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 José 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 València, Parc Científic de Paterna. C/ Catedrático José 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 M2: 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. (2), X and M should satisfy the ized by the same unitary matrix cl, and we can build its following relation explicit form:

053001-2 CP SYMMETRIES AS GUIDING POSTS: REVAMPING … PHYS. REV. D 100, 053001 (2019) pffiffiffi U† GU ¼ ðeiα;eiβ;eiγÞ cl diag Ur ¼ U23ðπ=4; 0ÞU12ðarcsinð1= 3; 0Þ cl TBM qffiffi iα iβ iγ † 0 1 → G ¼ U ðe ;e ;e ÞU ; ð Þ 2 1 cldiag cl 10 pffiffi 0 B 3 3 C B C ¼ B − p1ffiffi p1ffiffi p1ffiffi C ð13Þ where we used the fact that the eigenvalues of a unitary @ 6 3 2 A matrix must have norm 1, so we write them as eiα, eiβ, and p1ffiffi − p1ffiffi p1ffiffi eiγ, with α, β, and γ real. As a side remark that will become 6 3 2 important in Sec. IV, note that the previous argument implies that if we impose a flavor symmetry with non- where Uijðθ; ϕÞ is a complex rotation in the ðijÞ-plane of degenerate eigenvalues then the mixing matrix would be angle θ and phase ϕ. For example, uniquely determined up to permutation of its column vectors. This holds since, if the eigenvalues of G are 0 1 iα iβ iγ 2 θ θe−iϕ 0 ðe ;e ;e Þ with α ≠ β ≠ γ, then both M and G are B cos sin C B iϕ C diagonalized by the same unitary matrix. U12ðθ; ϕÞ¼@ − sin θe cos θ 0 A: ð14Þ However, the above argument does not hold if the 001 eigenvalues of G are at least partially degenerate, since in this case in a basis in which G is diagonal, the matrix M2 may not be diagonal. There will be a subspace generated by We can generalize this Ansatz so as to include nonzero the eigenvectors associated to the degenerate eigenvalues Majorana phases, thus defining the “complex TBM mix- which will not be diagonal in general. We will exploit this ing” pattern as feature in Sec. IV. As a final comment, let us mention that there is a relation pffiffiffi Uc ¼ U23ðπ=4; σÞU12ðarcsinð1= 3Þ; ρÞ between flavor symmetries and CP symmetries. If two CP TBM 0 1 iδ1 iδ2 iδ3 T qffiffi symmetries X1 ¼ U diagðe ;e ;e ÞU and X2 ¼ −iρ cl cl B 2 epffiffi 0 C U ðeiδ4 ;eiδ5 ;eiδ6 ÞUT 3 3 cldiag cl are preserved by the charged B C B iρ −iσ C lepton sector, then a flavor symmetry can be induced by ¼ B − pe ffiffi p1ffiffi epffiffi C: ð15Þ B 6 3 2 C successively performing two CP transformations @ A iðρþσÞ iσ e pffiffi − pe ffiffi p1ffiffi 6 3 2 G ¼ X X ¼ U ðeiα;eiβ;eiγÞU† ; ð Þ 1 2 cldiag cl 11 ðρ; σÞ¼ð0; 0Þ with α ¼ δ1 − δ4, β ¼ δ2 − δ5 and γ ¼ δ3 − δ6. In other Note that, apart from the choice , there are words, applying two CP symmetries automatically implies other choices for the phases ρ and σ which also lead to real the existence of a flavor symmetry. mixing patterns: ðρ; σÞ¼ðπ; 0Þ; ð0; πÞ, and ðπ; πÞ. If the lepton mixing matrix is the complex TBM matrix U ¼ U III. CP AND FLAVOR SYMMETRIES OF lep cTBM, then the charged lepton mixing matrix will U ¼ U† TRIBIMAXIMAL MIXING be cl cTBM in the neutrino diagonal basis. We can explicitly build the charged lepton mass matrix diagonal- The goal of this paper is to modify the tribimaximal U† mixing (TBM) pattern based on the charged lepton CP ized by cTBM as symmetries. In this section we will use the results obtained 0 pffiffiffi 1 in Sec. II to extract the CP and flavor symmetries of the 2 2e−iρ 0 m2 Bpffiffiffi C celebrated TBM Ansatz [14]. This will be useful in the M2 ¼ e @ 2eiρ 10A following sections, in which we will start from a charged cTBM 3 000 lepton mass matrix satisfying the full TBM symmetry, and 0 pffiffiffi pffiffiffi 1 add a perturbation term which will satisfy only a partial 1 − 2e−iρ − 3e−iðρþσÞ m2 B pffiffiffi pffiffiffi C symmetry. þ μ @ − 2eiρ 2 6e−iσ A The standard TBM mixing pattern [14] is the Ansatz in 6 pffiffiffi pffiffiffi which the three mixing angles take the following values − 3eiðρþσÞ 6eiσ 3 0 pffiffiffi pffiffiffi 1 1 1 − 2e−iρ 3e−iðρþσÞ m2 B pffiffiffi pffiffiffi C sin θ12 ¼ pffiffiffi ; θ13 ¼ 0; θ23 ¼ π=4: ð12Þ τ B iρ −iσ C 3 þ − 2e 2 − 6e ð16Þ 6 @pffiffiffi pffiffiffi A 3eiðρþσÞ − 6eiσ 3 The vanishing of one of the mixing angles, in our case θ13, implies that the Dirac CP phase δCP of the lepton mixing matrix is unphysical [40]. Assuming zero Majorana phases, Using Eq. (5) we can easily extract the CP symmetries of we can write the “real TBM mixing” as, the matrix M2, i.e., all the matrices X that satisfy Eq. (4),

