A Group theory of A_{4}

# Generalised CP and A4 Family Symmetry

## Abstract

We perform a comprehensive study of family symmetry models based on combined with the generalised CP symmetry . We investigate the lepton mixing parameters which can be obtained from the original symmetry breaking to different remnant symmetries in the neutrino and charged lepton sectors. We find that only one case is phenomenologically viable, namely in the neutrino sector and in the charged lepton sector, leading to the prediction of no CP violation, namely and the Majorana phases and are all equal to either zero or . We then propose an effective supersymmetric model based on the symmetry in which trimaximal lepton mixing is predicted together with either zero CP violation or with non-trivial Majorana phases. An ultraviolet completion of the effective model yields a neutrino mass matrix which depends on only three real parameters. As a result of this, all three CP phases and the absolute neutrino mass scale are determined, the atmospheric mixing angle is maximal, and the Dirac CP can either be preserved with or maximally broken with and sharp predictions for the Majorana phases and neutrinoless double beta decay.

## 1 Introduction

After the measurement of the reactor mixing angle by the Daya Bay [1], RENO [2], and Double Chooz [3] reactor neutrino experiments, all three lepton mixing angles , , and both mass-squared differences and have been measured to reasonably good accuracy. Yet within the standard framework of three-neutrino oscillations, the Dirac CP phase and neutrino mass ordering still elude measurement so far. Furthermore, if neutrinos are Majorana particles, there exist two more unknown Majorana CP phases which may play a role in neutrinoless double-beta decay searches. Thus, determining the exact neutrino mass ordering and measuring the Dirac and Majorana CP violating phases are the primary goals of future neutrino oscillation experiments. The CP violation has been firmly established in the quark sector and it is natural to expect that CP violation occurs in the lepton sector as well. It is insightful to note that hints of a nonzero have begun to show up in global analysis of neutrino oscillation data [4, 5, 6].

What would we learn from the measurements of the lepton CP violating phases? What is the underlying physics? These questions are particularly imperative in view of foreseeable future experimental programs to measure the CP-violation in the neutrino oscillations sector. In the past years, much effort has been devoted to explaining the structure of the lepton mixing angles through the introduction of family symmetries. In this scheme, one generally assumes a non-abelian discrete flavour group which is broken to different subgroups in the neutrino and charged lepton sectors. The mismatch between these two subgroups leads to particular predictions for the lepton mixing angles. For recent reviews, see Ref. [7] and Ref. [8] for the model building and relevant group theory aspects, respectively. Motivated by this approach one can extend the family symmetry to include a generalised CP symmetry  [9] which will allow the prediction of both CP phases and mixing angles.

The possibility of combining a family symmetry with a generalised CP symmetry has already been discussed in the literature. For example, the simple reflection symmetry, which is a combination of the canonical CP transformation and the exchange symmetry, has been discussed and successfully implemented in a number of models where both atmospheric mixing angle and Dirac CP phase were predicted to be maximal[10, 11, 12]. Additionally in Ref. [13], the phenomenological consequences of imposing both an flavour symmetry and a generalised CP symmetry have been analysed in a model-independent way. They found that all lepton mixing angles and CP phases depend on one free parameter for the symmetry breaking of to in the neutrino sector and to some abelian subgroup of in the charged lepton sector. Concrete family models with a generalised CP symmetry have been constructed in Refs. [14, 15, 16] where the spontaneous breaking of the down to in the neutrino sector was implemented. Other models with a family symmetry and a generalised CP symmetry can also be found in Refs. [17, 18, 19]. In addition, there are other theoretical frameworks comprising both family symmetry and CP violation [20, 21, 22].

In this work, we study generalised CP symmetry in the context of the most popular family symmetry 4 (please see Ref. [25, 26] for a classification of the models on the market). The generalised CP transformation compatible with an family symmetry is clarified, and a model-independent analysis of the lepton mixing matrix is performed by scanning all of the possible remnant subgroups in the neutrino and charged lepton sectors. We construct an effective model, where non-renormalisable operators are involved. The lepton mixing is predicted to be trimaximal pattern in the model, and the Dirac phase is trivial or nearly maximal. Furthermore, this effective model is promoted to a renormalisable one in which the higher order operators are under control.

