Lepton and Quark Mixing from a Stepwise Breaking of Flavor and CP
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. We summarize and conclude in Sec. V. identity matrix in the used basis. Residual symmetries in the different fermion sectors of II. BASICS ABOUT Δð384Þ AND CP the theory are Abelian subgroups of Gf, possibly together CP We use as Gf the group Δð384Þ, which belongs to the with the symmetry. From their mismatch, fermion series Δð6n2Þ, n integer. This group can be written as mixing arises. In particular, quark mixing is due to the G ðZ8 × Z8Þ ⋊ S3 with S3 being the permutation group of mismatch of the residual group u in the up quark and the three distinct objects. It can be described with four one, called Gd, in the down quark sector, while lepton generators, a, b, c, and d, that fulfil the relations 1We could have chosen any of the other faithful, irreducible, a3 ¼ e; b2 ¼ e; c8 ¼ e; d8 ¼ e; complex three-dimensional representations of Δð384Þ. Note that 2 −1 −1 −1 this can lead to the residual symmetries having different gen- ðabÞ ¼ e; cd ¼ dc; aca ¼ c d ; erators in terms of the elements of the flavor group in order to CP ada−1 ¼ c; bcb−1 ¼ d−1; bdb−1 ¼ c−1; ð Þ achieve the same results for mixing angles and phases; 1 compare also [11,12].
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⋆ mixing comes from the mismatch of Gν and Gl, the residual with denoting complex conjugation. This implies that symmetries in the neutrino and charged lepton sector, also the other elements of the Klein group commute with respectively. the CP transformation. For Gl, we use the generator a. The contributions Uν and Ul to the lepton mixing matrix U III. LEPTON SECTOR PMNS, arising from the neutrino and charged lepton sector, respectively, are determined by the unitary matrices diag- In the first step of symmetry breaking, we preserve G a Klein group contained in Δð384Þ together with a CP onalizing the generators of the residual symmetries ν;1 G 2 symmetry, represented by a CP transformation in Eq. (4), and l, respectively. They are of the form in the neutrino sector, while a Z3 symmetry is conserved 0 s−k s−k 1 among charged leptons, ω16 0 −iω16 1 B kþs kþs C Gν;1 ¼ Z2 × Z2 × CP and Gl ¼ Z3; ð5Þ Uνðk; sÞ¼pffiffiffi @ ω16 0 iω16 A and 2 pffiffiffi 8−s 4 0 2ω 0 see Fig. 1. We choose as generators of the Klein group c 0 18 and abck and leave k as well as the parameter s, labeling the 1 ωω2 1 CP B 2 C symmetry, unspecified for the moment. As can be Ul ¼ pffiffiffi @ 1 ω ω A; ð7Þ checked explicitly, the generators of the Klein group 3 11 1 and the CP transformation commute in the representation 3, i.e., 2πi=16 2πi=3 ω16 ¼ e ω ¼ e 4 ⋆ 4 with and , where we have chosen a Xð3ÞðsÞðcð3Þ Þ − cð3Þ Xð3ÞðsÞ¼0 and particular ordering of the columns, and thus ordering of the Xð3ÞðsÞðað3Þbð3Þcð3ÞkÞ⋆ − að3Þbð3Þcð3ÞkXð3ÞðsÞ¼0; neutrino and charged lepton masses, that can entail a viable description of the data on lepton mixing parameters.3 The ð Þ 6 form of the lepton mixing matrix is then
† U ;1ðk; sÞ¼Ul Uνðk; sÞ PMNS 0 pffiffiffi 1 s 8−s s 2ω16 cos ϕk 2ω8 −2ω16 sin ϕk 1 B pffiffiffi pffiffiffi pffiffiffi C πk ¼ pffiffiffi B s 8−s s C ϕ ¼ : ð Þ @ −ω16ðcos ϕk þ 3 sin ϕkÞ 2ω8 −ω16ð 3 cos ϕk − sin ϕkÞ A with k 8 6 pffiffiffi pffiffiffi pffiffiffi 8 s 8−s s −ω16ðcos ϕk − 3 sin ϕkÞ 2ω8 ω16ð 3 cos ϕk þ sin ϕkÞ