053001-3 PENG CHEN et al. PHYS. REV. D 100, 053001 (2019) 0 pffiffiffi 1 0 pffiffiffi pffiffiffi 1 † iδ1 iδ2 iδ3 X ¼ Uc diagðe ;e ;e ÞUc 2 2 0 1 − 2 − 3 TBM0 pffiffiffi 1 TBM eiα B pffiffiffi C eiβ B pffiffiffi pffiffiffi C 2 2eiρ 0 G ¼ @ 2 10A þ @ − 2 2 6 A iδ 3 6 pffiffiffi pffiffiffi e 1 Bpffiffiffi C ¼ @ 2eiρ e2iρ 0A 000 − 3 6 3 3 0 pffiffiffi pffiffiffi 1 000 1 − 2 3 0 pffiffiffi pffiffiffi 1 eiγ B pffiffiffi pffiffiffi C e−2iρ − 2e−iρ − 3e−iðρ−σÞ þ @ − 2 2 − 6 A: ð20Þ iδ 6 pffiffiffi pffiffiffi e 2 B pffiffiffi pffiffiffi C þ @ − 2e−iρ 2 6eiσ A 3 − 6 3 6 pffiffiffi pffiffiffi − 3e−iðρ−σÞ 6eiσ 3e2iσ 0 pffiffiffi pffiffiffi 1 M2 ¼ M2 e−2iðρþσÞ − 2e−iðρþ2σÞ 3e−iðρþσÞ Note that if the mass matrix is real then and eiδ3 B pffiffiffi pffiffiffi C there is no difference between Eq. (4) and Eq. (9). This is þ @− 2e−iðρþ2σÞ 2e−2iσ − 6e−iσ A: 6 pffiffiffi pffiffiffi why in the real TBM case the flavor symmetry matrices 3e−iðρþσÞ − 6e−iσ 3 are identical to the CP symmetry matrices. This statement ðρ; σÞ¼ð0; 0Þ ð Þ not only applies to the case , which we are 17 calling “real TBM,” but also to ðρ; σÞ¼ð0; πÞ, ðρ; σÞ¼ ðπ; 0Þ, and ðρ; σÞ¼ðπ; πÞ. The CP symmetries which correspond to the real TBM Although the TBM Ansatz provides an interesting limit in Eq. (13) can be constructed from Eq. (17) just by starting point, note that neither the real nor the complex going to the limit ρ → 0 and σ → 0. In this case the CP variants of the TBM mixing are viable lepton mixing matrix takes the simpler form patterns. Indeed, recent reactor measurements [15–17] have established that θ13 is nonzero with high significance. In the same spirit as [30] here we show that, starting from the † iδ1 iδ2 iδ3 X ¼ Ur diagðe ;e ;e ÞUr cTBM matrix in the charged lepton sector, and using TBM0 pffiffiffi 1 TBM0 pffiffiffi pffiffiffi 1 2 2 0 1 − 2 − 3 the generalized CP symmetries, one can systematically eiδ1 B pffiffiffi C eiδ2 B pffiffiffi pffiffiffi C construct and analyze realistic neutrino mixing matrices ¼ @ 2 10A þ @ − 2 2 6 A 3 6 pffiffiffi pffiffiffi with nonzero reactor angle. An appealing feature of this 000 − 3 6 3 method is that the resulting mixing patterns will share many 0 pffiffiffi pffiffiffi 1 1 − 2 3 properties with the original TBM Ansatz, while avoiding iδ e 3 B pffiffiffi pffiffiffi C the unwanted θ13 ¼ 0 prediction. þ @ − 2 2 − 6 A: ð18Þ 6 pffiffiffi pffiffiffi As a starting point we will assume neutrinos to be 3 − 6 3 Majorana-type and will work in a basis in which neutrinos are diagonal. We will then start with the complex TBM matrix of Eq. (15). The real TBM matrix can always be We now turn to the residual flavour symmetries of the obtained from it by simply taking the limit ρ, σ → 0.In U† mixing matrix cTBM. Again we will extract all the what follows we will take this limit at various stages of our matrices G, labeled by the three real parameters α, β, discussion. and γ, that satisfy Eq. (9). Using Eq. (10), we find that these matrices take the form IV. CHARGED LEPTON MASS MATRIX 0 pffiffiffi 1 CONSERVING TWO CP SYMMETRIES 2 2e−iρ 0 eiα B pffiffiffi C We will start our analysis by taking as a starting point a G ¼ @ 2eiρ 10A U† 3 charged lepton mass matrix diagonalized by cTBM, which 000 will be of the form shown in Eq. (16). We will then add a 0 pffiffiffi pffiffiffi 1 small perturbation which only preserves two remnant CP 1 − 2e−iρ − 3e−iðρþσÞ eiβ B pffiffiffi pffiffiffi C symmetries, and study the resulting mixing pattern, namely, þ @ − 2eiρ 2 6e−iσ A 6 pffiffiffi pffiffiffi − 3eiðρþσÞ 6eiσ 3 M2 ¼ M2 þ δM2; ð Þ 0 pffiffiffi pffiffiffi 1 cTBM 21 1 − 2e−iρ 3e−iðρþσÞ iγ e B pffiffiffi pffiffiffi C 2 þ @ − 2eiρ 2 − 6e−iσ A: ð19Þ where δM is the perturbation matrix and is therefore 6 pffiffiffi pffiffiffi 3eiðρþσÞ − 6eiσ 3 expected to be small. As explained in the previous section, a CP symmetry in the charged lepton sector compatible U with cTBM will be of the form shown in Eq. (17) and will As before, we can recover the real TBM limit just by going satisfy Eq. (4). We impose, in the perturbation term, two to the limit ρ → 0 and σ → 0. such symmetries given by

053001-4 CP SYMMETRIES AS GUIDING POSTS: REVAMPING … PHYS. REV. D 100, 053001 (2019)

