PHYSICAL REVIEW D 100, 075036 (2019) Lepton and quark mixing from a stepwise breaking of flavor and CP † Claudia Hagedorn * and Johannes König CP3-Origins, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark (Received 1 June 2019; published 30 October 2019) We explain all features of lepton and quark mixing in a scenario with the flavor symmetry Δð384Þ and a charge conjugation-parity (CP) symmetry, where these are broken in several steps. The residual symmetry in the neutrino and up quark sector is a Klein group and CP, while a Z3 and a Z16 symmetry are preserved among charged leptons and down quarks, respectively. If the Klein group in the neutrino sector is further broken down to a single Z2 symmetry, we obtain predictions for all lepton mixing parameters in terms of one real quantity, whose size is determined by the value of the reactor mixing angle. The Dirac and Majorana phases are fixed, in particular sin δ ≈−0.936. A sum rule, relating these CP phases and the reactor and atmospheric mixing angles, θ13 and θ23, is given. In the quark sector, we have for the Cabibbo angle θC ¼ sin π=16 ≈ 0.195 after the first step of symmetry breaking. This is brought into full accordance with experimental data with the second step of symmetry breaking, where either the Z16 group is broken to a Z8 symmetry in the down quark sector or the Klein group to one Z2 symmetry only among up quarks. The other two quark mixing angles are generated in the third and last symmetry breaking step, when the residual symmetries in the up and/or down quark sector are further broken. If this step occurs among both up and down quarks, the amount of CP violation in the quark sector is determined by the lepton sector and q explaining the current neutrino oscillation data entails that the Jarlskog invariant JCP is in very good q agreement with experimental findings. Last, a sum rule is derived that contains the CP phase δ and θC of the quark sector and the lepton mixing parameters θ13, θ23, and δ. DOI: 10.1103/PhysRevD.100.075036 I. INTRODUCTION analyses focus on the lepton sector [7,10–18] and only rather few attempt to (also) determine the mixing in the Most of the free parameters in the Standard Model (SM) quark sector by the mismatch of residual symmetries in up of particle physics are related to fermion masses and and down quark sectors [5,6,19–24]. mixing. In particular, the size of the mixing angles in We consider as Gf the discrete group Δð384Þ and a CP the quark and lepton sectors and their striking difference as symmetry that corresponds to an automorphism of Gf well as the amount of charge conjugation-parity (CP) [10,25–27]. Being primarily interested in fermion mixing violation remain unexplained. Approaches with discrete, in this study, we focus on the three generations of left- non-Abelian flavor symmetries Gf have been successfully handed (LH) lepton doublets Li and quark doublets Qi, employed in order to determine the mixing angles of i ¼ 1, 2, 3, not specifying the transformation properties of leptons and quarks; for reviews on flavor symmetries, – the right-handed (RH) fields. Furthermore, we assume that see [1 4]. Those, where different residual symmetries neutrinos are Majorana particles, whose mass is generated remain preserved in the charged lepton (down quark) from the Weinberg operator. The three generations of LH and neutrino (up quark) sectors, can determine all mixing CP – lepton doublets and quark doublets both transform as a angles and the Dirac-type phase [5 7]. Even more faithful, irreducible, complex three-dimensional represen- predictive are approaches with a flavor and a CP symmetry tation 3 of Δð384Þ. The symmetries Gf and CP are broken [8–10], since these allow to fix all CP phases with the help by an unspecified mechanism to different residual sym- of the choice of the residual symmetries. Many of the metries in the neutrino and up quark sector, charged lepton as well as down quark sector. This first step of symmetry *[email protected] breaking leads to tri-bimaximal (TB) mixing [28,29] in the † [email protected] lepton sector and generates the Cabibbo angle θC ¼ sin π=16 ≈ 0.195 in the quark sector [6,11]. In a further Published by the American Physical Society under the terms of step of symmetry breaking, where the residual symmetry the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to among neutrinos is reduced, the reactor mixing angle is the