We see, in particular, that the sine of the reactor mixing k ¼ 0: ð10Þ angle θ13 is given by This can be considered as a reasonable result after the first 2 step of symmetry breaking. 2θ ¼ 2ϕ : ð Þ sin 13 3 sin k 9 In the second step, the Klein group in the neutrino sector 4 is broken to a Z2 symmetry, generated by abc , while the k k ¼ 0 Taking into account the admitted choices of , ,1,2, CP symmetry is preserved 3, we find that the matrix in Eq. (8) becomes the TB mixing 4 matrix for the choice Gν;2 ¼ Z2 × CP: ð11Þ The residual symmetry among charged leptons remains 2 U CP We note that we constrain ν additionally such that the untouched; see Fig. 1. In this way, the contribution to transformation Xð3ÞðsÞ becomes the identity matrix after apply- † ⋆ lepton mixing, arising from the neutrino sector, becomes ing Uν, i.e., UνXð3ÞðsÞUν is the identity matrix. 3 less constrained, i.e., an additional rotation in the (23)- As is well known, exchanging the second and third columns θ of Ul leads to the atmospheric mixing angle changing octant and plane through an angle , undetermined by the symmetries the Dirac phase sign. of the theory and in the range between 0 and π, has to be 4This result can be understood by considering the group arising taken into account. The plane of the rotation is fixed by the from the combination of the generators of the Klein group and of fact that the unbroken generator abc4 in the representation the Z3 symmetry, preserved among neutrinos and charged 4 3 reads after applying Uνðk ¼ 0;sÞ, leptons, respectively, as one generates with the elements c , ab a Δð24Þ n ¼ 2 S , and the group ( ) that is isomorphic to 4. The † 4 latter group is well known for leading to TB mixing [7],ifa Uνðk ¼ 0;sÞ að3Þbð3Þcð3Þ Uνðk ¼ 0;sÞ¼diagð−1; 1; 1Þ: Z residual Klein group among neutrinos and a 3 symmetry in the ð Þ charged lepton sector is present. 12
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FIG. 1. Stepwise breaking of Δð384Þ and CP. The stepwise breaking of the flavor and CP symmetry in the different sectors of the theory and the corresponding results for lepton and quark mixing are shown. There are two steps of symmetry breaking in the lepton sector, while three steps are necessary in the quark sector in order to generate all three quark mixing angles. The particular choice of the residual symmetries in the second and third steps in the quark sector corresponds to the minimal viable case that leads to a strong correlation between the amount of CP violation in the lepton and in the quark sector. The results for quark mixing angles and the q Jarlskog invariant JCP can be found in Sec. IV, while those for the lepton mixing parameters are detailed in Sec. III. We thus arrive at a lepton mixing matrix of the form 2θ 1 1 − 3 2θ 2θ ¼ cos ¼ sin 13 ≈ 0 318: ð Þ sin 12 2 2 . 