† iδ1 iδ2 iδ3 ααβ αβα X1 ¼ Uc diagðe ;e ;e ÞUc ; In the following, we will consider the cases and . TBM TBM βαα † iδ iδ iδ We discard the third scenario, the case, as it leads to the X2 ¼ U diagðe 4 ;e 5 ;e 6 ÞU ; ð22Þ cTBM cTBM prediction θ13 ¼ 0 and is therefore not viable. where in general the δi can be different. As explained ααβ in Sec. II,twoCP symmetries generate automatically a A. Case † 2 2 flavor symmetry Gl ¼ X1X2 satisfying Gl M Gl ¼ M and We start our analysis by imposing the flavor symmetry G ¼ U† ðeiα;eiα;eiβÞU given by l cTBMdiag cTBM into the perturbation 2 term δM . Note that this implies δ1 − δ4 ¼ δ2 − δ5 ¼ α, G ¼ U† ðeiα;eiβ;eiγÞU ; ð Þ β ¼ δ − δ ≠ α CP l cTBMdiag cTBM 23 while 3 6 . Instead of six parameters, going to the ααβ case restricts the situation to five CP parameters. with α ¼ δ1 − δ4, β ¼ δ2 − δ5, and γ ¼ δ3 − δ6. It is clear It is clear that the eigenvector associated to the eigenvalue iβ 2 that e of Gl will also be an eigenvector of M .However,two independent eigenvectors of Gl with the degenerate eigen- † iα iβ iγ iα U G U ¼ ðe ;e ;e Þ: ð Þ value e will span a subspace of eigenvectors of Gl of cTBM l cTBM diag 24 which only a particular combination is also eigenvector of † 2 † G U M . Since Gl is diagonalized by U , then, after getting One sees that l is diagonalized by cTBM and its cTBM eigenvalues are eiα, eiβ, eiγ. As shown in Sec. II, when rid of the unphysical phases via redefinition of the phases of α ≠ β ≠ γ the eigenvalues of Gl are nondegenerate and the charged lepton fields we get U† G M2 cTBM will diagonalize both l and . In this case the U ¼ U† U ¼ U lepton mixing matrix would be lep cl ν cTBM 2 2 † Uc ðMc þ δM ÞUc with θ13 ¼ 0, hence inconsistent with experiment. Here we TBM0 TBM TBM 1 2 2 iϕ study the particular scenarios in which Gl is partially M11 δM e 0 degenerate, i.e. B 2 −iϕ 2 C ¼ @ δM e M22 0 A; ð30Þ 2 G ¼ U† P ðeiα;eiα;eiβÞPTU ; ð Þ 00mτ l cTBM ldiag l cTBM 25 where α ≠ β and Pl is a permutation matrix which para- M2 M2 δM2 ϕ metrizes the three possible orderings, ðeiα;eiα;eiβÞ, where 11, 22, , and are real parameters. Notice ðeiα;eiβ;eiαÞ ðeiβ;eiα;eiαÞ that the form of the mass matrix is not dependent on the ,or . These permutation matrices α β can be given as: particular values of and , only on the choice of the degenerate (12) sector in this case. Then the imposition of 0 1 0 1 δ1−δ2 100 010 X1 implies ϕ ¼ 2 with δ3 being completely unphysical. B C B C Regarding the CP labels, note that only the combination P123 ¼ @ 010A;P231 ¼ @ 001A; δ1 − δ2 is physical, but not the two phases δ1 and δ2 001 100 CP 0 1 independently. As a consequence, the generalized 001 symmetries enforce the charged lepton mass matrix with B C perturbation to be of the following form, P312 ¼ @ 100A; ð26Þ 010 2 2 † Uc ðM þ δM ÞU and correspond to three possible different cases respec- TBM cTBM cTBM 0 δ −δ 1 2 2 i 1 2 tively, as follows: M11 δM e 2 0 B C 2 −iδ1−δ2 2 ¼ @ 2 A: ð31Þ † iα iα iβ δM e M22 0 caseααβ∶ Pl ¼ P123 → Gl ¼ Uc diagðe ;e ;e ÞUc ; TBM TBM 2 00mτ ð27Þ

αβα∶ P ¼ P → G ¼ U† ðeiα;eiβ;eiαÞU ; case l 231 l cTBMdiag cTBM As a consistency check, we can see that imposing X2 does ð28Þ not add any new information, since we have already imposed the flavor symmetry parametrized by α ¼ δ1 − † iβ iα iα δ4 ¼ δ2 − δ5 and β ¼ δ3 − δ6. We can now see that the caseβαα∶ Pl ¼ P312 → Gl ¼ U diagðe ;e ;e ÞUc : cTBM TBM mass matrix in Eq. (30) can be diagonalized by ð29Þ iδ1 iδ2 iδ3 T diagðe 2 ;e 2 ;e 2 ÞU12ðθ; 0Þ with

053001-5 PENG CHEN et al. PHYS. REV. D 100, 053001 (2019)

2δM2 1 −iδ1 −iδ2 −iδ3 2θ ¼ − ; δM2 ¼ − ðm2 − m2Þ 2θ; U12ðθ; 0Þdiagðe 2 ;e 2 ;e 2 Þ tan M2 − M2 2 μ e sin 22 11 −iδ1 −iδ2 −iδ3 ¼ ðe 2 ;e 2 ;e 2 ÞU ðθ; δÞ; ð Þ 1 diag 12 35 M2 ¼ ½m2ð1 þ 2θÞþm2ð1 − 2θÞ; 11 2 e cos μ cos 1 with δ ¼ðδ2 − δ1Þ=2. Since the phases on the left side of a M2 ¼ ½m2ð1 − 2θÞþm2ð1 þ 2θÞ: ð Þ 22 2 e cos μ cos 32 mixing matrix can be absorbed by redefinitions of the charged lepton fields, only the combination of CP labels Note that θ will always be in the first or fourth quadrant and given by δ ≡ ðδ2 − δ1Þ=2 will be physically meaningful. is expected to be small, since we are perturbing the mass The lepton mixing matrix can be written as matrix. Therefore the charged lepton diagonalization matrix is given by U ¼ U ðθ; δÞU : ð Þ lep 12 cTBM 36 † iδ1 iδ2 iδ3 T U ¼ U ðe 2 ;e 2 ;e 2 ÞU ðθ; 0Þ: ð Þ cl cTBMdiag 12 33 where we remind that θ is a free real parameter and δ is the U ¼ U† U ¼ CP Consequently the lepton mixing matrix lep cl ν meaningful label. Now we can proceed to extract the U† mixing parameters from the lepton mixing matrix in cl will be given by Eq. (36). In the simplifying limit ρ → 0 and σ → 0, i.e. δ δ δ † −i 1 −i 2 −i 3 U ¼ U ðθ; 0Þ ðe 2 ;e 2 ;e 2 ÞU : ð Þ when the CP symmetries of the real TBM matrix Ur lep 12 diag cTBM 34 TBM U† and not cTBM are imposed, the mixing parameters are Moreover, we can exploit the relation given by the following expressions

2 2 2 sin θ 2 2 þ 2 sin 2θ cos δ 2 cos θ sin θ13 ¼ ; sin θ12 ¼ ; sin θ23 ¼ ; 2 3cos2θ þ 3 cos2θ þ 1 signðsin 2θÞð2cos2θ þ 2Þ sin δ sin δCP ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð5 þ 3 cos 2θ − 4 sin 2θ cos δÞð2 þ 2 sin 2θ cos δÞ ð 2θÞð 2θ þð6 2θ − 2Þ δÞ ð2 2θ þ 2Þ δ δ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisign sin sin cos cos ; δ ¼ cos sin ; cos CP tan CP 2 ð5 þ 3 cos 2θ − 4 sin 2θ cos δÞð2 þ 2 sin 2θ cos δÞ sin 2θ þð6cos θ − 2Þ cos δ 3 sin θð5 cos θ þ 3 cos 3θÞ sin δ þ 6sin2θcos2θ sin 2δ sin 2ϕ12 ¼ ; ð5 þ 3 cos 2θ − 4 sin 2θ cos δÞð1 þ sin 2θ cos δÞ 9sin22θsin2δ cos 2ϕ12 ¼ 1 − ; ð5 þ 3 cos 2θ − 4 sin 2θ cos δÞð1 þ sin 2θ cos δÞ 8 sin 2δcos2θ − 4 sin δ sin 2θ 16cos2θsin2δ sin 2ϕ13 ¼ ; cos 2ϕ13 ¼ 1 − : ð37Þ 5 þ 3 cos 2θ − 4 sin 2θ cos δ 5 þ 3 cos 2θ − 4 sin 2θ cos δ