The remainder of this paper is organised as follows. In Section 2, we present the general CP transformations consistent with the family symmetry. In Section 3, we perform a thorough scan of leptonic mixing parameters which can be obtained from the remnant symmetries of the underlying combined symmetry group . We find that only one case out of all possibilities is phenomenologically viable. This case predicts both Dirac and Majorana phases to be trivial. In Section 4 we specify the structure of the model at leading order, and the required vacuum alignment is justified. In subsection 4.3, we analyse the subleading Next-to-Leading-Order (NLO) corrections induced by higher dimensional operators and phenomenological predictions of the model are presented. In Section 5, we address the ultraviolet completion of the model which significantly increases the predictability of the theory such that all the mixing angles, CP phases and the absolute neutrino mass scale are fixed. We conclude in Section 6. The details of the group theory of are collected in Appendix A and Appendices B-D contain the implications of preserving other subgroups of different than and . Finally, Appendix E describes the diagonalisation of a general symmetric complex matrix.

## 2 Generalised CP transformations with family symmetry

### 2.1 General family symmetry group

In general, it is nontrivial to combine the family symmetry and the generalised CP symmetry together because the definition of the generalised CP transformations must be compatible with the family symmetry. Thus, the generalised CP transformations are subject to certain consistency conditions [27, 13, 28]. Namely, for a set of fields in a generic irreducible representation of , it transforms under the action of as

 φ(x)\lx@stackrelGf⟶ρr(g)φ(x),g∈Gf, (2.1)

where denotes the representation matrix for the element in the irreducible representation , the generalised CP transformation is of the form

 φ(x)\lx@stackrelCP⟶Xrφ∗(x′), (2.2)

where and the obvious action of CP on the spinor indices is omitted for the case of being spinor. Here we are considering the “minimal” theory in which the generalised CP transforms the field into its complex conjugate , and the transformation into another field with is beyond the present scope since both and would be required to be present in pair and correlated with each other in that case. Notice that should be a unitary matrix to keep the kinetic term invariant. Now if we first perform a CP transformation, then apply a family symmetry transformation, and finally an inverse CP transformation is followed, i.e.

 φ(x)\lx@stackrelCP⟶Xrφ∗(x′)\lx@stackrelGf⟶Xrρ∗r(g)φ∗(x′)\lx@stackrelCP−1⟶Xrρ∗r(g)X−1rφ(x), (2.3)

the theory should still be invariant since it is invariant under each transformation individually. To make the theory consistent the resulting net transformation should be equivalent to a family symmetry transformation of some family group element , i.e.

 Xrρ∗r(g)X−1r=ρr(g′),g′∈Gf, (2.4)

where the elements and must be the same for all irreducible representations of . Eq. (2.4) is the important consistency condition which has to be fulfilled in order to impose both generalised CP and family symmetry invariance simultaneously. It also implies that the generalised CP transformation maps the group element into and that the family group structure is preserved under this mapping. Therefore Eq. (2.4) defines a homomorphism of the family symmetry group . Notice that in the case where is a faithful representation, the elements and have the same order, the mapping defined in Eq. (2.4) is bijective, and thus the associated CP transformation becomes an automorphism [28]. It is notable that both and also satisfy the consistency equation of Eq. (2.4) for a generalised CP transformation , where is real and is any element of . Therefore the possible form of the CP transformation is only determined by the consistency equation up to an overall arbitrary phase and family symmetry transformation for a given irreducible representation . In the following, we investigate the generalised CP transformations consistent with an family symmetry for different irreducible representations, i.e. .

### 2.2 A4 family symmetry

The group can be generated by two generators and , which are of orders two and three, respectively (see Appendix A for the details of the group theory of ). To include a generalised CP symmetry consistent with an family symmetry, it is sufficient to only impose the consistency condition in Eq. (2.4) on the group generators:

 Xrρ∗r(S)X−1r=ρr(S′),Xrρ∗r(T)X−1r=ρr(T′). (2.5)

To do this, we start with the faithful triplet representation . Then the order of and will be 2 and 3, respectively. Therefore and can only belong to certain conjugacy classes of . Namely,

 S′∈3C2,T′∈4C3∪4C23 (2.6)

It is remarkable that the consistency condition of Eq. (2.4) must hold for all representations simultaneously. However, because of the models constructed in later sections, we assume that our theory contains only one of the nontrivial singlet irreducible representations (either or ) in the flavon sector and further restrict ourselves to a minimal case where there exists only one flavon transforming under that nontrivial singlet irreducible representation (in addition to other flavons transforming under the and representations). However, in these models there does exist a and in the matter sector. Yet, additional symmetry forbids the interchanging of these fields under the generalised CP symmetry. Therefore we have chosen to define a generalised CP symmetry without the interchanging of fields transforming under conjugate representations, e.g. fields transforming under and representations. Then, the element can further be constrained by these nontrivial singlet representations and , where the corresponding generalised CP transformations are numbers with absolute value equal to 1, and then we have