16 2 þ cos θ 3 1 − sin θ13 U ;2ðsÞ¼U ;1ðk ¼ 0;sÞR23ðθÞ with PMNS 0PMNS 1 10 0 This sum rule is well known [35] and follows from the fact B C that the lepton mixing matrix in Eq. (13) leads to TM1 R23ðθÞ¼@ 0 cos θ sin θ A: ð13Þ mixing [36]. In contrast, the atmospheric mixing angle θ23 0 − sin θ cos θ also depends on the value of s, pffiffiffi U ðsÞ When using PMNS;2 as lepton mixing matrix, we 2 1 2 6 sin 2θ 3πs sin θ23 ¼ 1 − cos : ð17Þ assume that the diagonal elements of the neutrino mass 2 5 þ cos 2θ 8 matrix are positive semidefinite after the application of U ðk ¼ 0;sÞR ðθÞ ν 23 . If this is not given, additional signs θ I I Using Eq. (14) and that is small, see Eq. (15), we can will appear in the Majorana invariants 1 and 2, defined in θ θ U ðsÞ relate 23 and 13, Eq. (20); compare discussion in [10,15]. From PMNS;2 , we read off that pffiffiffi 2 1 3πs sin θ23 ≈ − 2 cos sin θ13: ð18Þ 1 2 8 2θ ¼ 2θ: ð Þ sin 13 3 sin 14 3πs This relation can be used to express cosð 8 Þ in terms of the 2 Thus, the experimental data on the reactor mixing angle are two lepton mixing angles θ13 and θ23. Requesting sin θ23 2 5 3σ accommodated well, sin θ13 ≈ 0.022 [30], for to be within the experimentally preferred range [30] for θ ≈ 0.26 excludes several choices of s: s ¼ 0, s ¼ 2, s ¼ 3, 2 s ¼ 5, and s ¼ 6, while s ¼ 1 leads to sin θ23 very close to θ ≈ 0.26: ð15Þ the lower 3σ bound. The remaining two values of s, s ¼ 4 and s ¼ 7, lead to maximal atmospheric mixing and 2 Also the value of the solar mixing angle is fixed, since it is sin θ23 ≈ 0.579, respectively. strongly correlated with the reactor one The results for the amount of CP violation in the lepton sector can be quantified with the Jarlskog invariant JCP and 5We note that there exists a second solution with θ being the Majorana invariants I1 and I2 [37] (see also [38–40]). replaced by π − θ. This is known from preceding analyses [15]. These are defined as
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J ¼ ½ðU Þ ðU Þ⋆ ðU Þ⋆ ðU Þ 3σ 6 CP Im PMNS 11 PMNS 13 PMNS 31 PMNS 33 which is disfavored at more than the level. We thus 1 consider ¼ 2θ 2θ 2θ θ δ; 8 sin 12 sin 23 sin 13 cos 13 sin s ¼ 7 ð25Þ I ¼ ½ðU Þ2 ððU Þ⋆ Þ2 1 Im PMNS 12 PMNS 11 2 2 4 s ¼ sin θ12cos θ12cos θ13 sin α; as most suitable choice for . Using the information on the two Majorana phases α and β, we can compute the range of I ¼ ½ðU Þ2 ððU Þ⋆ Þ2 2 Im PMNS 13 PMNS 11 the quantity mee, accessible in neutrinoless double beta 2 2 2 ¼ sin θ13cos θ12cos θ13 sin β ð19Þ decay. Given that this scenario does not make predictions for the neutrino mass spectrum, we parametrize the latter in −2 terms of the lightest neutrino mass m0, 0≤m0 ≲4×10 eV and read for U ¼ U ;2ðsÞ in Eq. (13) PMNS PMNS in agreement with limits from cosmology [41], and the best Δm2 fit values of the two mass squared differences sol and 1 3πs Δm2 JCP ¼ − pffiffiffisin2θ sin ; atm, which we take for the two different neutrino mass 6 6 8 orderings from [30]. Assuming neutrinos follow normal 2 3πs 2 3πs ordering, we obtain I ¼ − 2θ I ¼ − 2θ : 1 9cos sin 4 and 2 9sin sin 4 −3 −2 2.86 × 10 eV ≲ mee ≲ 1.94 × 10 eV and ð20Þ −2 −2 2.43 × 10 eV ≲ mee ≲ 3.11 × 10 eV ð26Þ For θ ≪ 1, we can extract simple formulae for the CP for neutrino masses with inverted ordering. We note, phases δ, α, and β, in particular, that for neutrinos with normal ordering mee cannot vanish and has a minimum value for m0 ≈ 3πs 3πs 3 62 10−3 sinδ ≈−sin ; sinα ¼ sinβ ¼ −sin ; ð21Þ . × eV and that for inverted ordering the minimum 8 4 of mee is also larger than the lower bound expected when using the current information from neutrino oscillation compare also [15]. With the help of Eq. (18) and the first experiments at the 3σ level, cf. also Fig. 9 in [42]. relation in Eq. (21), we can relate the Dirac phase δ and the This mixing pattern belongs, according to the classifi- two lepton mixing angles θ13 and θ23, cation in [15], to Case 3 b.1). In [15], very similar numerical results have been found for Δð6n2Þ with 2 2 2 2 n ¼ 8, Gl ¼ Z3 generated by a and Z2 × CP, preserved ð1 − 2sin θ23Þ ≈ 8sin θ13cos δ: ð22Þ 4 4 in the neutrino sector, with bc d being the generator of Z2 7 and the CP transformation given as bð3Þcð3Þ dð3ÞX0ð3Þ in This sum rule turns out to be a very good approximation the representation 3. This choice of residual symmetries is of the exact one, given in [35] for TM1 mixing. related to the one used in the present study by the similarity Furthermore, we can express the Majorana phases α and transformation a; see [15] for details, and thus the results β in terms of the Dirac phase δ and the two lepton mixing coincide, up to a different definition of the angle θ. angles θ13 and θ23, As all data on lepton mixing can be satisfactorily CP described and predictions for phases are made after 3πs 3πs the second step of symmetry breaking, we conclude the α ¼ β ¼ −2 sin sin sin 8 cos 8 study of the lepton sector at this point. We note, however, 2 that in concrete models that realize this symmetry 1 − 2sin θ23 ≈ sin δ pffiffiffi : ð23Þ breaking sequence we expect that higher order corrections 2 sin θ13 at a certain point break all residual symmetries; see, e.g., [43,44]. Using Eq. (20) and θ ≈ 0.26 together with the choice s ¼ 7, we find IV. QUARK SECTOR
1 In the first step of symmetry breaking, the same residual sin δ ≈−0.936; sin α ¼ sin β ¼ pffiffiffi : ð24Þ symmetry is preserved in the up quark sector as in the 2
6 δ A possibility to cure this is to include an additional permu- The value of is hence close to the best fit value, obtained tation of the second and third generations of charged leptons. from the global fit in [30]. The choice s ¼ 4 would in Since the atmospheric mixing angle is maximal for s ¼ 4, this contrast correspond to maximal CP violation δ ¼ π=2 permutation does not affect its value.
075036-5 CLAUDIA HAGEDORN and JOHANNES KÖNIG PHYS. REV. D 100, 075036 (2019) neutrino one and a Z16 subgroup remains intact in the down leading order description of quark mixing, has already been quark sector, i.e., made in [11]. In the second step of symmetry breaking in the quark Gu;1 ¼ Gν;1 ¼ Z2 × Z2 × CP and Gd;1 ¼ Z16; ð27Þ sector, the leading order result of the Cabibbo angle is brought into full agreement with the experimental data [32]. as also shown in Fig. 1. In order to obtain a value for the This can be achieved in two different ways. One possibility Cabibbo angle close to the measured one at this step of is that the residual symmetry Gu;1 is broken to symmetry breaking, we have to choose the generator of 2ð4−kÞ 1 Gu;2 ¼ Z2 × CP; ð31Þ Gd;1 as abd with k ¼ 0,1,2,3forGu;1 ¼ Gν;1. The form of the contribution of the up quark sector to the quark 4 where the remaining Z2 symmetry is generated by c ; see U ðk; sÞ mixing matrix is the same as ν in Eq. (7), up to the Fig. 1. The latter element reads in the representation 3 after ordering of the columns of the mixing matrix that depends applying the matrix Uuðk; sÞ, on the up quark masses. Indeed, we choose in the following a slightly different ordering for the columns of Uuðk; sÞ † 4 Uuðk; sÞ cð3Þ Uuðk; sÞ¼diagð−1; −1; 1Þ; ð32Þ than of Uνðk; sÞ, i.e.,