author(s) and the published article’s title, journal citation, induced and all lepton mixing parameters become functions and DOI. Funded by SCOAP3. of one real quantity, whose size is determined by the reactor 2470-0010=2019=100(7)=075036(10) 075036-1 Published by the American Physical Society CLAUDIA HAGEDORN and JOHANNES KÖNIG PHYS. REV. D 100, 075036 (2019) mixing angle θ13. Accommodating the measured value of with e being the neutral element of the group [34]. The the atmospheric mixing angle θ23 well and the indication group Δð384Þ has several subgroups, among them Klein for close to maximal CP violation in neutrino oscillations groups Z2 × Z2, Z3 and Z16 symmetries [11]. It possesses [30], one CP symmetry is singled out that leads to in total 13 Klein groups: 12 of these are conjugate to each 2 4 k 4 2 k sin θ23 ≈ 0.579 and sin δ ≈−0.936. For this choice of other and can be generated by c and abc , d and a bd , 4 k k CP symmetry, the Majoranapffiffiffi phases α and β have to fulfil and ðcdÞ and bc d for k ¼ 0, 1, 2, 3, respectively, while j sin αj¼jsin βj¼1= 2. Both Majorana phases fulfil a the remaining one is normal and can be obtained from c4 4 nontrivial relation, involving the CP phase δ and the two and d as generators. There are 64 Z3 symmetries contained i j mixing angles θ13 and θ23. The quantity mee, measurable in in Δð384Þ, and these can be described by ac d for neutrinoless double beta decay, becomes strongly con- 0 ≤ i; j ≤ 7. The 12 Z16 subgroups can be generated by 2lþ1 2lþ1 2 2lþ1 strained, e.g., for positive sin α and sin β we find mee ≳ bd , abd , and a bc for l ¼ 0,1,2,3.AllZ3 and −3 2.86 × 10 eV for neutrino masses with normal ordering Z16 groups are conjugate to each other, respectively. For a and for inverted ordering that mee is detectable with the comprehensive list of subgroups and their properties, next generation of experiments [31]. In the quark sector, see [11,16]. either the residual group in the up or the down quark sector As is known, CP symmetries are automorphisms of Gf is reduced at the second step of symmetry breaking, so that [10,25–27] and a large class of these have been discussed the Cabibbo angle is brought into full accordance with in [15] for the groups Δð6n2Þ. In the present study, we experimental data [32]. Breaking the residual symmetries consider the ones corresponding to the automorphisms among up and/or down quarks even further eventually gives composed by rise to the remaining two quark mixing angles. If this −1 −1 breaking occurs in only one of the two sectors, the amount a → a; b → b; c → c ; and d → d ð2Þ of CP violation in quark mixing will depend on phases, that abcsd2s 0 ≤ s ≤ 7 are in general not specified further by the residual sym- and the group transformation , . L metries. If instead the latter are broken in the down as well The three generations of LH lepton doublets i and Q i ¼ 1 as the up quark sector, a strong correlation between CP quark doublets i, , 2, 3 are assigned to the same violation in the quark and in the lepton sector can be faithful, irreducible, complex three-dimensional represen- Jq tation 3, which can be represented by the generators að3Þ, established and the Jarlskog invariant CP [33] can be bð3Þ cð3Þ Jq ≈ 3 29 10−5 , , determined to be CP . × , in very good agree- 0 1 0 1 0 1 ment with experimental data [32]. This strong correlation 010 001 ω8 00 q leads to a sum rule, relating the CP phase δ and the B C B C B 7 C að3Þ¼@001A;bð3Þ¼@010A;cð3Þ¼@ 0 ω8 0A Cabibbo angle θC in the quark sector to the lepton mixing parameters δ, θ13, and θ23. These symmetry breaking 100 100 001 sequences differ in several aspects from the symmetry ð3Þ breaking pattern that has been proposed in [23,24] in order 2 2πi=8 1 to describe lepton and quark mixing with the help of a and dð3Þ¼að3Þ cð3Það3Þ with ω8 ¼ e . In the rep- flavor group Δð6n2Þ, n integer, and CP. resentation 3, the mentioned type of CP symmetry corre- The remainder of the paper is organized as follows: in sponds to the CP transformation Xð3Þ, Δð384Þ Sec. II, we present basic information about and the s 2s employed CP symmetry. In Secs. III and IV, we show the Xð3ÞðsÞ¼að3Þbð3Þcð3Þ dð3Þ X0ð3Þ; 0 ≤ s ≤ 7; ð4Þ results for fermion mixing, arising from the different steps with X0ð3Þ representing the CP symmetry induced by the of symmetry breaking, in the lepton and quark sector, automorphism in Eq. (2) and being of the form of the respectively.