 ρ1′,1′′(T′)=X1′,1′′ρ∗1′,1′′(T)X−11′,1′′=ρ∗1′,1′′(T)=ω∓2 (2.7)

Consequently, the element can only be in the conjugacy class . In summary, the consistency equation applied to our “minimal” case restricts and to

 S′∈3C2,T′∈4C23. (2.8)

For the simple case of and in the -dimensional representation, the associated CP transformation satisfying Eq. (2.4) can be found straightforwardly:

 X0=⎛⎜⎝100010001⎞⎟⎠≡\mathbbm13, (2.9)

which is the canonical CP transformation. The remaining eleven possible choices for and lead to different solutions for . These solutions are listed in Table 1 and can be neatly summarised in a compact way:

 X3=ρ3(g),g∈A4. (2.10)

For the singlet representations , and , we take

 X1,1′,1′′=ρ1,1′,1′′(g),g∈A4. (2.11)

Therefore the generalised CP transformation consistent with an family symmetry is of the same form as the family group transformation, i.e.

 Xr=ρr(g),g∈A4. (2.12)

Now that we have found all generalised CP transformations consistent with the family symmetry,5 we proceed by investigating their implications on lepton masses and mixings.

## 3 General analysis of lepton mixing from preserved family and CP symmetries

### 3.1 General family symmetry

To obtain definite predictions for both the lepton mixing angles and CP violating phases from symmetry, we impose the family symmetry and the generalised CP symmetry simultaneously at high energies. Then the family symmetry is spontaneously broken to the and subgroups in the neutrino and the charged lepton sector respectively, and the remnant CP symmetries from the breaking of are and , respectively. The mismatch between the remnant symmetry groups and gives rise to particular values for both mixing angles and CP phases. As usual, the three generations of the left-handed (LH) lepton doublets are unified into a three-dimensional representation of . The invariance under the residual family symmetries and implies that the neutrino mass matrix and the charged lepton mass matrix satisfy

 ρT3(gνi)mνρ3(gνi)=mν,gνi∈Gν, ρ†3(gli)mlm†lρ3(gli)=mlm†l,gli∈Gl. (3.1)

where the charged lepton mass matrix is given in the convention in which the left-handed (right-handed) fields are on the left-hand (right-hand) side of . Moreover, the neutrino and the charged lepton mass matrices are constrained by the residual CP symmetry via

 XT3νmνX3ν=m∗ν,X3ν∈HνCP, X†3lmlm†lX3l=(mlm†l)∗,X3l∈HlCP. (3.2)

Since there are both remnant family and CP symmetries, the corresponding consistency equation similar to Eq. (2.4) has to be satisfied. Namely, the elements of and of should satisfy

 Xrνρ∗r(gνi)X−1rν=ρr(gνj),gνi,gνj∈Gν, Xrlρ∗r(gli)X−1rl=ρr(glj),gli,glj∈Gl. (3.3)

Given a set of solutions and , we can straightforwardly check that and are solutions as well. The invariance conditions of Eqs. (3.1)-(3.2) allow us to reconstruct the mass matrices and , and eventually determine the lepton mixing matrix . Furthermore, if two other residual family symmetries and are conjugate to and under the element , i.e.

 G′ν=hGνh−1,G′l=hGlh−1, (3.4)

then the associated residual CP symmetries and are related to and as

 Hν′CP=ρr(h)HνCPρTr(h),Hl′CP=ρr(h)HlCPρTr(h), (3.5)

and the corresponding neutrino and charged lepton mass matrices are of the form

 m′ν=ρ∗3(h)mνρ†3(h),m′lm′†l=ρ3(h)mlm†lρ†3(h). (3.6)

Therefore, the remnant subgroups and lead to the same mixing matrix as and do.

Having completed a general discussion of the implementation of a generalised CP symmetry with a family symmetry, we now concentrate on the case of interest in which the family symmetry and a generalised CP symmetry consistent with is imposed. Thus, the theory respects the full symmetry . In the following, we perform a model independent study of the constraints that these symmetries impose on the neutrino mass matrix, the charged lepton mass matrix and the PMNS matrix by scanning all the possible remnant symmetries and . We begin this study with an analysis of the neutrino sector.