0 s−k s−k 1 meaning that it allows for a rotation R12ðθuÞ in the (12)- ω16 −iω16 0 7 plane through an angle θu. The absolute values of the 1 B kþs kþs C Uuðk; sÞ¼pffiffiffi @ ω iω 0 A: ð28Þ elements of the resulting quark mixing matrix become 2 16 16 pffiffiffi 8−s 002ω8 † jV ;2;uj¼jðUuðk;sÞR12ðθuÞÞ Ud; 1ðkÞj CKM 0 1 In addition, the form of the mixing matrix Ud that encodes jcosðπ=16 ∓ θuÞj jsinðπ=16 ∓ θuÞj 0 the contribution from the down quark sector to the quark B C ¼ @ jsinðπ=16 ∓ θuÞj jcosðπ=16 ∓ θuÞj 0A: mixing matrix is 001 0 1 1−2k 1−2k ω16 −ω16 0 ð33Þ 1 B C Ud; 1ðkÞ¼pffiffiffi @ 110A; ð29Þ 2 pffiffiffi Choosing θu≈ ∓0.030 depending on the sign in Eq. (33), 002 we achieve jVusj¼0.22452, corresponding to the exper- imental best fit value [32]. The other possibility is to reduce where Ud; 1ðkÞ corresponds to the choice of Gd;1 as 8 Gd;1 to residual symmetry among down quarks. The absolute values of the elements of the resulting quark mixing matrix Gd;2 ¼ Z8; ð34Þ read for both choices 2ð4−kÞ 1 2 2 1−2k † generated by ðabd Þ ¼ðcd Þ , respectively, jV ;1j¼jUuðk; sÞ Ud; 1ðkÞj CKM 0 1 while the residual symmetry Gu;1 in the up quark sector cos π=16 sin π=16 0 remains intact. The form of these generators in the B C 3 U ðkÞ ¼ @ sin π=16 cos π=16 0 A ð30Þ representation is after applying d; 1 001 † 2 1−2k Ud; 1ðkÞ ðcð3Þdð3Þ Þ Ud; 1ðkÞ 0 1 k 1−2k and are independent of . As can be read off, the size of the ω8 00 Cabibbo angle θC is sin π=16 ≈ 0.195 which represents a B C ¼ @ 0 ω 1−2k 0 A: ð35Þ reasonably good leading order approximation to the exper- 8 imentally measured value [32]. Combining the generators 00ið−1Þkþð1 1Þ=2 of Gu;1 and Gd;1 leads for all choices of k to a group that has 128 elements and is called ⟦128; 67⟧ in the computer This shows that two of their eigenvalues are degenerate for program GAP [45,46]. This group can be written as all admitted values of k, such that the contribution of the ðZ8 × Z8Þ ⋊ Z2 and only has irreducible representations 7 of dimension one and two. We have explicitly checked that Like in the case of Gν;2, the fact that Gu;2 contains CP as the representation 3 of Δð384Þ decomposes into a complex symmetry constrains the additional unitary matrix, leaving Eq. (32) invariant, to be a rotation with one free real parameter only. one- and a faithful, irreducible, complex two-dimensional 8 We could also consider G ¼ Z4 or G ¼ Z2, since these representation in the group ⟦128; 67⟧. The observation that d;2 d;2 choices will lead to the same additional matrix U12ðθd; ψ dÞ,as breaking Δð384Þ to a Z16 subgroup and a Klein group can also their generators have two degenerate eigenvalues in the lead to one mixing angle sin π=16 ≈ 0.195, suitable for a representation 3.
