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Lepton Flavor Mixing and CP Symmetry

###### Abstract

The strategy of constraining the lepton flavor mixing from remnant CP symmetry is investigated in a rather general way. The neutrino mass matrix generally admits four remnant CP transformations which can be derived from the measured lepton mixing matrix in the charged lepton diagonal basis. Conversely, the lepton mixing matrix can be reconstructed from the postulated remnant CP transformations. All mixing angles and CP violating phases can be completely determined by the full set of remnant CP transformations or three of them. When one or two remnant CP transformations are preserved, the resulting lepton mixing matrix would depend on three real parameters or one real parameter respectively in addition to the parameters characterizing the remnant CP, and the concrete form of the mixing matrix is presented. The phenomenological predictions for the mixing parameters are discussed. The conditions leading to vanishing or maximal Dirac CP violation are studied.

## 1 Introduction

The origin of flavor mixing is one of longstanding open questions in particle physics. Firstly motivated by the well-known tri-bimaximal mixing [Harrison:2002er], a considerable effort has been devoted to understanding lepton mixing from a discrete flavor symmetry which is spontaneously broken down to two different residual subgroups in the neutrino and the charged lepton sectors. Please see Refs. [Altarelli:2010gt, Ishimori:2010zr, Grimus:2012dk, King:2013eh, King:2014nza] for review of discrete flavor symmetries and their application in model building aspects. So far a complete classification of all possible lepton mixing which could be derived from a finite flavor symmetry group under the hypothesis of Majorana neutrino has been accomplished [Fonseca:2014koa]. Among the complete list of mixing patterns achievable, only the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix with the second column being (1,1,1)^{T}/\sqrt{3} can be compatible with experimental data, and the Dirac CP-violating phase is fixed to trivial. Moreover, the Majorana phases are indeterminate as the neutrino masses are unconstrained by flavor symmetry.

On the experimental side, the precise measurements of the reactor mixing angle \theta_{13} by T2K [Abe:2011sj], MINOS [Adamson:2011qu], DOUBLE-CHOOZ [Abe:2011fz], RENO [Ahn:2012nd] and DAYA-BAY [An:2012eh] reactor neutrino experiments is one of the most significant discoveries in recent years. The sizable \theta_{13}\sim 9^{\circ} opens the gateway to access two remaining unknown parameters in the neutrino sector: the neutrino mass hierarchy and the leptonic Dirac CP phase \delta_{CP}. If neutrinos are Majorana particles, there are two additional Majorana CP phases which can play a critical role in the neutrinoless double beta decay, and we know nothing about their values so far. The T2K collaboration reported a weak evidence for nonzero \delta_{CP}\sim 3\pi/2 [Abe:2013hdq], and some indications of nontrivial \delta_{CP} are starting to appear in global analysis of neutrino oscillation data [Capozzi:2013csa, Forero:2014bxa, Gonzalez-Garcia:2014bfa]. Needless to say, probing CP violation in the lepton sector would help deepen our understanding of the universe. Some long-baseline neutrino oscillation experiments such as LBNE [Adams:2013qkq], LBNO [::2013kaa] and Hyper-Kamiokande [Abe:2011ts] have been proposed to precisely measure the lepton mixing parameters in particular the Dirac CP phase \delta_{CP}.

