# Lepton Mixing Parameters from \Delta(48) Family Symmetry and Generalised CP

###### Abstract

We provide a systematic and thorough exploration of the \Delta(48) family symmetry and the consistent generalised CP symmetry. A model-independent analysis of the achievable lepton flavor mixing is performed by considering all the possible remnant symmetries in the neutrino and the charged lepton sectors. We find a new interesting mixing pattern in which both lepton mixing angles and CP phases are nontrivial functions of a single parameter \theta. The value of \theta can be fixed by the measured reactor mixing angle \theta_{13}, and the excellent agreement with the present data can be achieved. A supersymmetric model based on \Delta(48) family symmetry and generalised CP symmetry is constructed, and this new mixing pattern is exactly reproduced.

## 1 Introduction

It is well-known that CP (CP is a combination of charge conjugation symmetry and parity symmetry) is not an exact symmetry of the nature. CP violation in the quark sector has been firmly established in oscillations and decays of K and B mesons [pdg]. In the lepton sector, the precise measurement of the reactor mixing angle \theta_{13} [An:2012eh] opens the door to measure the leptonic CP violation. Measurement of the Dirac CP-violating phase \delta_{\rm{CP}} has become one of the primary physical goals of the next-generation neutrino oscillation experiments. The origin of CP violation is a longstanding fundamental question in particle physics. In the Standard Model, violation of CP occurs in the flavor sector. It is conceivable that promoting CP to a symmetry at high energies which is then broken allows to impose constraints on the neutrino and charged lepton mass matrices [Ecker:1981wv, Grimus:1995zi].

In the past years, non-abelian discrete groups have been widely used to explain the structure of lepton mixing angles, please see Refs. [Altarelli:2010gt] for recent reviews. It seems natural to combine the discrete family symmetry with the CP symmetry to predict both lepton mixing angles and CP phases simultaneously. However, the interplay between family and CP symmetries should be carefully treated [Feruglio:2012cw, Holthausen:2012dk, Chen:2014tpa]. In the presence of a family symmetry, in many cases it is impossible to define CP in the naive way, i.e., \phi\rightarrow\phi^{*}, but rather a nontrivial transformation in flavor space is needed [Feruglio:2012cw, Holthausen:2012dk]. A typical example is the so-called \mu-\tau reflection symmetry [Harrison:2002kp, Grimus:2003yn, Farzan:2006vj], which interchanges a muon (tau) neutrino with a tau (muon) antineutrino in the charged lepton mass basis. In order to consistently define CP transformations in the context of non-abelian discrete family groups, certain consistency conditions must be satisfied [Ecker:1981wv, Grimus:1995zi, Feruglio:2012cw, Holthausen:2012dk]. In fact, it has been established that all generalized CP symmetries are outer automorphisms of the symmetry group [Grimus:1995zi, Holthausen:2012dk].

Combining family symmetry with generalised CP symmetry, one can obtain lepton mixing angles compatible with the current experimental data and in the meantime predict CP-violating phases. This interesting idea has been explored to some extent in the past. The generalised CP symmetry has been implemented within S_{4} [Ding:2013hpa, Feruglio:2013hia, Luhn:2013lkn, Li:2013jya] , A_{4} [Ding:2013bpa], and T^{\prime} [Girardi:2013sza] family symmetries, and some concrete models have also been constructed. In these models, the full symmetry is generally spontaneously broken down to a cyclic subgroup in the charged lepton sector and to Z_{2}\times H^{\nu}_{\rm{CP}} in the neutrino sector. The surviving symmetries constrain the neutrino mass matrix and charged lepton mass matrix, leading to predictions for CP-violating phases as well as constraints on mixing angles. Typically, the Dirac CP-violating phase is predicted to take simple values such as 0, \pi, or \pm\pi/2. A comprehensive analysis of the generalised CP within \Delta(96) family symmetry is recently performed in the semi-direct approach [Ding:2014ssa], and some new interesting mixing patterns are found. The generalised CP has also been investigated for an infinite series of finite groups \Delta(6n^{2}) [King:2014rwa], where the full Klein symmetry is assumed to be preserved in the neutrino sector such that the Dirac CP phase can only be 0 or \pi. There are also other approaches in which family symmetries and CP violation appear together [Mohapatra:2012tb, Branco:1983tn, Chen:2009gf, Antusch:2011sx].