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q down quark sector to quark mixing includes an additional breaking, we find that JCP crucially depends on the phases unitary transformation U12ðθd; ψ dÞ of the form that are contained in the unitary matrices, arising from the last 0 1 step of symmetry breaking (and possibly the second one, if in −iψ cos θd sin θde d 0 thattheresidualsymmetryofthedownquarksectoris B iψ C reduced). The value of such phases is in general not con- U12ðθd; ψ dÞ¼@ − sin θde d cos θd 0 A ð36Þ strained by the residual symmetries. However, in a concrete 001 model, these can take specific values due to fixed phases, originating from the vacuum expectation values of some with free parameters θd and ψ d. The resulting quark mixing flavor (and CP) symmetry breaking fields; see [44].One matrix is example is to assume that in the second and third steps only the residual symmetry in the up quark sector is broken, i.e., † jV ;2;dj¼jUuðk; sÞ Ud; 1ðkÞU12ðθd; ψ dÞj after the second step the quark mixing matrix is of the form as CKM 0 1 j cos ζjjsin ζj 0 in Eq. (33) and scenario (ii) is realized in the third step. We B C find ¼ @ j sin ζjjcos ζj 0 A;
001 q π q q sinθ12 ≈ sin ∓θu ; θ23 ≈jθu;23j; θ13 ≈jθu;13j; 2ζ ¼ π=8 2θ ∓ π=8 2θ ψ : 16 with cos cos cos d sin sin d sin d 1 π ð37Þ Jq ¼ ∓2θ 2θ 2θ θ ψ and CP 8sin 8 u sin u;13 sin u;23 cos u;23 sin
This shows that effectively only one real parameter is with ψ ¼ψ u;13 −ψ u;23; ð38Þ responsible for achieving jVusj in full accordance with the experimental results and possible numerical values of θd where θu;13, ψ u;13 and θu;23, ψ u;23 parametrize the unitary and ψ d can be obtained in a similar way as for θu. We thus transformations, arising from the third step of symmetry conclude that both versions of the second step of symmetry breaking, analogously to θd, ψ d for the matrix U12ðθd; ψ dÞ in breaking in the quark sector, together with the two different Eq. (36). We note that only the difference ψ of the two phases − þ possible choices Gd;1 and Gd;1, lead to the same phenom- ψ u;13 and ψ u;23 enters and that its value is essential for the enology. For reasons of minimality, we do not consider the determination of the amount of CP violation in the quark case where Gu;1 and Gd;1 are both broken to Gu;2 and Gd;2, sector. In order to accommodate the best fit values for the q respectively. quark mixing angles θij and the Jarlskog invariant [32],we In the third step of symmetry breaking the two remaining choose the parameters as9 quark mixing angles are generated together with the CP phase such that all experimental data can be accommodated θu ≈ ∓ 0.030; jθu;23j ≈ 0.042; well [32]. We thus assume that this step of symmetry jθu;13j ≈ 0.00365 and j sin ψj ≈ 0.95; ð39Þ breaking induces in the quark mixing matrix two additional angles acting in the (13) and (23) planes. We can envisage showing that the breaking in the second and third steps is different scenarios for this last step: small. Indeed, it is almost an order of magnitude smaller than (i) only the residual flavor symmetry in the up quark in the lepton sector; compare Eq. (15). sector is broken, while leaving the CP symmetry in If instead one of the remaining quark mixing angles this sector together with the residual group among comes from the breaking of the residual symmetry in the up down quarks intact; quark and the other one from that in the down quark sector, (ii) the residual symmetry in the up quark sector is Jq k s broken completely, while the one in the down quark CP always depends on the choice of the parameters and . sector is still preserved; Minimal viable cases, in which only one phase is due to the (iii) the symmetry in the up quark sector remains symmetry breaking, are encountered, if the second step of symmetry breaking takes place in the up quark sector. Out untouched, while the residual one among down of the four possible cases, depending on the choice of Gd;1 quarks is broken; as well as q q (iv) the residual symmetries in both, up and down quark, and on whether θ23 or θ13 is dominantly generated in the up sectors are broken. quark sector, two are particularly interesting. In these cases, q With one of the four different quark mixing matrices, that θ23 is due to the symmetry breaking in the up quark sector. can possibly arise from the second step of symmetry They lead to a viable quark mixing matrix, even if no phase breaking, see Eqs. (33) and (37), the different versions of the third step lead to one of the following results for 9 The signs of θu;23 and θu;13 are not uniquely fixed, but quark mixing: if the residual symmetry is only broken in constrained through the requirement to eventually obtain the q one of the two sectors in the third step of symmetry correct sign for JCP for a fixed value of ψ.