ρ σ 2 2 The expressions for the general scenario in which and cos θ23cos θ13 ¼ 1=2 ; are nonzero and the initial symmetry is the complex TBM 2 2 † 3 cos 2θ12ð2 − 3cos θ13Þþcos θ13 mixing matrix Uc are particularly lengthy, specially δ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð Þ TBM cos CP 2 39 concerning the CP parameters. However, we can recover 6 sin 2θ12 sin θ13 2cos θ13 − 1 the general scenario just by doing the following substitutions in the previous equations This is because, if we only look at the oscillation observables, all the CP parameters (ρ, σ, δ1, δ2 and δ3) are either unphysical for oscillations (δ3 and σ), or can be δ → δ − 2ρ; ϕ12 → ϕ12 þ ρ; ϕ13 → ϕ13 þ ρ þ σ: ð38Þ rewritten in terms of just one meaningful CP label, 0 δ ≡ ðδ2 − δ1Þ=2 − 2ρ, while the expressions of the oscillation parameters (θij and δCP) will take the same form as in As a result, the modified variant of the complex TBM Eq. (37) just replacing δ by δ0. matrix predicts the same relations between mixing angles From Eq. (37) we see that both θ13 and θ23 only depend and the Dirac CP phase as the modified variant of the real on the free parameter θ. Hence the possible values of the TBM case. These predictions can be neatly summarized as free parameter θ are strongly constrained due to a very good

053001-6 CP SYMMETRIES AS GUIDING POSTS: REVAMPING … PHYS. REV. D 100, 053001 (2019)

FIG. 1. Correlations between θ13 and θ23 when two CP symmetries are preserved by the charged lepton sector. For the ααβ case (blue dashed line), they come from Eq. (39) (left) and the αβα case (red dashed line), they come from Eq. (46) (left). The boxes represent the 2 2 3σ and 1σ allowed ranges for normal ordered neutrino masses [1]. The star is the best-fit value of sin θ13 and sin θ23. Note that fixing θ13 in its 3σ allowed range greatly restricts the allowed values of the atmospheric angle θ23, nearly maximal in both cases. The ααβ and αβα cases correspond to first and second octant, respectively.

measurement of the reactor angle θ13. This implies a very perform a numerical analysis and randomly scan over θ and sharp prediction for the atmospheric angle θ23 independent the CP label δ in the ranges ½−π=2; π=2 and ½0; 2π of the particular values of the CP labels δ1 and δ2. The good respectively, keeping only the points for which the lepton determination of θ13 [1,41–43] and the fact that the mixing angles θij and δCP are consistent with experimental atmospheric mixing angle θ23 is a slowly varying function data at 3σ level. The resulting predictions for the lepton of θ13 in the allowed 3σ range, leads to a tight prediction, mixing angles and CP violation phases are displayed in θ23 ∈ ½44.28°; 44.43°, as shown in Fig. 1. Fig. 3. We observe strong correlations among the solar θ θ 2 Turning to the solar angle 12 Eq. (37) indicates that 12 angle sin θ12, the Dirac CP phase δCP and the Majorana CP δ and the Dirac phase CP depend on both the free phase ϕ12 and ϕ13 in this case. parameter θ and the CP parameter δ. The correlation The situation changes for the Majorana phases, since 2 between sin θ12 and δCP is displayed in Fig. 2, where nonzero ρ and σ will shift the Majorana phases, as can be the values of the CP label δCP are indicated by the color seen in Eq. (38). Therefore, the difference between the shadings. Requiring the solar angle θ12 in the experimen- symmetries of the real and the complex versions of TBM tally preferred 3σ range [1], we can read out the allowed can be seen only in neutrinoless double beta decay experi- region of δCP is 1.37 < δCP=π < 1.62. Furthermore, we ments, as discussed in Sec. VI B.

FIG. 2. Correlations between the mixing angle θ12 and the CP phase δCP when two CP symmetries are preserved by the charged lepton sector. For the ααβ case (left panel), they come from Eq. (39) (right), and in the αβα case (right panel), they come from Eq. (46) (right). The boxes are the 3σ and 1σ allowed ranges respectively for normal ordered neutrino masses [1]. The star is the best-fit value of 2 sin θ12 and δCP. The reactor angle θ13 is assumed to lie in the 3σ region of the global fit [1].The shaded colour indicates the value of the CP label δ.

053001-7 PENG CHEN et al. PHYS. REV. D 100, 053001 (2019)

FIG. 3. Correlations between mixing angles and CP phases when two CP symmetries are preserved by the charged lepton sector in the case ααβ. We treat both parameters θ and δ as random numbers, the three lepton mixing angles and Dirac CP phase δCP are required to lie within their 3σ ranges [1]. The blue, green and magenta regions correspond to the CP violation phases δCP, ϕ12, and ϕ13, respectively.

B. Case αβα where θ is expected to be small. Using similar arguments to U ¼ Following an analogous argument as in the ααβ case, we those of the previous case, the lepton mixing matrix lep U† U can now study the case in which the flavor symmetry is cl ν is given by G ¼ U† ðeiα;eiβ;eiαÞU l cTBMdiag cTBM. In this case the perturbation δM2 satisfies

† U ðM2 þ δM2ÞU U ¼ U13ðθ; δÞUc ; ð42Þ cTBM c c lep TBM 0 TBM TBM 1 2 2 iðδ1−δ3Þ M 0 δM e 2 B 11 C B 2 C ¼ B 0 mμ 0 C; ð40Þ @ A where we have defined δ ≡ ðδ3 − δ1Þ=2. Extracting the 2 −iðδ1−δ3Þ 2 ρ → 0 δM e 2 0 M33 mixing parameters in the real TBM case where and σ → 0, we find them to be 2 2 2 2 where M11, M22, δM , and M33 are real parameters and mμ is the muon mass. Equation (40) can be diagonalized by iδ =2 iδ =2 iδ =2 T diagðe 1 ;e 2 ;e 3 ÞU13ðθ; 0Þ , with 2θ 2 − 2 2θ δ 2θ ¼ sin ; 2θ ¼ sin cos ; sin 13 2 sin 12 3 2θ þ 3 2δM2 1 cos 2θ ¼ − ; δM2 ¼ − ðm2 − m2Þ 2θ 1 tan 2 2 τ e sin 2 M − M 2 sin θ23 ¼ ; ð43Þ 22 11 cos2θ þ 1 1 M2 ¼ ½m2ð1 þ 2θÞþm2ð1 − 2θÞ; 11 2 e cos τ cos 1 M2 ¼ ½m2ð1 − 2θÞþm2ð1 þ 2θÞ; ð Þ 33 2 e cos τ cos 41 while the CP phases are given by