### 3.2 Neutrino sector from a subgroup of A4⋊HCP

As shown in Appendix B, the case is not phenomenologically viable. To resolve this issue, we assume that the underlying symmetry is broken into 6 in the neutrino sector [13]. Since the three subgroups in Eq. (A.6) are related by conjugation as and , it is sufficient to only consider , where the element of should satisfy

 Xrνρ∗r(S)X−1rν=ρr(S). (3.7)

It is found that only 4 of the 12 non-trivial CP transformations are acceptable7,

 HνCP={ρr(1),ρr(S),ρr(T2ST),ρr(TST2)}. (3.8)

Thus, the neutrino mass matrix is constrained by

 ρT3(S)mνρ3(S)=mν, (3.9) XT3νmνX3ν=m∗ν, (3.10)

where Eq. (3.9) is the invariance condition under , and it implies that the neutrino mass matrix is of the form

 mν=α⎛⎜⎝2−1−1−12−1−1−12⎞⎟⎠+β⎛⎜⎝100001010⎞⎟⎠+γ⎛⎜⎝011110101⎞⎟⎠+ϵ⎛⎜⎝01−11−10−101⎞⎟⎠, (3.11)

where , , and are complex parameters, and they are further constrained by the remnant CP symmetry shown in Eq. (3.10). In order to diagonalise the neutrino mass matrix in Eq. (3.11), we first apply the tri-bimaximal transformation to yield

 m′ν=UTTBmνUTB=⎛⎜⎝3α+β−γ0−√3ϵ0β+2γ0−√3ϵ03α−β+γ⎞⎟⎠, (3.12)

where

 UTB=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝√231√30−1√61√3−1√2−1√61√31√2⎞⎟ ⎟ ⎟ ⎟ ⎟⎠. (3.13)

Now we return to the investigation of the residual CP symmetry constraint of Eq. (3.10). Two distinct phenomenological predictions arise for the different choices of :

• For this case, we see that we can straightforwardly solve Eq. (3.10) and find that all four parameters , , and are real. Then can be further diagonalised by

 U′Tνm′νU′ν=diag(m1,m2,m3),U′ν=R(θ)P, (3.14)

where is a unitary diagonal matrix with entries or which renders the light neutrino masses positive, and

 R(θ)=⎛⎜⎝cosθ0sinθ010−sinθ0cosθ⎞⎟⎠ (3.15)

is a rotation matrix with

 tan2θ=√3ϵβ−γ. (3.16)

This diagonalisation reveals that the light neutrino masses are given by

 m1=∣∣∣3α+sign((β−γ)cos2θ)√(β−γ)2+3ϵ2∣∣∣, m2=|β+2γ|, m3=∣∣∣3α−sign((β−γ)cos2θ)√(β−γ)2+3ϵ2∣∣∣. (3.17)

We conclude that this case is acceptable.

• In this case, it can be seen that the of Eq. (3.11) is purely imaginary, and the remaining parameters , and are real. Then the hermitian combination turns out to be of the form:

 m′†νm′ν=diag(−9α2+(β−γ)2+3ϵ2,(β+2γ)2,−9α2+(β−γ)2+3ϵ2), (3.18)

which implies . Clearly, this is not consistent with the experimental observation that the three light neutrinos have different masses. Note that the generalised CP transformations are not symmetric in the chosen basis, and hence we confirm the argument of Ref. [13] that non-symmetric CP transformations consistent with the remnant family symmetry in the neutrino sector lead to partially degenerate neutrino masses.

Since the remaining choices or are related to the discussed case by conjugation, the corresponding remnant CP symmetry is or , respectively, where is given by Eq. (3.8). Then their corresponding neutrino mass matrices are of the form or , respectively, with given in Eq. (3.11). Now that we have finished a systematic discussion of the effects of the residual flavour and CP symmetries on the neutrino mass matrix, we turn to analyse their effects on the charged lepton mass matrix.

### 3.3 Charged lepton sector from a subgroup of A4⋊HCP

In Appendices C and D we consider the cases and and show that they are not phenomenologically viable. Here we consider the successful case that is one of the subgroups shown in Eq. (A.7). Since the four subgroups are conjugate to each other, i.e.

 (TST2)ZT3(TST2)−1=ZST3,(T2ST)ZT3(T2ST)−1=ZTS3,SZT3S=ZSTS3, SZST3S=ZTSS,(T2ST)ZST3(T2ST)−1=ZSTS3,(TST2)ZTS3(TST2)−1=ZSTS3, (3.19)

we choose for demonstration. Then the combined symmetry group is broken to in the charged lepton sector. The element of should satisfy the consistency equation8

 Xrlρ∗r(T)X−1rl=ρr(T2). (3.20)

It is found that the remnant CP transformation can be

 Missing \left or extra \right (3.21)

Similar to the neutrino mass matrix, the charged lepton mass matrix must respect both the residual family symmetry and the generalised CP symmetry , i.e.

