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Physics Letters B

www.elsevier.com/locate/physletb

Tri-bimaximal-Cabibbo mixing

S.F. King

School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, UK

ARTICLE INFO

Article history: Received 8 August 2012 Received in revised form 13 September 2012

Accepted 8 October 2012 Available online 11 October 2012 Editor: T. Yanagida

ABSTRACT

Recent measurements of the lepton mixing angle 913 by the Daya Bay and RENO reactor experiments are consistent with the relationship 913 & QC/-f2 where 9C is the Cabibbo angle. We propose Tri-Bimaximal-Cabibbo (TBC) mixing, in which sin 913 = sin9C/\f2, sin 623 = 1/V2 and sin 912 = 1/V3. We show that TBC mixing may arise approximately from tri-bimaximal, bi-maximal or Golden Ratio neutrino mixing, together with Cabibbo-like charged lepton corrections arising from a Pati-Salam gauge group, leading to predictions for the CP-violating phase of S & ±90°, ±180°, ±75°, respectively. Alternatively, we show that TBC neutrino mixing may realised accurately using the type I see-saw mechanism with partially constrained sequential right-handed neutrino dominance, assuming a family symmetry which is broken by a flavon common to quarks and neutrinos.

© 2012 Published by Elsevier B.V.

1. Introduction

913 = 9° ± 1C

012 = 34° ± 1°, d23 = 45° ± 5°.

It is one of the goals of theories of particle physics beyond the Standard Model to predict quark and lepton masses and mixings, or at least to relate them. While the quark mixing angles are known to all be rather small, by contrast two of the lepton mixing angles, the atmospheric angle 623 and the solar angle 612, are identified as being rather large [1]. This is usually interpreted as demonstrating that quark mixing is very different from lepton mixing. However the smallest remaining angle, the reactor angle 613, has recently been measured and its value shown to be not that small. We shall discuss the implications of the observation that the smallest lepton mixing angle, 613, may be related to the largest quark mixing angle, the Cabibbo angle, 6C, providing a possible link between lepton and quark mixing.

Early indications for 613 from global fits were given in [2]. Direct evidence for 813 was first provided by T2K, MINOS and Double CHOOZ [3-5]. Global fits including these results were subsequently given in [6,7]. Recently, Daya Bay [8] have measured,

sin2 2913 = 0.092 ± 0.016(stat.) ± 0.005(syst.), while, shortly afterwards, RENO [9] have measured,

sin2 2913 = 0.113 ± 0.013(stat.) ± 0.019(syst.).

The global fits for the solar and atmospheric in [6,7], together with the above Daya Bay and RENO results for the reactor angle, lead to the approximate one sigma ranges for the lepton mixing angles,

The above results from the Daya Bay and RENO reactor experiments are consistent with a remarkable relationship between the smallest lepton mixing angle 013 and the largest quark mixing angle, 0C, namely 013 & 0C/42, where the Cabibbo angle 0C & 13° implies Qi3 & 9.2°. In Section 2 we combine this relation with maximal atmospheric mixing and trimaximal solar mixing to give Tri-Bimaximal-Cabibbo (TBC) mixing in which sin 613 = sin 0C/42, sin 023 = 1/V2 and sin 012 = 1/V3. In Section we show how approximate TBC mixing may emerge from tri-bimaximal, bimaximal or Golden Ratio neutrino mixing, by invoking Cabibbo-like charged lepton corrections, leading to approximate predictions for the CP-violating phase of S & ±90°, ±180°, ±75°, respectively. The required Cabibbo-like charged lepton mixing may be present in Pati-Salam models with a particular Clebsch structure. Alternatively, in Section 4, we show how accurate TBC neutrino mixing may arise from the type I see-saw mechanism with partially constrained sequential right-handed neutrino dominance. This may be realised in models with a family symmetry where a misaligned flavon is common to both the neutrino and quark sectors. The summary and conclusions are given in Section 5.

