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Revamping Tribimaximal Mixing. II PHYSICAL REVIEW D 100, 053001 (2019) CP symmetries as guiding posts: Revamping tribimaximal mixing. II. † ‡ ∥ Peng Chen,1,* Salvador Centelles Chuliá ,2, Gui-Jun Ding,3, Rahul Srivastava,2,§ and Jos´e W. F. Valle2, 1College of Information Science and Engineering, Ocean University of China, Qingdao 266100, China 2AHEP Group, Institut de Física Corpuscular—C.S.I.C./Universitat de Val`encia, Parc Científic de Paterna. C/ Catedrático Jos´e Beltrán, 2 E-46980 Paterna (Valencia)—SPAIN 3Interdisciplinary Center for Theoretical Study and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China (Received 3 July 2019; published 5 September 2019) In this follow up of arXiv:1812.04663 we analyze the generalized CP symmetries of the charged lepton mass matrix compatible with the complex version of the tribimaximal (TBM) lepton mixing pattern. These symmetries are used to “revamp” the simplest TBM Ansatz in a systematic way. Our generalized patterns share some of the attractive features of the original TBM matrix and are consistent with current oscillation experiments. We also discuss their phenomenological implications both for upcoming neutrino oscillation and neutrinoless double beta decay experiments. DOI: 10.1103/PhysRevD.100.053001 I. INTRODUCTION part of the truth, providing a valid starting point for building viable patterns of lepton mixing [18–20]. The structure of the lepton sector and the properties of Flavor symmetries can be implemented within different neutrinos stand out as a key missing link in particle physics, approaches. For example, one can build specific theories whose understanding is required for the next leap forward. from scratch [21–23]. Alternatively, one can adopt a model- Neutrino oscillation studies have already given us a first independent framework based on the imposition of residual hint for CP violation in the lepton sector [1], though further symmetries, irrespective of how the mass matrices actually experiments are needed to improve our measurement of arise from first principles [24–30]. As a step in this leptonic CP violation [2–4]. Moreover, we need informa- direction one may consider complexified versions of the tion on the elusive Majorana phases that would show up in standard tribimaximal (TBM) neutrino mixing pattern. By the description of lepton number violating processes such partially imposing generalized CP symmetries one can as neutrinoless double-beta decay [5,6], whose discovery construct non-trivial variants of the standard TBM Ansatz would establish the self-conjugate nature of neutrinos. (or any other) in a systematic manner. Depending on the Improving the current experimental sensitivities [7–12] type and number of preserved CP symmetries, one obtains is, again, necessary for the next step. several different mixing matrices. Such “revamped” var- It is a timely moment to make theory predictions for θ CP iants of the TBM Ansatz have in general non-zero 13 as mixing parameters and violating phases characterizing well as CP violation, as currently indicated by the the lepton sector. The most promising tool is to appeal to oscillation data. Examples of this procedure have already symmetry considerations [13]. As benchmarks we have been given in [30,31]. The prospects for probing various ideas such as mu-tau symmetry and the tribimaximal neutrino mixing scenarios in present and future oscillation (TBM) neutrino mixing [14]. Thanks to the reactor mea- experiments have been discussed extensively in the liter- surements of nonzero θ13, these benchmarks cannot be the – – ature [4,19,32 38]. final answer [15 17]. However, they capture an important This paper is a follow-up of Ref. [30]. Throughout this work we will adopt the basis in which the neutrino mass *[email protected] matrix is diagonal and therefore the leptonic mixing matrix † [email protected] U ¼ U† U ‡ lep , where cl is the matrix which diagonalizes the [email protected] cl § charged lepton mass matrix. Here we take the famous [email protected] ∥ [email protected] Ansatz of TBM mixing [14] in the charged lepton sector as U ¼ U† → U ¼ U† ¼ U a starting point, i.e., cl TBM lep cl TBM Published by the American Physical Society under the terms of and seek for solutions which satisfy only a partial sym- the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to metry with respect to the full TBM symmetries. The paper the author(s) and the published article’s title, journal citation, is structured as follows. We start by briefly discussing and DOI. Funded by SCOAP3. remnant generalized CP symmetries in