If the signal of CP violation is observed in future neutrino oscillation apparatuses, the paradigm of the flavor symmetry would be disfavored. Moreover, in light of the hints for maximal Dirac CP violation \delta_{CP}\sim 3\pi/2, it is imperative and significative to be able to understand the observed lepton mixing angles and meanwhile predict the values of CP phases from certain underlying principles. It is notable that CP symmetry was found to impose strong constraints on the fermion mass matrices nearly thirty years ago [Ecker:1981wv, Grimus:1995zi]. A typical simple CP transformation is the so-called \mu-\tau reflection under which \nu_{\mu} and \nu_{\tau} transform into the CP-conjugate of each other [Harrison:2002kp, Grimus:2003yn, Farzan:2006vj]. A neutrino mass matrix fulfilling the \mu-\tau reflection symmetry immediately gives rise to both maximal atmospheric mixing angle \theta_{23} and maximal CP violation \cos\delta_{CP}=0. In recent years, it is found that the \mu-\tau reflection can naturally appear when CP symmetry is imposed together with the widely studied S_{4} flavor symmetry [Mohapatra:2012tb, Feruglio:2012cw, Ding:2013hpa]. Furthermore, phenomenological models in which the desired breaking patterns of the flavor and CP symmetries are achieved dynamically have been constructed [Ding:2013hpa, Feruglio:2013hia, Luhn:2013lkn, Li:2013jya, Li:2014eia]. The interplay between CP symmetry with the flavor symmetries A_{4} [Ding:2013bpa] and T^{\prime} [Chen:2009gf, Girardi:2013sza] has been investigated as well. When CP symmetry is combined with \Delta(48) [Ding:2013nsa] or \Delta(96) [Ding:2014ssa] flavor symmetry, CP transformations distinct from \mu-\tau reflection can be produced such that \delta_{CP} can be non-maximal. It turns out that both mixing angles and CP phases depend on only one common free parameter in that case. Recently the possible lepton mixing patterns derived from CP symmetry and the \Delta(6n^{2}) or \Delta(3n^{2}) flavor symmetry group series have been analyzed [King:2014rwa, Hagedorn:2014wha, Ding:2014ora], the experimentally preferred values of the mixing angles can be accommodated very well, and the corresponding phenomenological implications in neutrinoless double decay are discussed [Ding:2014ora]. Note that it is highly nontrivial to consistently define the CP symmetry in the context of a finite flavor symmetry [Holthausen:2012dk, Chen:2014tpa]. There are more than one theoretical approachs dealing with flavor symmetry and CP violation [Branco:1983tn].

In this work, we shall only concentrate on CP symmetry, and show that the lepton mixing matrix can be reconstructed from the remnant CP symmetry. We will derive the explicit form of the PMNS matrix when one or two residual CP transformations are preserved. Phenomenological implications for the lepton flavor mixing parameters are discussed in detail. Compared with flavor symmetry paradigm, both mixing angles and CP phases can be predicted by remnant CP, and the observed value of the reactor mixing angles can be easily accommodated.

The paper is organized as follows. In section 2, remnant flavor symmetry and remnant CP transformations of the neutrino mass matrix are analyzed in the charged lepton diagonal basis. In section 3, we show that the lepton mixing matrix can be reconstructed from the presumed remnant CP transformations. If two (or one) remnant CP transformations are preserved, the explicit form of the PMNS matrix is derived, and it depends on one (or three) free real parameters in addition to the parameters of the remnant CP. In section LABEL:sec:phenomenology, the phenomenological predictions for the lepton mixing parameters are discussed, and we search for conditions of zero or maximal Dirac CP violation. Finally we summarize our results in section LABEL:sec:conclusion.

## 2 Remnant symmetries of the mass matrices

In this section, we shall clarify the remnant flavor symmetry and remnant CP symmetry of the lepton mass matrices. We shall assume throughout this paper that the neutrinos are Majorana particles. The lepton mass terms obtained after symmetry breaking are of the following form:

\mathcal{L}_{mass}=-\overline{l}_{R}m_{l}l_{L}+\frac{1}{2}\nu^{T}_{L}C^{-1}m_{% \nu}\nu_{L}+h.c.\,, | (2.1) |

where C is the charge-conjugation matrix, l_{L}\equiv(e_{L},\mu_{L},\tau_{L})^{T} and l_{R}\equiv(e_{R},\mu_{R},\tau_{R})^{T} denote the three generation left and right-handed charged lepton fields respectively, and \nu_{L}\equiv(\nu_{eL},\nu_{\mu L},\nu_{\tau L})^{T} is the three left-handed neutrino fields. The Majorana neutrino mass matrix m_{\nu} is symmetric. Since the mixing matrix only relates to left-handed fermions in standard model, as usual we construct the hermitian mass matrix \mathcal{M}_{l}\equiv m^{\dagger}_{l}m_{l} which connects left-handed charged leptons on both sides. We denote the unitary diagonalization matrix of \mathcal{M}_{l} and m_{\nu} by U_{l} and U_{\nu} respectively, i.e.,