In our recent paper [Ding:2013nsa], we propose to use \Delta(48) as the family symmetry and extend it to include the generalized CP symmetry. As we shall see later, the group \Delta(48), which has been overlooked in the literature, has a large automorphism group of order 384. Hence \Delta(48) provides us more choices for generalised CP transformations than some popular family symmetries A_{4}, S_{4}, etc. As a consequence, we find a new interesting mixing pattern, which is denoted as patter D in [Ding:2013nsa], is admissible by neutrino oscillation experiments. This mixing texture can fit the experimental data quit well and predict the Dirac-type CP violation neither vanishing nor maximal, i.e., \delta_{\rm{CP}}\neq 0, \pi, \pm\pi/2.

This paper is devoted to a comprehensive analysis of lepton flavor mixing within the context of the \Delta(48) family symmetry combined with generalized CP symmetry. In section 2, we discuss the structure of the automorphism group of \Delta(48) and present the generalized CP transformations consistent with the \Delta(48) family symmetry. In section LABEL:sec:model_independent_analysis, we perform a systematic scan of lepton mixing within the framework of \Delta(48)\rtimes H_{\rm{CP}} by analyzing all possible residual symmetries in the neutrino and the charged lepton sectors. We find 10 different cases, and subsequently we investigate the corresponding phenomenological predictions for the lepton mixing parameters which depend on one single free parameter \theta. In particular, a new interesting mixing pattern (pattern D in Ref. [Ding:2013nsa]) is found. Both mixing angles and CP phases are nontrivial functions of \theta, and three leptonic mixing angles in the experimentally preferred range can be achieved for certain values of the parameter \theta. In section LABEL:sec:model_construction, we construct a supersymmetric model with both \Delta(48) family symmetry and generalised CP symmetry. This model gives rise to the new mixing pattern we mentioned above, and its phenomenological predictions are discussed. In addition, the required vacuum alignment is justified. We summarize the main results of our paper in section LABEL:sec:conclusions. Some details of the group theory of \Delta(48) are contained in Appendix LABEL:sec:appendix_A_group_theory, and the Clebsch-Gordan coefficients in the chosen basis are reported. Appendix LABEL:sec:appendix_B lists the vacuum alignment invariant under the remnant family and CP symmetries in cases II, IV, VI and VIII.

## 2 Generalized CP transformations consistent with \Delta(48)

In this section, we are going to discuss all the possible generalized CP transformations which are consistent with \Delta(48). Before that, we give a brief review of the consistent definition of the generalized CP transformation in the context of discrete family symmetries. It is nontrivial to impose the generalized CP symmetry on a theory in the presence of a family symmetry G_{f}. For a field multiplet

\Phi=\left(\phi_{R},\phi_{P},\phi^{*}_{P},\phi_{C},\phi^{*}_{C}\right)^{T}\,, | (2.1) |

where the subscript R, P and C denote that the fields \phi are in the real, pseudo-real and complex representations of G_{f}, respectively. Under the action of the family symmetry, the field \Phi transforms as

\Phi\lx@stackrel{{\scriptstyle g}}{{\longrightarrow}}\rho(g)\Phi,\qquad g\in G% _{f}\,, | (2.2) |

where \rho is a representation of the group element g. Depending on the component fields \phi_{R}, \phi_{P}, \phi^{*}_{P}, \phi_{C} and \phi^{*}_{C}, the representation \rho is generally reducible with

\rho(g)=\left(\begin{array}[]{ccccc}\rho_{R}(g)&&&&\\ &\rho_{P}(g)&&&\\ &&\rho^{*}_{P}(g)&&\\ &&&\rho_{C}(g)&\\ &&&&\rho^{*}_{C}(g)\\ \cr\omit\span\omit\span\omit\span\omit\span\@@LTX@noalign{ }\omit\\ \end{array} |