053001-8 CP SYMMETRIES AS GUIDING POSTS: REVAMPING … PHYS. REV. D 100, 053001 (2019)

signðsin 2θÞð2 cos2 θ þ 2Þ sin δ sin δCP ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð5 þ 3 cos 2θ þ 4 sin 2θ cos δÞð2 − 2 sin 2θ cos δÞ ð 2θÞð 2θ − ð6 2 θ − 2Þ δÞ −2ð 2 θ þ 1Þ δ δ ¼ − pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisign sin sin cos cos ; δ ¼ cos sin ; cos CP tan CP 2 ð5 þ 3 cos 2θ þ 4 sin 2θ cos δÞð2 − 2 sin 2θ cos δÞ sin 2θ − 2ð3 cos θ − 1Þ cos δ −3 sin θð5 cos θ þ 3 cos 3θÞ sin δ þ 6 sin2 θ cos2 θ sin 2δ sin 2ϕ12 ¼ ; ð5 þ 3 cos 2θ þ 4 sin 2θ cos δÞð1 − sin 2θ cos δÞ 9 sin2 2θ sin2 δ cos 2ϕ12 ¼ 1 − ; ð5 þ 3 cos 2θ þ 4 sin 2θ cos δÞð1 − sin 2θ cos δÞ 8 sin δ cos2 θ þ 4 sin δ sin 2θ 16 cos2 θ sin2 δ sin 2ϕ13 ¼ ; cos 2ϕ13 ¼ 1 − : ð44Þ 5 þ 3 cos 2θ þ 4 sin 2θ cos δ 5 þ 3 cos 2θ þ 4 sin 2θ cos δ

Similar to the previous case, the general scenario in which ρ their predictions for neutrinoless double beta decay, as and σ are nonzero can be recovered by making the discussed in Sec. VI B. following substitutions

δ → δ − 2ρ − 2σ; ϕ → ϕ þ ρ; ϕ → ϕ þ ρ þ σ: V. CHARGED LEPTON MASS MATRIX 12 12 13 13 CONSERVING ONE CP SYMMETRY ð45Þ We will now study the case in which only one CP symmetry is preserved in the perturbation term. The CP Again, as in the previous case, the good measurement of U symmetries compatible with cTBM in the charged sector θ13 severely restrict the allowed values of the free parameter are θ. This gives a very sharp prediction for the atmospheric θ ∈ ½45 56 ; 45 71 angle 23 . ° . ° , as can be seen in Fig. 1. † X ¼ U ðeiδ1 ;eiδ2 ;eiδ3 ÞU ð Þ Moreover, we find the following analytical correlations cTBMdiag cTBM 47 between the physical parameters where δi are CP parameters labeling the CP transforma- 2 2 sin θ23cos θ13 ¼ 1=2 ; tion. The charged lepton squared mass matrix satisfies the 2 2 relation 3 cos 2θ12ð3cos θ13 − 2Þ − cos θ13 cos δCP ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð46Þ 6 2θ θ 2 2θ − 1 sin 12 sin 13 cos 13 X†M2X ¼ M2; ð48Þ which also hold in the case where ρ and σ are nonzero. where M2 ¼ M2 þ δM2. Then, the matrix form of Notice that, in contrast to the ααβ case, this time θ23 falls cTBM U M2U† in the second octant, although again very close to the cTBM cTBM can be written as maximal mixing value. It is worth noting that the two cases show a very similar behavior, as can be seen from U ðM2 þ δM2ÞU† cTBM cTBM cTBM Figs. 1 and 2. In both cases the angle θ23 is very close to 0 1 2 2 iðδ1−δ2Þ 2 iðδ1−δ3Þ M δM e 2 δM e 2 the maximal mixing value. As seen in Fig. 1 the first B 11 12 13 C ααβ αβα B iðδ −δ Þ iðδ −δ Þ C octant corresponds to the case, while the case is 2 − 1 2 2 2 2 3 ¼ B δM e 2 M δM e 2 C; associated to the second octant. The correlation between @ 12 22 23 A iðδ −δ Þ iðδ −δ Þ θ12 and δCP is similar to the previous case but of opposite 2 − 1 3 2 − 2 3 2 δM13e 2 δM23e 2 M33 shape. From here we can read off the allowed range for the Dirac CP phase, i.e., 1.379π ≤ δCP ≤ 1.634π. The ð49Þ numerical results for the lepton mixing and CP phase 2 2 2 2 parameters for the real TBM case are shown in Fig. 4, where M11, M22, M33, and δMij are real parameters, where the three mixing angles and the Dirac CP phase 3σ unconstrained by the residual symmetry. The matrix are required to lie inside the current range [1]. Similar U M2U† cTBM cTBM in Eq. (49) can be diagonalized by to the ααβ case, the modified variants of the complex and δ iδ1 iδ2 i 3 T ðe 2 ;e 2 ;e 2 ÞO O real TBM obey the same relations given in Eq. (46) and diag 3×3. Here 3×3 is a real orthogonal 2 hence their predictions for the three mixing angles and matrix, its matrix elements are determined by the M11, 2 2 2 the Dirac CP phase are the same. However they differ in M22, M33, and δMij [30], and it can be parametrized as

053001-9 PENG CHEN et al. PHYS. REV. D 100, 053001 (2019)

FIG. 4. Correlations between mixing angles and CP phases when two CP symmetries are preserved by the charged lepton sector for the αβα case. The blue, green, and magenta regions correspond to δCP, ϕ12, and ϕ13, respectively.

O ¼ U ðθ ; 0ÞU ðθ ; 0ÞU ðθ ; 0Þ: ð Þ 3×3 23 1 13 2 12 3 50 and therefore the lepton mixing matrix can be written as † −iδ1 −iδ2 −iδ3 U ¼ U U ¼ O ðe 2 ;e 2 ;e 2 ÞU : ð Þ Then, one can write lep cl ν 3×3diag cTBM 52

† iδ1 iδ2 iδ3 T Extracting the mixing angles in the simple limit of real U ¼ U ðe 2 ;e 2 ;e 2 ÞO ; ð Þ cl cTBMdiag 33 51 TBM, i.e., ρ ¼ 0 ¼ σ yields

1 δ − δ 2θ ¼ 1 þ 2θ θ 2 3 − 2θ 2θ ; sin 13 2 sin 2 sin 3 cos 2 cos 2cos 3

2ð1 − 2θ ð θ δ1−δ3 þ θ δ2−δ3Þþ 2θ 2θ δ1−δ2Þ 2 sin 2 cos 3 cos 2 sin 3 cos 2 sin 3cos 2 cos 2 sin θ12 ¼ ; δ2−δ3 2 2 3ð1 − sin 2θ2 sin θ3 cos 2 þ cos θ2cos θ3Þ −2 δ2−δ3 2θ θ θ − 2 2θ 2θ þ 2θ θ 2θ 2 2 cos 2 sin 1 cos 2 cos 3 cos 1cos 3 sin 1 sin 2 sin 3 sin θ23 ¼ sin θ1 − : ð53Þ δ2−δ3 2 2 2ð1 − sin 2θ2 sin θ3 cos 2 þ cos θ2cos θ3Þ

One sees that in this case there is no predictivity for the oscillation as well as neutrinoless double beta decay mixing angles, since all parameters are completely free. experiments. We will focus on the case where two remnant Since the expressions for the CP parameters are too lengthy CP symmetries are preserved in the charged lepton sector, to be enlightening, we will not show them here. as this is the most predictive situation.