U^{\dagger}_{l}\mathcal{M}_{l}U_{l}=\text{diag}\left(m^{2}_{e},m^{2}_{\mu},m^{% 2}_{\tau}\right),\qquad U^{T}_{\nu}m_{\nu}U_{\nu}=\text{diag}\left(m_{1},m_{2}% ,m_{3}\right)\equiv m_{diag}\,, | (2.2) |

where the light neutrino masses m_{i}(i=1,2,3) are real and non-negative. The lepton mixing matrix is the mismatch between neutrino and charged lepton diagonalization matrices,

U_{PMNS}=U^{\dagger}_{l}U_{\nu}\,. | (2.3) |

Without loss of generality we shall choose to work in the basis where \mathcal{M}_{l} is diagonal. Then U_{l} would reduce to a unit matrix and the lepton mixing completely comes from the neutrino sector with U_{PMNS}=U_{\nu}. The general form of m_{\nu} is

m_{\nu}=U^{*}_{PMNS}\;m_{diag}\;U^{\dagger}_{PMNS}\,. | (2.4) |

Firstly let’s determine the remnant flavor symmetries G_{\nu} and G_{l} of the neutrino and charged lepton mass terms. A unitary transformation \nu_{L}\rightarrow G_{\nu}\nu_{L} of the left-handed Majorana neutrino leads to the transformation of the neutrino mass matrix m_{\nu}\rightarrow G^{T}_{\nu}m_{\nu}G_{\nu}. G_{\nu} is a flavor symmetry if and only if m_{\nu} is invariant, i.e.,

G^{T}_{\nu}m_{\nu}G_{\nu}=m_{\nu} | (2.5) |

Substituting the expression of m_{\nu} in Eq. (2.4) into this invariant condition and considering that the three light neutrino masses m_{i} are non-degenerate ^{4}^{4}4Here we assume that the three light neutrino masses are non-vanishing. If the lightest neutrino is massless, then one diagonal entry “\pm 1” could be replaced with an arbitrary phase factor in Eq. (2.6)., we obtain

U^{\dagger}_{PMNS}G_{\nu}U_{PMNS}=\text{diag}\left(\pm 1,\pm 1,\pm 1\right)\,. | (2.6) |

As an overall -1 factor of G_{\nu} is irrelevant, there are essentially four solutions for G_{\nu},

G_{i}=U_{PMNS}\;d_{i}U^{\dagger}_{PMNS},\qquad i=1,2,3,4\,, | (2.7) |

where

\displaystyle d_{1}=\text{diag}\left(1,-1,-1\right),\qquad d_{2}=\text{diag}% \left(-1,1,-1\right), | |||

\displaystyle d_{3}=\text{diag}\left(-1,-1,1\right),\qquad d_{4}=\text{diag}% \left(1,1,1\right)\,. | (2.8) |

It is easy to see that G_{4} is exactly a trivial identity matrix, and we can further check that

G^{2}_{i}=1,\qquad G_{i}G_{j}=G_{j}G_{i}=G_{k}~{}~{}\text{with}~{}~{}i\neq j% \neq k\neq 4. | (2.9) |

Hence the residual flavor symmetry of the neutrino mass matrix is Z_{2}\times Z_{2} Klein group. On the other hand, given the remnant Klein symmetry in the neutrino sector and the associated 3-dimensional unitary representation matrices, one can straightforwardly construct the diagonalization matrix U_{\nu}. Similarly, the remnant flavor symmetry G_{l} of the charged lepton mass matrix satisfies