 ρ†3(T)mlm†lρ3(T)=mlm†l, ρ†3(1)mlm†lρ3(1)=(mlm†l)∗, (3.22)

where from Eq. (3.21) has been taken. For the value or , the resulting constraint is equivalent to Eq. (3.22). One can easily see that is diagonal in this case,

 mlm†l=diag(m2e,m2μ,m2τ), (3.23)

where , and are the electron, muon and tau masses, respectively. For the other choices and , the corresponding residual CP symmetry and the mass matrix follow from the general relations Eq. (3.5) and Eq. (3.6) immediately with and , respectively.

### 3.4 Lepton mixing from A4⋊HCP broken to GνCP≅ZS2×HνCP and GlCP≅ZT3⋊HlCP

In the context of family symmetry and its extension of including generalised CP symmetry, a specific lepton mixing pattern arises from the mismatch between the symmetry breaking in the neutrino and the charged lepton sectors. In this section, we perform a comprehensive analysis of all possible lepton mixing matrices obtainable from the implementation of an family symmetry and its corresponding generalised CP symmetry by considering all possible residual symmetries and discussed in previous sections.

Immediately we can disregard the cases predicting partially degenerate lepton masses. Therefore, breaking to the subgroups or will be neglected in the following. Furthermore, in order that the elements of and give rise to the entire family symmetry group , we take to be one of the subgroups shown in Eq. (A.7). Then, there are combinations for and . However, we find that all of these are conjugate to each other9. As a result, all possible symmetry breaking chains of this kind lead to the same lepton mixing matrix . This important point is further confirmed by straightforward calculations which are lengthy and tedious.

Without loss of generality, it is sufficient to consider the representative values and , and the original symmetry is broken to in the neutrino sector and in the charged lepton sector, where 10 and . In this case, is diagonal as shown in Eq. (3.23). Therefore, no rotation of the charged lepton fields is needed to get to the mass eigenstate basis, and the lepton mixing comes completely from the neutrino sector. In the PDG convention [29], the PMNS matrix is cast in the form

 UPMNS=Vdiag(1,eiα212,eiα312), (3.24)

with

 V=⎛⎜⎝c12c13s12c13s13e−iδCP−s12c23−c12s23s13eiδCPc12c23−s12s23s13eiδCPs23c13s12s23−c12c23s13eiδCP−c12s23−s12c23s13eiδCPc23c13⎞⎟⎠. (3.25)

where we use the shorthand notation and , is the Dirac CP phase, and are the Majorana CP phases. Using this PDG convention we find that the resulting PMNS matrix is:

 UPMNS=UTBR(θ)P=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝2√6cosθ1√32√6sinθ−1√6cosθ+1√2sinθ1√3−1√6sinθ−1√2cosθ−1√6cosθ−1√2sinθ1√3−1√6sinθ+1√2cosθ⎞⎟ ⎟ ⎟ ⎟ ⎟⎠P, (3.26)

where as shown previously is a unitary diagonal matrix with entries or and and are given in Eq. (3.13) and Eq. (3.15). Hence, the lepton mixing angles and CP phases are

 sinδCP=sinα21=sinα31=0, sin2θ13=23sin2θ,sin2θ12=12+cos2θ=13cos2θ13,sin2θ23=12[1+√3sin2θ2+cos2θ], (3.27)

which implies the three CP phases , , , and therefore there is no CP violation in this case. Note that the same results are found in Ref. [13].

To summarise the arguments of the preceding section, if one imposes the symmetry , which is spontaneously broken to certain residual family and CP symmetries in order to obtain definite predictions for mixing angles and CP phases, then only the symmetry breaking of to in the neutrino sector and in the charged lepton sector can lead to lepton mixing angles in the experimentally preferred range. However, there is no CP violation in this case. This is consistent with the result found for for the case where with [14]. For it was possible to achieve maximal CP violation for the case with . This case is not directly accessible for since the generator is absent, although it is accidentally present at LO in the models that we now discuss.

## 4 Model with A4 and generalised CP symmetries

Guided by the general analysis of previous sections, we construct an effective model in this section. The predictions of Eq. (3.27) are realised if the remnant CP is preserved otherwise the Dirac CP phase is approximately maximal. The model is based on , which is supplemented by the extra symmetries . The auxiliary symmetry