075036-7 CLAUDIA HAGEDORN and JOHANNES KÖNIG PHYS. REV. D 100, 075036 (2019) is induced by the symmetry breaking in the down quark θu≈ ∓0.030; jθu;23j ≈ 0.042 and jθd;13j ≈ 0.00375; sector in the third step. We display this symmetry breaking ð44Þ scenario in Fig. 1. We get as result for the mixing angles q and JCP, demonstrating that also in this case only a small breaking is needed at the second and third steps in the quark sector.
q π q Using the values of k and s that are most suitable for the sinθ12 ≈ sin ∓ θu ; θ23 ≈ jθu;23j; 16 description of the lepton sector, k ¼ 0 and s ¼ 7, and π choosing the signs of the parameters θu;23 and θd;13 θq ≈ θ ∓ θ ; 13 d;13 cos 16 u accordingly, we arrive at q 1 π π q −5 π JCP ¼ ∓ sin ∓ 2θu cos ∓ θu J ≈ 3 35 10 ð2 ∓ 1Þ þ ψ : ð Þ 8 8 16 CP . × × cos d;13 45 16 3πs π 2θ 2θ ϕ þ ∓ þ ψ × sin d;13 sin u;23 sin k 8 16 d;13 This formula shows that even in the limit, in which no phase ψ d;13 is generated in the third step of symmetry ð40Þ q breaking, the value of JCP is correctly accommodated
q −5 þ depending on Gd;1. Extracting a formula for the CP phase JCP ≈ 3.29 × 10 for Gd;1 q δ itself, we find q −5 − and JCP ≈ 2.79 × 10 for Gd;1 ð46Þ 3πs π þ q that is in the experimentally preferred 1σ range for Gd;1 and j sin δ j ≈ sin ϕk þ ∓ þ ψ d;13 : ð41Þ 8 16 − q −5 3σ range for Gd;1, compare JCP ¼ð3.18 0.15Þ × 10 [32]. The vanishing of the phase ψ d;13 can be explained in a This relation is very interesting, since it allows to relate the concrete model, if, e.g., the operators relevant for the q CP phase δ with the lepton mixing parameters δ, θ13, and symmetry breaking in the down quark sector do only θ23. Since k is set to zero after the first step of symmetry contain flavor (and CP) symmetry breaking fields that breaking in the lepton sector, see Eq. (10), we focus on this acquire real vacuum expectation values. In order to achieve q case. Using Eq. (18) and the first relation in Eq. (21),we the current experimental best fit value of JCP, the phase find ψ d;13 should take a value
þ − 3πs π ψ d;13 ≈ 0.13 for G and ψ d;13 ≈−0.27 for G ; ð47Þ j δqj≈ ∓ þ ψ d;1 d;1 sin sin 8 cos 16 d;13 3πs π respectively. All shown numerical estimates agree well with χ2 þ cos sin ∓ þ ψ d;13 the results of an analysis, based on the experimental best 8 16 2 fit values and 1σ errors found in [32]. In this χ analysis, we
have also studied quark mixing, arising from the other ≈ sin δ cosðθC ∓ ψ d;13Þ possible realizations of the second and third symmetry 2 breaking step, and found that all of them can lead to a viable 1 − 2sin θ23 pffiffiffi sinðθC ∓ ψ d;13Þ ; ð42Þ description of the experimental data except for one case, 2 2 sin θ13 where no CP violation can be induced, since the only symmetry that is reduced in the second and third steps is the where we have also inserted the Cabibbo angle θC, whose residual flavor symmetry in the up quark sector. value is given by θC ¼ sin π=16 ≈ π=16 after the first We have thus shown that quark mixing can be success- step of symmetry breaking in the quark sector; compare fully described with three steps of symmetry breaking. Eq. (30). This relation simplifies further, if we consider a While the breaking in the first step is unique, different case, where ψ d;13 vanishes, namely options arise at the second and third steps. As explained, these can lead to different results. In particular, the case, 1 − 2 2θ where only the residual symmetry in either the up quark or q sin 23 jsinδ j ≈ sinδcosðθCÞ pffiffiffi sinðθCÞ : ð43Þ the down quark sector is broken at the third step, has to be 2 2 θ sin 13 distinguished from the case, where symmetry breaking occurs in both sectors in the third step; see Fig. 1. In the q The experimental best fit values of the quark mixing angles former case, JCP depends on phases that originate from the are achieved for (second and) third step of symmetry breaking, whereas in