Sec. II and the CP 2470-0010=2019=100(5)=053001(15) 053001-1 Published by the American Physical Society PENG CHEN et al. PHYS. REV. D 100, 053001 (2019) U† X†M2X ¼ M2Ã: ð Þ and flavor symmetries of the TBM mixing matrix in 4 Sec. III. We will then focus in the cases in which the mass matrix satisfies two CP symmetries in Sec. IV and one CP If this is the case, we say that X is a remnant CP symmetry symmetry in Sec. V. We also discuss the phenomenological of the mass matrix M2. As shown in several previous works – X U predictions from these matrices and give a brief sum-up [24 26] [30], can be written in terms of cl as discussion at the end. X ¼ U ðeiδ1 ;eiδ2 ;eiδ3 ÞUT ; ð Þ cldiag cl 5 II. PRELIMINARIES where the δi are arbitrary real parameters which label In this section, we shall briefly review the remnant the CP transformation. In other words, given a mass matrix CP generalized symmetry and flavor symmetry of the one can extract an infinite set of X matrices satisfying charged lepton sector. We assume that the neutrinos are Eq. (4) labeled by the three real parameters δ1, δ2 and δ3. Majorana particles. The neutrino and charged lepton mass Equation (5) allows us to build all the possible generalized term can be written as CP symmetries from a given mass matrix or mixing pattern. 1 As stated before, one can also invert the logic and L ¼ −l¯ m l þ νT CTm ν þ : :; ð Þ construct the mixing matrix given a remnant CP symmetry mass R l L 2 L ν L H c 1 matrix X. If the squared mass matrix M2 satisfies Eq. (4) it C l l can be shown [24–26] [30] that, after Takagi decomposing where is the charge-conjugation matrix, L and R are the T three generations of the left and right-handed charged lepton the symmetric CP transformation as X ¼ ΣΣ , the result- Σ U fields, respectively. In this paper we work in the neutrino ing relation between and cl is diagonal basis without loss of generality. As a consequence, U ¼ ΣOT Q−1=2; ð Þ mν is the diagonal neutrino mass matrix and νL stands for the cl 3×3 6 three generation left-handed neutrino mass eigenstates. O Q A generalized CP symmetry combines the canonical CP where 3×3 is a generic real orthogonal matrix and is a transformation with a flavor symmetry. Under the action of general unitary diagonal matrix. If Uν is the unitary matrix a generalized CP transformation, a generic fermion multi- that diagonalizes the neutrino mass matrix, then the lepton plet field ψ transforms as mixing matrix is given as 0 T U ¼ U† U : ð Þ ψ → iXγ Cψ¯ ; ð2Þ lep cl ν 7 where X is a unitary symmetric matrix in the flavor space As mentioned above, we shall work in the neutrino mass which characterizes the generalized CP transformation. diagonal basis throughout this paper. Hence Uν is a unit Notice that for the conventional CP transformation X is matrix and the lepton mixing matrix can be written as U ¼ U† just the identity matrix and does not mix the flavours. lep cl. Studying the remnant generalized symmetries, i.e., the We can also follow a similar procedure to obtain the surviving generalized CP symmetries after spontaneous flavor symmetries of the mass matrix M2. We define a symmetry breaking, provides a powerful method to study remnant flavor transformation as the mixing pattern of leptons. Given a mixing pattern one can extract all the remnant CP symmetries of the corre- ψ → Gψ; ð8Þ sponding lepton mass matrices [25,39]. On the other hand, one can also invert the procedure and extract the possible where again G is a 3 × 3 unitary matrix acting in flavor mixing matrices (up to some freedom) that respect one or space. If this transformation leaves the squared mass matrix more generalized CP symmetries. In this work we will invariant, then we say that G is a residual flavor symmetry study some aspects of such generalized CP transformations of the mass matrix and it satisfies acting on the charged lepton sector. m † 2 2 We denote the charged lepton mass matrix as cl, and the G M G ¼ M : ð9Þ M2 ≡ m† m Hermitian mass matrix cl cl can be diagonalized U by a unitary transformation cl, We can see from Eq. (9) that the flavor symmetries are all the matrices G which commute with M2 and therefore they U† M2U ¼ ðm2;m2;m2Þ; ð Þ share a common basis of eigenvectors. Since the eigenval- cl cl diag e μ τ 3 ues of M2 are the squared charged lepton masses, which are 2 where me, mτ, and mτ are the charged lepton masses. If the non-degenerate, in the basis in which M is diagonal G will charged lepton mass term is invariant under the generalized also be diagonal. Therefore both G and M2 are diagonal- 2 U CP transformation of Eq.
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