A. Neutrino oscillations VI. PHENOMENOLOGICAL IMPLICATIONS The observation of neutrino oscillations indicates that In this section we shall study the phenomenological neutrinos are massive and that neutrino flavor eigenstates implications of the above CP symmetries for neutrino mix with each other. The three lepton mixing angles and the

053001-10 CP SYMMETRIES AS GUIDING POSTS: REVAMPING … PHYS. REV. D 100, 053001 (2019)

FIG. 5. The appearance probability Pμe (left panel) and the CP asymmetry Aμe (right panel) as functions of the beam energy when the baseline is fixed to 295 km (T2K experiment). In both panels the predicted band in cyan is the generically expected region obtained by varying the oscillation parameters within their 3σ ranges assuming normal ordering of neutrino masses [1], while the yellow band is the prediction for the ααβ neutrino mixing pattern, see text for explanation. neutrino mass-squared differences have been precisely The oscillation probability asymmetry between neutrinos measured. However, we have not yet established with high and antineutrinos is defined as: significance whether CP is violated in the lepton sector. Moreover, we still do not know whether the neutrino mass Pμe − Pμ¯ e¯ Aμe ¼ : ð55Þ spectrum has normal ordering (NO) or inverted ordering Pμe þ Pμ¯ e¯ (IO), nor whether the atmospheric angle lies in the first or second octant. The upcoming reactor and long-baseline In order to illustrate our points we focus, for definiteness, experiments such as JUNO, DUNE, T2HK should be able on the revamped ααβ scenario. We display the results for to shed light on these issues and they expect to bring us the neutrino appearance oscillation probability Pμe and the increased precision on the oscillation parameters θ12, θ23, CP asymmetry Aμe in Figs. 5–7. These are given in terms of and δCP. As shown in previous sections, lepton mixing the neutrino energy at a fixed distance of L ¼ 295 km, parameters are predicted to lie in narrow regions and L ¼ 810 km, and L ¼ 1300 km, corresponding to the correlations among the mixing parameters are obtained baselines of the T2K, NOvA, and DUNE experiments, when two residual CP symmetries are preserved. This respectively. would translate into phenomenological implications for the Note that our generalized CP symmetries lead to expected neutrino and antineutrino appearance probabil- correlations involving the mixing and CP violation param- ities in neutrino oscillation experiments. eters. In particular, the three mixing angles and the CP The νμ → νe neutrino oscillation probability in matter phase can be given in terms of only two parameters. This can be expanded to second order in the mass hierarchy translates into restrictions on the attainable ranges for the 2 2 – parameter α ≡ Δm21=Δm31 and the reactor angle sin θ13,as neutrino oscillation probabilities. In Figs. 5 7 the cyan follows from [44]: bands are the generically expected regions obtained when the oscillation parameters are varied within their current 3σ 2AΔ 2ðA − 1ÞΔ ranges, while the yellow bands denote the predictions for P ¼ α2 22θ c2 sin þ 4s2 s2 sin μe sin 12 23 A2 13 23 ðA − 1Þ2 the ααβ neutrino mixing pattern. The solid black lines correspond to the current best fit point predictions, where þ 2αs 2θ 2θ ðΔ þ δ Þ 13 sin 12 sin 23 cos CP we have chosen the Dirac phase δCP ¼ 1.5π. As a final AΔ ðA − 1ÞΔ comment let us mention that, by looking at Eqs. (37) and sin sin ; ð Þ × A A − 1 54 (38) one sees that the oscillation results obtained by taking real or complex TBM as the starting point before revamp- Δm2 L ρ σ Δ ¼ 31 A ¼ 2EV L ing, i.e., taking and in Eq. (15) to be nonzero, are with 4E and Δm2 , where is the baseline 31 essentially the same. length and E is the energy of the neutrino beam. V ≃ 7 56 The matter-induced effective potential is . × B. Neutrinoless double decay 10−14 ρ Y Y ¼ 0 5 g=cm3 e eV where e . is the electron fraction The neutrinoless double beta (0νββ) decay ðA; ZÞ → and a constant matter density ρ ¼ 3 g=cm3 is assumed. ðA; Z þ 2Þþ2e− is the unique probe of the Majorana The oscillation probability for antineutrinos is related to nature of neutrinos [45]. There are many experiments that for neutrinos by Pμ¯ e¯ ¼ PμeðδCP → −δCP;V → −VÞ. currently searching for 0νββ decay, or in various stages

053001-11 PENG CHEN et al. PHYS. REV. D 100, 053001 (2019)

FIG. 6. Same as Fig. 5 when the baseline is fixed to 810 km (NOvA experiment), see text for explanation.

FIG. 7. Same as Fig. 5 when the baseline is fixed to 1300 km (DUNE experiment), see text for explanation. of planning and construction. The sensitivity to this rare mixing angles disappear. It is easy to understand this process should improve significantly, with good prospects behavior from Eq. (38). It is worth noting that in the real for probing the whole region of parameter space associated TBM case, i.e., ρ ¼ 0 ¼ σ, the analytical expression of the with the inverted ordering spectrum. The 0νββ decay effective Majorana mass mee simplifies to, provides another test of the CP symmetries of TBM. ααβ 1 For our Ansatz in the general scenario where the rTBM iδ 2 iδ 2 m ¼ jm1ðsinθ − 2e cosθÞ þ 2m2ðe cosθ þ sinθÞ phases ρ and σ are nonzero, we find the analytical ee;ααβ 6 m 2 expression of the effective Majorana mass ee is given by þ 3m3sin θj: ð57Þ 1 cTBM 2iρ iðδ−ρÞ 2 Limiting ourselves to the real TBM case we can not only m ¼ jm1e ðsin θ − 2e cos θÞ ee;ααβ 6 generate CP violation as shown in the previous sections, iðδ−ρÞ 2 −2iσ 2 þ 2m2ðe cos θ þ sin θÞ þ 3m3e sin θj: but also obtain very stringent predictions for neutrinoless double beta decay. The allowed values of mee are shown in ð Þ 56 Fig. 8, where the oscillation parameters are required to lie in the currently preferred 3σ ranges [1]. One sees that the Notice that ρ, σ, and δ are CP parameters, i.e., they label the real TBM prediction for mee in the IO case corresponds to CP symmetries respected by the mass matrix, while θ is a the upper boundary of the generic IO region, very close to completely free parameter, which represents the degree up the sensitivities of the upcoming 0νββ decay experiments. to which the mass matrix is not determined by the remnant Notice the existence of a lower bound for mee in the real CP symmetry. If all the three labels ρ, σ, and δ are treated as TBM scenario also for the NO case, where generically it is free parameters, i.e., scanning over the full class of mass absent due to possible cancellations. matrices which allow some preserved CP symmetry of the We can also study the predictions for neutrinoless double complex TBM Ansatz, there is no prediction for the 0νββ beta decay within the αβα scenario. The effective mass mee decay, since the correlations between Majorana phases and parameters in this case is given by