2. Tri-bimaximal-Cabibbo mixing

The recent data is consistent with the remarkable relationship, sin 0C X

42 42'

E-mail address: king@soton.ac.uk.

where X = 0.2253 ± 0.0007 [1] is the Wolfenstein parameter. This relationship is an example of "Cabibbo Haze" [10], the general

0370-2693/$ - see front matter © 2012 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.physletb.2012.10.028

hypothesis that the Cabibbo angle is an expansion parameter for lepton as well as quark mixing, where it appears as the Wolfenstein parameter in the parametrisation of the CKM matrix. It was proposed earlier in the context of "Quark-Lepton Complementarity" (QLC) in which 612 + 6C = 45° [11], and even earlier still in [12]. For related approaches see [13]. Our approach in Section 3 relies on maximal atmospheric mixing but the solar angle is determined by "Sum Rules" [14], which differ from the QLC relation. These examples illustrate that the value of the solar angle is independent of the relation in Eq. (4). On the other hand, phenomenology is consistent with a trimaximal solar angle as in Eq. (3), and furthermore the approach in Section 4 suggests a trimaximal solar angle. It is therefore natural to combine Eq. (4) with TB mixing, as discussed below.

In terms of the combination measured by the reactor neutrino experiments, Eq. (4) implies,

sin2 2013 = 2X2 ( 1 - y )« 0.099,

in excellent agreement with the recent Daya Bay and RENO results above. Furthermore the above ansatz implies a reactor angle of

013 « -= :

'< 9.2°

where 6C & 13° is the Cabibbo angle.

Apart from the reactor angle, the measured and fitted atmospheric and solar angles are in good agreement with the ansatz of Tri-Bimaximal (TB) mixing [15]. We are therefore led to combine the relation in Eq. (4) with TB mixing to yield Tri-Bimaximal-Cabibbo (TBC) mixing:

In terms of the TB deviations parameters defined in [16], this corresponds to r = X with s = a = 0. Using the second order expansion in [16], Eq. (7) then leads to the following approximate form of the mixing matrix,

Utbc -

-3 o - 4 x2)

-3 0 - 1 xeis)

~2 ' - (2 - 1X2 )

fi(1 - 4X2)

--6(1 + Xe'S^ V-6 (1 - XeiS%) --3(1 + 1 XeiS%) -(1 - 4X2))

+ O(X3), corresponding to the mixing angles,

0i3 « 9.2°, 012 = 35.26°

023 = 45°.

3. TBC mixing from charged lepton corrections

In a typical convention (see e.g. [18]) the PMNS matrix may be constructed as,

U = UeU vt

where in models, Ue is related to the left-handed rotations involved in diagonalising the charged lepton mass matrix Me, and Uv is related to the matrix that diagonalises the left-handed Majorana neutrino mass matrix mv. A particular model typically has a preference for a particular basis in which Me and mv take a particular form, leading to Ue and Uv having also particular forms which may be separately parameterised by e and v mixing angles in analogy with the PDG parameterisation. Here we show how Eq. (6) can simply arise from a zero neutrino mixing angle 6V3 & 0 with Cabibbo-like charged lepton corrections 6^ & 0c.

In many grand unified theories (GUTs) (for some recent examples see e.g. [17]) the charged lepton mixing angles are dominated by 6e2 ^ 6e3,62e3. Furthermore, assuming 6\2 ^ 6V3, it has been widely observed that charged lepton corrections then imply [18,19],

613 &—: • (11)

Note that the factor of 1/V2 arises from maximal atmospheric neutrino mixing. In order to achieve Eq. (6) we need only assume that the dominant charged lepton angle 61e2 is equal to the Cabibbo angle,

0(2 « 0c .

Then Eqs. (11) and (12) imply, 0C_ -2

in agreement with Eq. (6).

(12) (13)

3.1. Simple neutrino mixing patterns

Below we give three classic examples of simple patterns of mixing in the neutrino sector which all have 6j3 = 0 and 62V3 = 45°, namely Tri-Bimaximal (TB) neutrino mixing [15], Bi-Maximal (BM) neutrino mixing (see e.g. [20,21] and references therein), and the Golden Ratio (GR) neutrino mixing [22,23]. They all lead to solar mixing angle Sum Rules [14] involving the physical CP-violating oscillation phase 8 [1],

u 0C 012 « 0Ï2 + -:

'01, + - cos 5,

where 6^ = 35.26°, 45°, 31.7° for the case of TB, BM, GR1 neutrino mixing respectively, where we have used the prediction in Eq. (13). Note that the Sum Rule is subject to (typically small) corrections due to renormalisation group running and canonical normalisation effects [25]. Given the prediction in Eq. (13) for 613, the sum rule in Eq. (14) then yields a favoured range of cos 8 in each case which may be tested in proposed neutrino experiments [26]. We discuss this in more detail below for each of the three cases:

3.1.1. TB mixing in the neutrino sector

-L sv = _L

-3' s23 = -2'

together with charged lepton corrections

sÏ3 = 0

0C « X » 013,02e3 yields to first order in 012 [14],

-, s12 + ^ cos S,

42 V3 V3

This is not quite of the TBC form in Eq. (7) due to the large deviation in the solar angle, leading to the approximate linear relation between the solar angle and cos 8,

612 & 35.26° + 9.2° cos 8. (18)

Thus TB neutrino mixing implies that cos 8 & 0 or 8 & ±90° in order that the solar angle does not deviate to much from its TB value.

1 Note that there is an alternative version of GR mixing where cos6V3 = ^/2 and

¡2 = 36° [24]. This leads to the Sum Rule 612 & 36° + 9.2° cos 8, numerically similar

to the case of TB neutrino mixing.

3.1.2. Bi-maximal neutrino mixing

± su =--2 , s23 =-2,

together with charged lepton corrections

si3 = 0

0Î2 « 0C « X » 013 ,023 yields to first order in 01e2 [14],

s13 « —, s12 « —+ - cos S,

42 V2 2

S23 • (21)

Again this is not quite of the TBC form in Eq. (7) due to the large deviation in the solar angle, leading to the approximate linear relation between the solar angle and cos S,

O12 « 45° + 9.2° cos 8.

Thus BM neutrino mixing implies that cos 8 &—1 or 8 & ±180° in order to achieve a solar angle 612 & 35.8° which lies within two sigma above the global best fit value. Note that this is a very distinct prediction from the above case of TB neutrino mixing where we predict 8 &±90°.

3.1.3. Golden Ratio neutrino mixing 1

sv13 = 0

/1+02 '

where the Golden Ratio is 0 = (1 + 45)/2, together with charged lepton corrections

0Î2 « 0C « X » 0[3,02e3

yields to first order in 01e2 the relation,

S13 ~ 7, S12 ~ —=- , -

42 + 02 + 02

cos S,

72 •

As before, this is not quite of the TBC form due to the deviation in the solar angle, leading to the approximate linear relation between the solar angle and cos 8,

012 « 31.7° + 9.2° cos b.

Thus GR neutrino mixing implies that cos 8 & 0.25 or 8 & ±75° in order to achieve a solar angle 612 & 34°. This is closer to the case of TB neutrino mixing where we predict 8 & ±90°, but is very different from BM neutrino mixing where we predict 8 &±180°.

3.2. ModelS with 6en ;

A crucial assumption of the above approach is that the dominant charged lepton mixing angle is equal to the Cabibbo angle, namely that 6^ & 6C, as stated in Eq. (12). This provides a connection between the charged lepton sector and the quark sector which may hint at some underlying quark-lepton unification. However traditional quark-lepton unification models involve different relations, for example the Georgi-Jarlskog (GJ) [27] prediction would be 6e2 & 6C/3. Other more recent studies which consider large values of 6je2 in GUT models [28,29] do not explain 6^ & 6C. It therefore requires some discussion about how this might be achieved in models.