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q the latter JCP crucially depends on the parameters k and s after the first step of symmetry breaking and brought into that determine the amount of CP violation in the lepton full accordance with experimental data with the second one. sector. This second case thus leads to a sum rule between In the third step, eventually the two other quark mixing q the CP phase δ and the Cabibbo angle θC in the quark angles are generated. For certain sequences of symmetry sector and the Dirac phase δ and the mixing angles θ13 and breaking, we find a close correlation between the amount of θ23 in the lepton sector; see Eqs. (42) and (43). In the limit, CP violation in the lepton and in the quark sector. This in which the second and third steps of symmetry breaking correlation can be expressed as a sum rule among the CP q do not introduce any phases as free parameters, the most phase δ and the Cabibbo angle θC in the quark sector and q δ suitable choice of k and s for leptons can indeed lead to JCP the Dirac phase and the reactor and atmospheric mixing in accordance with experimental data. Corrections to the angles θ13 and θ23 in the lepton sector. The most con- entertained symmetry breaking scenario might occur in an strained viable sequence is shown in Fig. 1, where explicit model, e.g., through higher-dimensional operators, determining the generators of the residual symmetries, but are usually suppressed. common to the neutrino and the up quark sector, with the help of neutrino oscillation data can lead to a value of q V. SUMMARY AND CONCLUSIONS JCP that lies in the experimentally preferred 1σ range, CP independent of further free parameters. We have presented a scenario with a flavor and a A realization of this symmetry-based scenario in a symmetry, where the mixing patterns for leptons and concrete model will be presented elsewhere [44]. This quarks arise from the stepwise breaking of these sym- model will have salient features beyond the successful metries to different residual subgroups in the different description of lepton and quark mixing in terms of residual sectors of the theory. In particular, the lepton mixing pattern symmetries and their breaking, such as the explanation of the originates from two steps of symmetry breaking, where the charged fermion mass hierarchies via higher-dimensional first one gives rise to TB mixing and the second one operators (fixing the ordering of rows and columns of the introduces one real quantity, whose size is determined by mixing matrices), the generation of light neutrino masses via the requirement to correctly accommodate the reactor the type-I seesaw mechanism, as well as the spontaneous mixing angle. Furthermore, a sum rule is derived that breaking of the flavor and CP symmetry. relates the Dirac and Majorana phases with the reactor and atmospheric mixing angles. One CP symmetry is singled ACKNOWLEDGMENTS out by the experimental data on the atmospheric mixing angle and the indications for δ close to 3π=2. In the quark The CP3-Origins centre is partially funded by the Danish sector, the Cabbibo angle is fixed to θC ¼ sin π=16 ≈ 0.195 National Research Foundation, Grant No. DNRF90.
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