053001-12 CP SYMMETRIES AS GUIDING POSTS: REVAMPING … PHYS. REV. D 100, 053001 (2019)

FIG. 8. The effective Majorana mass mee versus the lightest neutrino mass for the ααβ (left) and the αβα cases (right). The thick colored regions correspond to the predictions of the complex TBM scenario, while the thin magenta and purple bands are for the real TBM case, with ρ ¼ σ ¼ 0. The red and blue dashed lines delimit the most general allowed regions for IO and NO neutrino mass spectra, and are obtained by varying the mixing parameters over their 3σ ranges [1]. The most stringent current upper limit mee < 0.061 eV from KamLAND-ZEN [7] and EXO-200 [46] is shown by horizontal grey band. The vertical grey band refers to the current sensitivity of cosmological data from the Planck collaboration [47].

1 cTBM iδ iðρþσÞ 2 TBM Ansatz for the lepton mixing matrix in a systematic m ¼ jm1ð2 cos θe þ sin θe Þ ee;αβα 6 manner. The resulting generalized patterns share some of iðδ−ρÞ iσ 2 2 þ 2m2ðe cos θ − e sin θÞ þ 3m3sin θj; the attractive features of the original TBM matrix, while being consistent with current oscillation experiments indi- ð Þ 58 cating nonvanishing θ13. We have explicitly examined the case where two CP symmetries are preserved in the which simplifies to the following expression in the limit of charged lepton sector, the resulting predictions given in ρ ¼ 0 ¼ σ , Eqs. (39) and (46) and Figs. 1–4. We have also briefly 1 discussed some of the phenomenological implications of mcTBM ¼ jm ð2 θeiδ þ θÞ2 our new mixing patterns, both for neutrino oscillation as ee;αβα 6 1 cos sin well as neutrinoless double beta decay search experiments, iδ 2 2 þ 2m2ðe cos θ − sin θÞ þ 3m3sin θj: ð59Þ illustrated in Figs. 5–8. Finally, we mention that the same systematic procedure may be employed in order to revamp Imposing the current restrictions from neutrino oscillations other a priori unrealistic patterns of neutrino mixing. [1] one finds nearly identical results for the 0νββ decay amplitude parameter mee also for this case. Indeed, mee in the αβα scenario only differs from the ααβ case by a ACKNOWLEDGMENTS slightly different value for θ23 (see Fig. 1) and a slightly This work is supported by National Natural Science different correlation for θ12 vs δCP (see Fig. 2). The Foundation of China under Grants No. 11835013, atmospheric angle is nearly maximal in both cases, but No. 11522546, and No. 11847240 and China Postdoctoral in different octant. Science Foundation under Grant No. 2018M642700 and the Spanish Grants No. FPA2017-85216-P (AEI/FEDER, UE), VII. SUMMARY No. SEV-2014-0398 and No. PROMETEO/2018/165 Starting from the complex version of the tribimaximal (Generalitat Valenciana). We thank the support of the lepton mixing pattern we have examined the generalized Spanish Red Consolider MultiDark FPA2017-90566- CP symmetries of the charged lepton mass matrix. These REDC. S. C.C is also supported by the FPI Grant symmetries are employed in order to “revamp” the simplest No. BES-2016-076643.

053001-13 PENG CHEN et al. PHYS. REV. D 100, 053001 (2019)