Let us focus on the upper 2 x 2 block of the mass matrices, assuming an approximately diagonal up-type quark Yukawa matrix.

In order to achieve 0e2 « 0C we propose a structure (in a LR convention for mass matrices):

* Cd X

CeX Ce

where the factors of ced represents the effect of Clebsch coefficients in some unified model where the assumption of equal Clebsch factor in the (1, 2) and (2, 2) elements leads to the relation 6^2 & 6C. For example the choice ce/cd = 3 gives the mass relation m^ = 3ms at the GUT scale, which can be achieved in GUTs as in the original GJ mechanism [27]. Note that there are many ways to obtain the correct electron and down quark masses, which depend on the unspecified elements denoted by "*" above (assumed to be smaller than the (2, 2) element). For example, operators exist which contribute to either the charged lepton or the down quark mass matrix, but not both at the same time, in Pati-Salam models [30,31].

It is not the purpose of this Letter to provide a detailed model, but the general strategy is clear. One may start from some family symmetry GF which is capable of yielding a simple pattern of neutrino mixing such as TB, BM or GR, for example GF = A4, S4, A5 as discussed in many papers (for a review see e.g. [32]). Then one must extend such a model to include both the quarks and leptons by assuming a Pati-Salam gauge group, for example, with mass matrices of the above form, leading to Cabibbo-like charged lepton corrections. We emphasise that the key feature of this approach is a Cabibbo-like charged lepton correction 612 & 6C, starting from a zero neutrino mixing angle 6V3 & 0. Other approaches to obtain a large reactor angle are discussed in [33].

4. TBC mixing in the neutrino sector

In the previous section we showed how Eq. (4) could arise in cases with 6V3 & 0 and 6^ & 6C. In this section we show how it can arise in models with zero charged lepton corrections, namely 61v3 & 6c/42 and 6l2 & 0. In order to achieve this we need to explain two things: (i) the appearance of the Cabibbo angle in the neutrino sector, (ii) the factor of 42. In this section we show that these features may arise starting from the type I see-saw mechanism [34] with sequential right-handed neutrino dominance (SD) [35].

In the first subsection below we start by reviewing SD and show how it can give the factor of 42 in Eq. (4). Although SD is well known, it is instructive to go through these arguments to see how the factor of 42 arises in 613 from maximal atmospheric neutrino mixing, and to set the scene for the vacuum alignments which follow. In the subsequent subsection we show how these assumptions may be justified dynamically starting from a family symmetry, based on symmetry breaking flavons with particular vacuum alignments. In the final subsection we show how one of the flavons involving X must appear in both the neutrino and quark sectors in order to account for both the reactor angle and Cabibbo mixing in the quark sector.

4.1. Sequential right-handed neutrino dominance

First consider the case of single right-handed neutrino dominance where only one right-handed neutrino N3 of heavy Majorana mass M3 is present in the see-saw mechanism, namely the one responsible for the atmospheric neutrino mass m3 [35]. If the single right-handed neutrino couples to the three lepton doublets Li in the diagonal charged lepton mass basis as,

H (dLe + eLß + fLT)NCc,

where d, e, f are Yukawa couplings (assumed real for simplicity2), and H is the Higgs doublet, where it is assumed that d ^ e, f, so that the see-saw mechanism yields the atmospheric neutrino mass,

leaves the reactor angle unchanged from its value in Eq. (32), since A013 = 0, while Eqs. (36) and (38), together with maximal atmospheric mixing, implies trimaximal solar mixing,

+ f 2> M3 ■

(29) tan 012 :

where v = (H). Then the reactor and atmospheric angles are approximately given by simple ratios of Yukawa couplings [35],

^^2+72 '

tan 023 & f.

If the Yukawa couplings would satisfy the condition,

f)=*( 1.

then Eqs. (30) and (31) imply,

72 ■

tan 023 & 1

which predicts the desired relation in Eq. (4), with the factor of V2 arising from maximal atmospheric mixing. However we need to show that this relation is not spoiled when other right-handed neutrinos are included.