G^{\dagger}_{l}\mathcal{M}_{l}G_{l}=\mathcal{M}_{l}\,. | (2.10) |

Since \mathcal{M}_{l} is diagonal in the chosen basis and the three charged lepton masses are unequal, G_{l} can only be a unitary diagonal matrix, i.e.,

G_{l}=\text{diag}\left(e^{i\alpha_{e}},e^{i\alpha_{\mu}},e^{i\alpha_{\tau}}% \right)\,, | (2.11) |

where \alpha_{e,\mu,\tau} are arbitrary real parameters. Hence the charged lepton mass term generically admits a U(1)\times U(1)\times U(1) remnant flavor symmetry. Conversely, if G_{l} is diagonal with non-degenerate eigenvalues, \mathcal{M}_{l} would be forced to be real. The idea of residual symmetries G_{\nu} and G_{l} arising from some underlying discrete flavor symmetry group \mathcal{G}_{f} has been extensively explored, and many flavor models have been constructed [Altarelli:2010gt, Ishimori:2010zr, Grimus:2012dk, King:2013eh, King:2014nza].

In the following, we shall investigate the remnant CP symmetry of the lepton mass terms. They don’t receive enough attention they deserve in the past. The CP transformation of the left-handed neutrino fields is defined via

\nu_{L}(x)\lx@stackrel{{\scriptstyle CP}}{{\longmapsto}}iX_{\nu}\gamma^{0}C% \bar{\nu}^{T}_{L}(x_{P})\,, | (2.12) |

where x_{P}=(t,-\vec{x}), and X_{\nu} is a 3\times 3 unitary matrices acting on generation space. X_{\nu} is usually called generalized CP transformation in the literature [Ecker:1981wv, Grimus:1995zi, Branco:2011zb], since it is an identity matrix in conventional CP transformation. The Lagrangian of the neutrino mass term in Eq. (2.1) would be invariant if the neutrino mass matrix m_{\nu} fulfills

X^{T}_{\nu}m_{\nu}X_{\nu}=m^{\ast}_{\nu}\,. | (2.13) |

With the general form of m_{\nu} in Eq. (2.4), we obtain

\left(U^{\dagger}_{PMNS}X_{\nu}U^{\ast}_{PMNS}\right)^{T}m_{diag}\left(U^{% \dagger}_{PMNS}X_{\nu}U^{\ast}_{PMNS}\right)=m_{diag}\,, | (2.14) |

which yields

U^{\dagger}_{PMNS}X_{\nu}U^{\ast}_{PMNS}=\text{diag}\left(\pm 1,\pm 1,\pm 1\right) | (2.15) |

Therefore there are eight possibilities for X_{\nu}. However, only four of them are relevant, and they can chosen to be

X_{i}=U_{PMNS}\,d_{i}\,U^{T}_{PMNS},~{}~{}i=1,2,3,4\;. | (2.16) |

The remaining four can be obtained from the above chosen ones by multiplying an over -1 factor. Note that we can not distinguish X_{\nu} from -X_{\nu} since the minus sign can be absorbed by redefining the neutrino fields. Moreover, we see that the remnant CP transformations X_{i} are symmetric unitary matrices:

X_{i}=X^{T}_{i}\,, | (2.17) |

otherwise the light neutrino masses would be degenerate. The same constraint that the remnant CP transformations in the neutrino sector should be symmetric is also obtained in Ref. [Feruglio:2012cw] in another way. It is remarkable that the remnant flavor symmetry can be induced by the remnant CP symmetry. From Eq. (2.13), it is easy to obtain,

X^{\dagger}_{j}X_{i}^{T}m_{\nu}X_{i}X_{j}^{\ast}=m_{\nu}\,. | (2.18) |

This means that successively performing two CP transformations X_{i}X_{j}^{\ast} is equivalent to a flavor symmetry transformations. Concretely we have the following relations:

\displaystyle X_{2}X^{\ast}_{3}=X_{3}X^{\ast}_{2}=X_{4}X^{\ast}_{1}=X_{1}X^{% \ast}_{4}=G_{1}, | |||

\displaystyle X_{1}X^{\ast}_{3}=X_{3}X^{\ast}_{1}=X_{4}X^{\ast}_{2}=X_{2}X^{% \ast}_{4}=G_{2}, | |||

\displaystyle X_{1}X^{\ast}_{2}=X_{2}X^{\ast}_{1}=X_{4}X^{\ast}_{3}=X_{3}X^{% \ast}_{4}=G_{3}, | |||

\displaystyle X_{1}X^{\ast}_{1}=X_{2}X^{\ast}_{2}=X_{3}X^{\ast}_{3}=X_{4}X^{% \ast}_{4}=G_{4}=1\,. | (2.19) |

As a consequence, once we impose a set of generalized CP transformations onto the theory, there is always an accompanied flavor symmetry generated. Furthermore, Eq. (2.19) implies that any residual CP transformation can be expressed in terms of the remaining ones as follows,

X_{i}=X_{j}X^{\ast}_{m}X_{n},\qquad i\neq j\neq m\neq n\,. | (2.20) |

In other words, only three of the four remnant CP transformations are independent. In the same fashion, the CP transformation of the charged lepton fields is

l_{L}(x)\lx@stackrel{{\scriptstyle CP}}{{\longmapsto}}iX_{l}\gamma^{0}C\bar{l}% ^{\,T}_{L}(x_{P})\,, | (2.21) |

for the symmetry to hold, the mass matrices \mathcal{M}_{l} has to satisfy

X^{\dagger}_{l}\mathcal{M}_{l}X_{l}=\mathcal{M}^{*}_{l} | (2.22) |

In the chosen basis where \mathcal{M}_{l} is diagonal, X_{l} can only be a diagonal phase matrix, i.e.,

X_{l}=\text{diag}\left(e^{i\beta_{e}},e^{i\beta_{\mu}},e^{i\beta_{\tau}}\right% )\,, | (2.23) |

where \beta_{e,\mu,\tau} are real. In short, the remnant CP symmetry can be constructed from the mixing matrix, and its explicit form can be determined more precisely with the improving measurement accuracy of the mixing angles and CP phases. Before closing this section, we present the above discussed residual CP symmetry in an arbitrary basis:

X_{i}=U_{\nu}\,d_{i}\,U^{T}_{\nu}~{}~{}(i=1,2,3,4),\qquad X_{l}=U_{l}\,\text{% diag}\left(e^{i\beta_{e}},e^{i\beta_{\mu}},e^{i\beta_{\tau}}\right)U^{T}_{l}\,, | (2.24) |

which can be derived in exactly the same way. In the end, we conclude that the remnant symmetries can be constructed from the mixing matrix which can be measured experimentally. In the following section, we shall demonstrate that the lepton mixing matrix can be constructed from the postulated remnant CP transformations.

## 3 Reconstruction of lepton mixing matrix from remnant CP symmetries

As has been shown in section 2, residual CP symmetries can be derived from mixing matrix, and vice versa lepton mixing matrix can be constructed from the remnant CP symmetries in the neutrino and the charged lepton sectors. In concrete models, we can start from a set of CP transformations \mathcal{X_{CP}} which the Lagrange respects at high energy scale. Subsequently \mathcal{X_{CP}} is spontaneously broken by some scalar fields into different remnant symmetries in the neutrino and the charged lepton sectors. The misalignment between the two remnant symmetries is responsible for the mismatch of the rotations which diagonalize the neutrino and charged lepton matrices, and accordingly the PMNS matrix is generated. The remnant CP symmetries would be assumed hereinafter and we shall not consider how the required vacuum compatible with the remnant symmetries is dynamically achieved, since the resulting lepton mixing pattern is independent of vacuum alignment mechanism and there are generally more than one methods realizing the desired symmetry breaking in practical model building.