[1] P. F. de Salas, D. V. Forero, C. A. Ternes, M. Tórtola, and [20] M. H. Rahat, P. Ramond, and B. Xu, Asymmetric tribimax- J. W. F. Valle, Status of neutrino oscillations 2018: 3σ hint imal texture, Phys. Rev. D 98, 055030 (2018). for normal mass ordering and improved CP sensitivity, [21] K. S. Babu, E. Ma, and J. W. F. Valle, Underlying Að4Þ Phys. Lett. B 782, 633 (2018). symmetry for the neutrino mass matrix and the quark mixing [2] R. Acciarri et al. (DUNE Collaboration), Long-baseline matrix, Phys. Lett. B 552, 207 (2003). neutrino facility (LBNF) and deep underground neutrino [22] S. Morisi and J. W. F. Valle, Neutrino masses and mixing: experiment (DUNE), arXiv:1512.06148. A flavour symmetry roadmap, Fortschr. Phys. 61, 466 [3] R. Srivastava, C. A. Ternes, M. Tórtola, and J. W. F. (2013). Valle, Zooming in on neutrino oscillations with DUNE, [23] S. F. King and C. Luhn, Neutrino mass and mixing with Phys. Rev. D 97, 095025 (2018). discrete symmetry, Rep. Prog. Phys. 76, 056201 (2013). [4] N. Nath, R. Srivastava, and J. W. F. Valle, Testing general- [24] P. Chen, C.-C. Li, and G.-J. Ding, Lepton flavor mixing and ized CP symmetries with precision studies at DUNE, CP symmetry, Phys. Rev. D 91, 033003 (2015). Phys. Rev. D 99, 075005 (2019). [25] P. Chen, G.-J. Ding, F. Gonzalez-Canales, and J. W. F. Valle, [5] J. Schechter and J. W. F. Valle, Neutrino masses in SUð2Þ × Generalized μ − τ reflection symmetry and leptonic CP Uð1Þ theories, Phys. Rev. D 22, 2227 (1980). violation, Phys. Lett. B 753, 644 (2016). [6] J. Schechter and J. W. F. Valle, Neutrino oscillation thought [26] P. Chen, G.-J. Ding, F. Gonzalez-Canales, and J. W. F. Valle, experiment, Phys. Rev. D 23 (1981) 1666. Classifying CP transformations according to their texture [7] A. Gando et al. (KamLAND-Zen Collaboration), Search zeros: Theory and implications, Phys. Rev. D 94, 033002 for Majorana Neutrinos near the Inverted Mass Hierarchy (2016). Region with KamLAND-Zen, Phys. Rev. Lett. 117, 082503 [27] P. Chen, S. C. Chuliá, G.-J. Ding, R. Srivastava, and J. W. F. (2016); Phys. Rev. Lett. 117, 109903(A) (2016). Valle, Neutrino predictions from generalized CP sym- [8] M. Agostini et al. (GERDA Collaboration), Improved Limit metries of charged leptons, J. High Energy Phys. 07 on Neutrinoless Double-β Decay of 76Ge from GERDA (2018) 077. Phase II, Phys. Rev. Lett. 120, 132503 (2018). [28] L. L. Everett, T. Garon, and A. J. Stuart, A bottom-up [9] C. E. Aalseth et al. (Majorana Collaboration), Search for approach to lepton flavor and CP symmetries, J. High Neutrinoless Double-β Decay in 76Ge with the Majorana Energy Phys. 04 (2015) 069. Demonstrator, Phys. Rev. Lett. 120, 132502 (2018). [29] L. L. Everett and A. J. Stuart, Lepton sector phases and their [10] C. Alduino et al. (CUORE Collaboration), First Results roles in flavor and generalized CP symmetries, Phys. Rev. D from CUORE: A Search for Lepton Number Violation 96, 035030 (2017). via 0νββ Decay of 130Te, Phys. Rev. Lett. 120, 132501 [30] P. Chen, S. C. Chuliá, G.-J. Ding, R. Srivastava, and J. W. F. (2018). Valle, CP symmetries as guiding posts: Revamping tri-bi- [11] J. B. Albert et al. (EXO Collaboration), Search for Neu- maximal mixing. Part I, J. High Energy Phys. 03 (2019) trinoless Double-Beta Decay with the Upgraded EXO-200 036. Detector, Phys. Rev. Lett. 120, 072701 (2018). [31] P. Chen, S. C. Chuliá, G.-J. Ding, R. Srivastava, and J. W. F. [12] R. Arnold et al. (NEMO-3 Collaboration), Measurement of Valle, Realistic tribimaximal neutrino mixing, Phys. Rev. D the 2νββ decay half-life and search for the 0νββ decay of 98, 055019 (2018). 116Cd with the NEMO-3 detector, Phys. Rev. D 95, 012007 [32] P. Pasquini, S. C. Chuli, and J. W. F. Valle, Neutrino (2017). oscillations from warped flavor symmetry: Predictions for [13] H. Ishimori, T. Kobayashi, H. Ohki, Y. Shimizu, H. Okada, long baseline experiments T2K, NOvA and DUNE, Phys. and M. Tanimoto, Non-abelian discrete symmetries in Rev. D 95, 095030 (2017). particle physics, Prog. Theor. Phys. Suppl. 183, 1 (2010). [33] R. Srivastava, C. A. Ternes, M. Tórtola, and J. W. F. Valle, [14] P. F. Harrison, D. H. Perkins, and W. G. Scott, Tri-bimaximal Testing a lepton quarticity flavor theory of neutrino oscil- mixing and the neutrino oscillation data, Phys. Lett. B 530, lations with the DUNE experiment, Phys. Lett. B 778, 459 167 (2002). (2018). [15] Y. Abe et al. (Double Chooz Collaboration), Improved [34] S. S. Chatterjee, P. Pasquini, and J. W. F. Valle, Probing measurements of the neutrino mixing angle θ13 with the atmospheric mixing and leptonic CP violation in current Double Chooz detector, J. High Energy Phys. 10 (2014) and future long baseline oscillation experiments, Phys. Lett. 086; Erratum, J. High Energy Phys. 02 (2015) 74. B 771, 524 (2017). [16] F. P. An et al. (Daya Bay Collaboration), Measurement of [35] N. Nath, Consequences of μ − τ Reflection Symmetry at electron antineutrino oscillation based on 1230 days of DUNE, Phys. Rev. D 98, 075015 (2018). operation of the Daya Bay experiment, Phys. Rev. D 95, [36] K. Chakraborty, K. N. Deepthi, S. Goswami, A. S. 072006 (2017). Joshipura, and N. Nath, Exploring partial μ − τ reflection [17] M. Y.Pac (RENO Collaboration), Recent results from RENO, symmetry at DUNE and Hyper-Kamiokande, Phys. Rev. D Proc. Sci., NuFact2017 (2018) 038, [arXiv:1801.04049]. 98, 075031 (2018). [18] S. Morisi, M. Nebot, K. M. Patel, E. Peinado, and J. W. F. [37] P. Pasquini, Long-baseline oscillation experiments as a tool Valle, Neutrino mixing with revamped A4 flavor symmetry, to probe high energy flavor symmetry models, Adv. High Phys. Rev. D 88, 036001 (2013). Energy Phys. 2018, 1 (2018). [19] S. Chatterjee, M. Masud, P. Pasquini, and J. W. F. Valle, [38] G.-J. Ding, N. Nath, R. Srivastava, and J. W. F. Valle, Status Cornering the Revamped BMV Model with Neutrino and prospects of bi-large leptonic mixing, Phys. Lett. B 796, Oscillation Data, Phys. Lett. B 774, 179 (2017). 162 (2019).

053001-14 CP SYMMETRIES AS GUIDING POSTS: REVAMPING … PHYS. REV. D 100, 053001 (2019)

[39] P. Chen, C.-Y. Yao, and G.-J. Ding, Neutrino mixing from [44] E. K. Akhmedov, R. Johansson, M. Lindner, T. Ohlsson, CP symmetry, Phys. Rev. D 92, 073002 (2015). and T. Schwetz, Series expansions for three flavor neutrino [40] J. Schechter and J. W. F. Valle, Comment on the lepton oscillation probabilities in matter, J. High Energy Phys. 04 mixing matrix, Phys. Rev. D 21, 309 (1980). (2004) 078. [41] P. F. de Salas, S. Gariazzo, O. Mena, C. A. Ternes, and M. [45] J. Schechter and J. W. F. Valle, Neutrinoless double beta Tórtola, Neutrino mass ordering from oscillations and decay in SUð2Þ × Uð1Þ theories, Phys. Rev. D 25, 2951 beyond: 2018 status and future prospects, Front. Astron. (1982). Space Sci. 5, 36 (2018). [46] J. B. Albert et al. (EXO Collaboration), Search for Neu- [42] S. Gariazzo, M. Archidiacono, P. F. de Salas, O. Mena, C. A. trinoless Double-Beta Decay with the Upgraded EXO-200 Ternes, and M. Tórtola, Neutrino masses and their ordering: Detector, Phys. Rev. Lett. 120, 072701 (2018). Global data, priors and models, J. Cosmol. Astropart. Phys. [47] P. A. R. Ade et al. (Planck Collaboration), Planck 2015 03 (2018) 011. results. XIII. Cosmological parameters, Astron. Astrophys. [43] Valencia-Globalfit, http://globalfit.astroparticles.es/, 2018. 594, A13 (2016).

053001-15