According to sequential dominance (SD) [35] the solar neutrino mass and mixing are accounted for by introducing a second right-handed neutrino Nc2 with mass M2 which couples to the three lepton doublets Li in the diagonal charged lepton mass basis as,

H (aLe + bL„ + cLT)Nc2

where a, b, c are Yukawa couplings (assumed real for simplicity). Then the second right-handed neutrino is mainly responsible for the solar neutrino mass, providing

(a, b, c)2/M2 « (e, f )2/Ms,

which is the basic SD condition. Assuming this, then the see-saw mechanism leads to the solar neutrino mass,

m2 & (a2 + (C23b — S23C)2) — ■

and the solar neutrino mixing is approximately given by a simple ratios of Yukawa couplings [35],

tan 012 ^

(C23b - S23C)

So far the subdominant Yukawa couplings a, b, c are unconstrained. The presence of these couplings will in general affect the reactor angle and destroy the relation in Eq. (4), since there is an additional contribution of the form [35],

a(eb + fc) M3 (e2 + f 2)3/2 —2 .

It is clear that, if the lower two components of the two Yukawa column vectors are orthogonal (i.e. if eb + fc = 0) then the reactor angle will be unaffected. Choosing the subdominant Yukawa couplings to satisfy,

b I = a2

1 1 -1

Eqs. (32) and (39) realise TBC mixing as in Eq. (7), assuming no charged lepton corrections.

Note that, in the limit X ^ 0, Eqs. (31) and (38) are just the conditions of constrained sequential dominance (CSD) where the precise orthogonality of the columns was responsible for 013 = 0 [36], leading to TB mixing. Eqs. (31) and (38) with a general parameter e instead of X is referred to as partially constrained sequential dominance (PCSD), leading to Tri-Bimaximal-Reactor (TBR) mixing [37]. Although the above argument is only valid to leading order in X, and m2/m3, TBR mixing is in fact realised much more accurately than this as shown numerically [38] and analytically [39].

4.2. Family symmetry andflavons in the neutrino sector

In order to account for the equality of Yukawa couplings in Eqs. (31) and (38) such as a = b = — c and e = f, we need to introduce a family symmetry GF which is broken by flavons y with particular vacuum alignments.

Let us therefore introduce a discrete family symmetry G F which is broken by the VEVs of triplet flavon fields which are aligned as follows, where the alignment arises from the flavon superpotential in Section 2.3 of [39],

<V3> = v 3(1

<V2> = v2 ( 1

The idea is that these flavons are responsible for generating the columns of the neutrino Yukawa matrix in Eqs. (31) and (38), associated with the right-handed neutrinos N3 and N2, respectively.

For example, consider a very simple type I see-saw model, based on Gf = A4, as discussed in [39]. In this model we identify the left-handed lepton doublets L and flavons qn with A4 triplets, while the right-handed neutrinos Nic and Higgs doublets Hu are A4 singlets. The neutrino part of the effective Lagrangian reads,

nc + NcN

ViVi_\ My ) ■

where the mixing term Nc2Nc3is forbidden by a choice of appropriate messengers [39].

Inserting the Higgs and flavon VEVs, whose alignment is discussed in [39], leads to the Dirac and right-handed Majorana neutrino mass matrices below,

a2 a3X^ mD = 1 a2 a3 | v,

v —a2 a3

M2 0 0 M3

The full results including phases are discussed in [35].

The Dirac mass matrix mD in Eq. (42), emerging from a family symmetry, leads to the same result as the previous assumption that the Yukawa couplings satisfy the constraints in Eqs. (31), (38). The family symmetry therefore provides a justification for the previous assumption.

