). The relevant couplings for this correction are given by While Y σ 1 W W * can easily be stated in a closed form, Y σ 1 τ 2 τ * 2 and Y τ * 2 W W * are in general complicated functions of the potential parameters λ i , see Equations (D.4) and (D.5) in Appendix D, where we also give general expression of the couplings independently of the potential parameters. From these, in general basis dependent, couplings it is straightforward to construct basis invariant quantities via contractions. For that, indices should be contracted such that basis transformations cancel. 8 From the given couplings, we find two CP-odd basis invariant contractions The decay asymmetry ε σ 1 →W W * (cf. Equation (5.5)) receives contributions proportional to 2 Im I 1,2 = I 1,2 − I * 1,2 . For our choice of parameters one finds Im I 1 = + 0.090 v 4 g 6 and Im I 2 = − 0.126 v 4 g 6 , (5.9) clearly indicating the presence of CP violation. Inspecting the general expressions for the invariants (5.8) together with the general expressions of the couplings in Appendix D, we note that all contributing complex phases are parameter independent and arise from the projection of SU(3) Clebsch-Gordan coefficients onto the T 7 subgroup. However, the geometric phases are weighted by functions of the continuous parameters λ i of the potential. Therefore, the resulting phases of I 1 and I 2 depend on the potential parameters. Furthermore, we note that the two invariants are closely related to each other by the relation which is dictated by the T 7 Z 2 symmetry.
There are additional contributions to ε σ 1 →W W * from other one loop diagrams, for example those containing τ 1,3 or gauge bosons running in the loop. We have explicitly checked that these contributions do not cancel the asymmetry.
Note that all relevant couplings as well as the decay asymmetry are proportional to positive powers of v. Therefore, there is no physical CP violation when the SU(3) symmetry is restored by taking the limit v → 0.
6 Natural protection of θ = 0 in the broken phase Finally, let us observe how a possible θ-term [32][33][34] is affected if CP violation arises in the way described above. The usual topological term where G a µν is the field strength and G µν,a := 1 2 ε µνρσ G a ρσ its dual, is odd under parity or time-reversal transformations. Therefore, θ = 0 is enforced by the transformation (2.3a) and L θ is absent from the theory. Crucially, the transformation (2.3a) is not broken by the VEV. Thus, even though it is not possible to interpret the transformation (2.3a) as a physical CP transformation of the physical states in the broken phase, the transformation is unbroken and warrants that θ = 0, not only for the broken SU(3), but also for other gauge groups. We stress that our model is not realistic and does not even contain fermions. Nevertheless, it is tempting to speculate that the observed mechanism does provide a symmetry based solution to the strong CP problem in more realistic settings. symmetry continues to forbid θ QCD , while physical CP is violated in the flavor sector. Of course, the Z 2 must then also be preserved when the remaining T 7 symmetry is spontaneously broken. A detailed study of this new avenue in flavor model building is beyond the scope of this work, but will be explored elsewhere.

Summary and Discussion
We have studied how one obtains a T 7 toy model from a CP conserving SU(3) theory by spontaneous breaking. T 7 is a so-called type I group, i.e. it is not possible to impose a physical CP transformation on a T 7 model with generic field content while maintaining the interactions of the theory. This reflects the fact that there is no basis for T 7 in which all Clebsch-Gordan coefficients are real.
We paid particular attention to the fate of the SU(3)-CP transformation. We found that it is not spontaneously broken by the 15-plet VEV which breaks SU (3)  Stated in simple terms, CP is violated because there are T 7 states which emerge as complex linear combinations of certain SU (3) states. An example is the Z boson, which transforms as complex T 7 1 1 -plet but does not get conjugated under the T 7 outer automorphism transformation (cf. Figure 2). That is, the outer automorphism cannot be interpreted as a CP transformation at the level of T 7 , and physical CP is violated. We have demonstrated this explicitly by establishing a decay asymmetry in the decay of a complex scalar to to massive gauge bosons.
Our findings have interesting physical consequences. The definition of matter and antimatter, at least with respect to a CP mirror, is not universally possible for chains of groups and subgroups. Rather, the definition of matter and antimatter depends on the underlying unbroken symmetry. This has profound implications for cosmology, where the symmetries of the ground state change in the course of the evolution of the universe.
Interestingly, the θ parameters of SU(3) and other gauge groups remain forbidden by the outer automorphism, also in the broken phase. This may allow one to construct realistic models, in which CP is broken in the flavor sector, but θ QCD is forbidden by the outer automorphism.
In our analysis, we have restricted ourselves to only one simple Lie group, SU(3), and one type I symmetry, T 7 . It will be interesting to generalize the discussion to other groups with richer outer automorphism structure and include fermions in the discussion.
vorty on a related project during the early stages of this work. This work has been supported by the German Science Foundation (DFG) within the SFB-Transregio TR33 "The Dark Universe".

A Details of the quartic SU(3) invariants
There are in total 14 linearly independent SU (3)

B Details of SU(3) and the T 7 -diagonal basis
The basis we choose for the SU(3) triplet generators has been given by Fonseca [35] and it is implemented in the SusyNo Mathematica package [17]. The generators are given by In order to obtain a basis in which the T 7 elements A and B are block-diagonal we rotate the (ij-space) basis of the generators given in SusyNo according to The corresponding rotation matrix is given by Due to the degeneracy of T 7 representations in the 15-plet, we note that there is a degeneracy in this basis choice corresponding to rotations among T 7 triplets and among anti-triplets, respectively. We have chosen our basis such that the (anti-)triplet Goldstone modes reside only in one of the (anti-)triplets, thereby allowing for the simplest form of Equation (4.1).
In order to obtain a basis for the adjoint space in which the T 7 elements A and B are block-diagonal we rotate the (a-space) basis of the generators according to 9 The corresponding transformation matrix is given by This, of course, is consistent with (4.5), where C Details of T 7 The group T 7 has been used in flavor model building [36][37][38][39][40][41][42][43][44] also motivated by the fact that T 7 is the smallest finite subgroup of SU(3) which has an irreducible triplet representation. T 7 is the smallest non-Abelian finite subgroup of SU(3) which is not also a subgroup of SU(2) or SO(3) (cf. e.g. [16]). Further details of T 7 can, for example, be found in [45], whose conventions we follow. T 7 is implemented in GAP [46] as SmallGroup SG(21, 1).
The outer automorphism group of T 7 has order two, and is generated by the transformation u : (a, b) → (a 6 , b) . (C.1) The action of the outer automorphism on the irreps has already been given in (2.6), and it is clearly not class-inverting. We note that the chosen basis (3.2) is an eigenbasis of the outer automorphism, meaning that the consistency condition [9,47,48] U A * U −1 = A 6 , U B * U −1 = B , (C.2) is solved by U = 1. 9 The transformation of the gauge bosons is always given by t