As before we still stick to the charged lepton diagonal basis in the following. If four CP transformations X_{Ri}~{}(i=1,2,3,4) out of \mathcal{X}_{CP} are conserved by the neutrino mass matrix, where the subscript “R” denotes remnant. In order to be well-defined, X_{Ri} should be unitary matrices and satisfy:

X_{Ri}=X^{T}_{Ri},\qquad X_{Ri}X^{\ast}_{Rj}=X_{Rj}X^{\ast}_{Ri}=X_{Rm}X^{\ast% }_{Rn}=X_{Rn}X^{\ast}_{Rm},\qquad\left(X_{Ri}X^{\ast}_{Rj}\right)^{2}=1\,, | (3.1) |

for i\neq j\neq m\neq n. As shown in Eq. (2.19), a Z_{2}\times Z_{2} remnant flavor symmetry is generated with element of the form X_{Ri}X^{\ast}_{Rj}~{}(i\neq j). It is well-known that the lepton mixing matrix (except the Majorana phases) is fixed by the residual Klein group up to independent permutations of rows and columns. If the residual flavor symmetry originates from a finite flavor symmetry group at high energy, a complete classification of all possible PMNS matrix has been worked out [Fonseca:2014koa]. An added bonus here is that the Majorana CP phases can also be determined from the postulated remnant CP transformations although they are not constrained by the remnant flavor symmetry at all. If three CP transformations are preserved by the neutrino mass terms, the fourth one can be generated in the way shown in Eq. (2.20). Hence there are still four residual CP transformations. Given the explicit forms of remnant CP, the lepton mixing matrix can be straightforwardly calculated.

In the following, we shall investigate the most interesting case in which the neutrino mass matrix is invariant under the action of two residual CP transformations X_{R1} and X_{R2}. In concrete models, this situation can be realized in two different ways: only X_{R1} and X_{R2} belong to the beginning CP transformations \mathcal{X_{CP}} or \mathcal{X_{CP}} contains all the four remnant CP transformations, but two of them are broken at low energy. To avoid degenerate light neutrino masses, both X_{R1} and X_{R2} should be symmetric unitary matrices. Furthermore, a residual Z_{2} flavor symmetry is induced with the generator G_{R}\equiv X_{R1}X^{\ast}_{R2}=X_{R2}X^{\ast}_{R1}. It is easy to check that the following consistency equations are fulfilled,

X_{R1}G^{\ast}_{R}X^{-1}_{R1}=G_{R},\qquad X_{R2}G^{\ast}_{R}X^{-1}_{R2}=G_{R}\,. | (3.2) |

As we have X_{R2}=G_{R}X_{R1}, the neutrino mass matrix would be invariant under X_{R2} if it is invariant under both CP transformation X_{R1} and flavor transformation G_{R}. As a consequence, from now on we shall focus on the residual symmetry X_{R1} and G_{R} for convenience. Firstly we note that only one column of the diagonalization matrix U_{\nu}, which coincides with U_{PMNS} in the working basis, is fixed by the single Z_{2} residual flavor symmetry G_{R}. This column is exactly the unique eigenvector of G_{R} with eigenvalue \pm 1 if \det(G_{R})=\pm 1 [Ge:2011ih]. Using the freedom of redefining the charged lepton fields (or changing the basis further but keeping \mathcal{M}_{l} still diagonal), each element of this column can always set to be real and non-negative. Hence the column dictated by G_{R} can be parameterized as

v_{1}=\left(\begin{array}[]{c}\cos\varphi\\ \sin\varphi\cos\phi\\ \sin\varphi\sin\phi\\ \cr\@@LTX@noalign{ }\omit\\ \end{array} |