Using the type I see-saw formula we can express the light neutrino mass matrix as

X2 X X \

X 1 1 1 (43)

mv = mDMR xmTD

&( 1 1 -1) + =23

3 l-t -1 J 2

where we have written m3/2 & a^v2/M3 and m2/3 & a2v2/M2 from Eqs. (29) and (35). Eq. (43) leads to TBC mixing in Eq. (7), with the deviations being of order X2 (where phenomenologically we assume m2/m3 ~ X) multiplied by small coefficients, as discussed in [39]. This means that the order X2 corrections to the mixing matrix closely approximate to those shown in Eq. (8).

4.3. The charged fermion sector

In order to account for the appearance of the Wolfenstein parameter X in Eq. (31) it is necessary that the flavon involving X should be common to both the quark and neutrino sectors. We also need to justify that the charged lepton Yukawa matrix is diagonal.

In order to account for the diagonal charged lepton Yukawa structure we identify the right-handed charged leptons ec ,tc with A4 singlets, and distinguish them using Z4 symmetries as discussed in [39]. The resulting effective charged lepton Lagrangian then takes the form

-Hd(Lyrrc + Lp^^c + L(peec).

Inserting the flavon VEVs '0 \

<Pt > = v T ( 0 ), <pM> = v 1

<Pe> = ve ( 0 0

whose alignment is discussed in [39], leads to 1

■Hd(v t L3T c + vLdic + veLte^,

thus yielding a diagonal charged lepton mass matrix, which justifies ignoring charged lepton mixing angle corrections. In a more realistic model, the charged lepton mass hierarchy may be accounted for via the Froggatt-Nielsen mechanism [40].

The quark sector was not discussed in [39]. Here we suppose that the up-type quark mass matrix is diagonal, as for the charged leptons, which may be achieved by treating the quark doublets Q as A4 triplets and the right-handed up-type quark singlets U c as A4 singlets, and using the same flavons as in the charged lepton sector,

1 Hu (Q px tc + Q V^ + Q Veuc). (47)

Inserting the flavon VEVs leads to 1

Hu(v T Q 3tc + vQ 2cc + ve Q1Uc),

where, as before, the up-type quark mass hierarchy may be accounted for via the Froggatt-Nielsen mechanism [40].

The quark mixing must arise from the down-type quark mass matrix. In order to achieve this, the right-handed down-type quark singlets Dc are assigned as A4 triplets. The down-type quark Lagrangian takes the form, assuming that the diagonal contraction Q .Dc is forbidden by suitable messenger arguments,3

3 The messengers responsible for these operators are required to couple to the

bilinear pairs (Q.pb)i. (Vb Dc)i, (Q V3)i. (p3.Dc)] which means that the messen-

Ld ~ M^Hd(Q.Pb)1(Pb.Dc)1 + Hd(Q.(3)1 (p3.Dc)j

+ MLHd(Q.(P3)1((Pb.Dc)j + M-Hd(Q.p) (p3.Dc)1,

a3b ab3

where pb is a triplet flavon with an alignment, 0

<Pb> = vb ( S2

which may be achieved in a similar way to the flavon yT responsible for the tau lepton mass [39], here allowing for a small misalignment S2 ~ O(ms/mb). Such misalignments are generically expected to appear at higher order in all the flavon alignments, although here we only assume one particular correction in order to allow a reasonable fit to Vub/Vcb as discussed below. However the main point of the model is that the flavon q3 from the neutrino sector, which is responsible for the reactor angle, also appears in Eq. (49) and is responsible for Cabibbo mixing, as follows.

The down-type quark mass matrix arising from Eq. (49) is,

0 0 \ / X2 X X

S4 S2 1 + m33 X 1 1 1 1

S2 \ X

0 0 X" \ / 0 0 0

0 0 1 + mb3 0 0 1 0 1

0 0 1, / X

where we assume that the mass parameters map ~ Vd(cpa)(Vp)/ M2Qa/} satisfy mbb & mb and m33 & ms with m3b, mb3 < m33, leading to md ~ X2(m3b — mb3), since the determinant of the mass matrix vanishes in the limit m3b = mb3.

The fact that we have used the same flavon p3 in the down sector as in the neutrino sector implies the relation,

Vus = X + O (XS2),

which verifies that the Wolfenstein parameter X yields Cabibbo mixing in the quark sector. In other words, the same parameter that arises in the first component of the vacuum alignment of p3 is responsible for both neutrino reactor mixing and Cabibbo mixing, and we identify this parameter with the Wolfenstein parameter X, leading to the relation in Eq. (4). Assuming mbbS2 ~ m33, the mass matrix Md also implies Vub/Vcb ~ X/2, in reasonable agreement with experiment [1].

5. Summary and conclusion

To summarise, recent data from the Daya Bay and RENO reactor experiments is consistent with a remarkable relationship between the smallest lepton mixing angle, 813, and the largest quark mixing angle, 8c, namely 813 & 8C/V2. We have proposed a new mixing ansatz called Tri-bimaximal-Cabibbo (TBC) mixing which combines this relation with TB atmospheric and solar mixing. We then discussed two ways to achieve TBC mixing, summarised as follows:

gers a3 and ab are required to be colour triplets and antitriplets, which forbids them from coupling to ( Q .Dc ). The messengers are also assumed to be A4 singlets so that they lead to only A4 singlet contractions as indicated by the subscript (■ ■ )i. The flavon pb and messenger ab are odd under a Z2 symmetry, such that pb only couples to the messenger ab which is coloured and does not enter the lepton sector. The messenger mass is generated by a field Sb which is odd under Zb-1, giving

<sb>a ab.

(i) The first approach is based on Cabibbo-like charged lepton corrections 6^2 & 6C, starting from a zero neutrino mixing angle 6V3 & 0. The desired empirical factor of V2 in Eqs. (4), (6) then arises automatically from Eq. (11), assuming maximal atmospheric neutrino mixing. The suitable mixing patterns are therefore those with 6V3 = 0 and 623 = 45°. We have considered three such mixing patterns, namely Tri-Bimaximal (TB) neutrino mixing, Bi-Maximal (BM) neutrino mixing, and the Golden Ratio (GR) neutrino mixing, which each lead to the Sum Rule in Eq. (14) where 6¡2 = 35.26°, 45°, 31.7°, respectively. Given the prediction 613 & 9.2°, the Sum Rule then yields a favoured range of cos 8 in each case, namely 8 &±90°, ±180°, ±75°, respectively. These predictions are testable in future neutrino accelerator experiments [26]. We have indicated how such scenarios may be realised in Family Symmetry Models with Pati-Salam symmetry.

(ii) The second approach generates a neutrino mixing angle directly (with no charged lepton corrections), 6 v3 & 6c/42, using the type I see-saw mechanism with sequential dominance (SD), assuming a particular form of the Dirac neutrino Yukawa couplings in Eqs. (31) and (38). The desired empirical factor of V2 in this case arises automatically from Eq. (32), assuming maximal atmospheric neutrino mixing, and A613 = 0 in Eq. (37), which is satisfied if solar mixing is trimaximal as follows from Eq. (38). The conditions Eqs. (31) and (38) may be justified using family symmetry breaking flavons with particular vacuum alignments in the neutrino sector. The appearance of 6C in the flavon <p3 misalignment is justified by the fact that <p3 is responsible for Cabibbo mixing in the quark sector. The main prediction of the second approach is that, unlike the first approach, TBC mixing in Eqs. (7)-(9) is realised accurately, up to corrections of order X2 multiplied by small coefficients. However, as usual, there will be additional renormalisation group and canonical normalisation effects which will give additional corrections.

In conclusion, we have proposed the TBC mixing pattern in Eqs. (7) and (8) and shown how it can be realised in two very different approaches to quark and lepton mixing, with distinctive experimental predictions.

Acknowledgement

S.F.K. acknowledges partial support from the STFC Consolidated ST/J000396/1 and EU ITN grants UNILHC 237920 and INVISIBLES 289442.

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