We now turn to the neutrino sector. We start with the derivation of the mass matrix for the light neutrino states. The neutrino Yukawa matrix is obtained from Eq. (9.12) as
Yν = 0 aν2123 aν123 aν2123 −aν123 aν2123 . (9.37)
Additionally, we have a diagonal mass matrix for the two right-handed neutrinos from Eq. (9.13), MR= 2aR1 2 23 0 0 3aR2 2 123+ 3(w2−1)˜aR2˜ 2 23 . (9.38)
Using the seesaw relation
mν =−vu2YνMR−1Y T
ν , (9.39)
we obtain for the neutrino mass matrix
mν = m3 2 0 0 0 0 1 −1 0 −1 1 + m2 3 1 1 1 1 1 1 1 1 1 , (9.40) with m2 =−v2u a2ν22123 aR2 2 123+ (w2−1)˜aR2˜ 2 23 and m3 =−vu2 a2ν1 aR1 . (9.41) These two parameters can have either sign.
108 9. Implications for GUT Model Building and a Concrete Application
From the structure of mν, we obtain TB mixing in the neutrino sector,
θν13= 0◦ , θ23ν = 45◦ , θν12= arcsin√1
3 ≈35.3
◦. (9.42)
From the lepton sector we get the additional mixing contributions
θ13e = 0◦ , θe23= 0◦ , |θ12e |= c123123 c123123−i ˜c23˜23 ≈ 4.2◦ . (9.43)
There is also a complex phase introduced by the charged lepton Yukawa matrix which can be calculated to δe12= arctan ˜ c23˜23 c123123 ≈85.8◦ . (9.44) For the approximate calculation of the MNS mixing parameters at the GUT scale we can use [135, 136]: sMNS23 e−iδ23MNS ≈sν 23e−iδ ν 23 −θe 23, sMNS13 e−iδ13MNS ≈θν 13e− iδν 13 −sν 23θ e 12e− i(δν 23+δ12e ), sMNS12 e−iδ12MNS ≈sν 12e− iδν 12 −cν 23c ν 12θ e 12e− iδe 12 , (9.45)
where we have already discarded RG corrections which are small for the case of hierarchical neutrino masses [110, 137]. For simplicity, we first want to assume that m2 and m3 have
the same sign. In this case, the phases δν
ij are trivial. For the total leptonic mixing angles
we then obtain
θMNS12 ≈35.2◦ , θMNS13 ≈3.0◦ , θMNS23 ≈45◦ .
(9.46)
For the phases we haveδMNS
13 =δ12e −π ≈ −94.2◦,δ12MNS =−4.2◦ andδ23MNS = 0◦ from which
the final MNS phases can be calculated according to [135, 136]
δMNS =δ13MNS−δMNS12 ≈ −90◦ ,
α1 = 2(δ12MNS+δ23MNS) = 2δMNS12 ≈ −8.4◦ ,
α2 = 2δ23MNS ≈0◦ ,
(9.47)
whereα1 and α2 are the Majorana phases as in the PDG parameterisation where they are
contained in a diagonal matrix diag(eiα1/2,eiα2/2,1). This is not the whole story yet, since
we made the assumption that m2 and m3 have the same sign. If they have opposite signs,
we get α2 = 180◦, while the other phases remain the same.
Thus, in summary, the predictions of our model for the leptonic mixing parameters are compatible with the experimental 1σranges at low energy which are: θMNS
12 = (34.5±1.0)◦,
9.2 A GUT Flavour Model for Large tanβ 109
We note that with the mixing pattern of our model, i.e. TB mixing produced in the neutrino sector, and charged lepton mixing corrections from θe
12 only, the leptonic mixing
angles and the Dirac CP phase δMNS satisfy the lepton mixing sum rule [43, 134, 139, 140]
θ12MNS−θMNS13 cos(δMNS)≈arcsin(1/
√
3). (9.48)
The approximately maximal CP violation, i.e.δMNS ≈ −90◦, only leads to small deviations
of the solar mixing angle from its TB value of arcsin(1/√3), although the charged lepton corrections generate θMNS
13 ≈3.0◦.
The predictions of our model for the leptonic mixing angles and Dirac CP phase δMNS
stated in Eqs. (9.46) and (9.47) can be tested accurately by ongoing and future precision neutrino oscillation experiments [141].
The kinematic mass accessible in the single beta decay end-point experiment KATRIN is
m2β ≡m21c212c213+m22s212c213+m23s213 . (9.49) In our case, neutrino masses are strictly hierarchicalm3 m2 > m1 = 0 such that we can
determine the masses m2 and m3 from the mass squared differences and obtain
mβ2 = 3.2−+00..23×10−5 eV2 , (9.50) which is beyond the reach of the upcoming experiments.
The effective mass relevant for neutrinoless double beta decay reads
mee=|m1c212c 2 13e iα1 +m 2s212c 2 13e iα2 +m 3s213e 2 iδMNS|, (9.51) which is calculated to mee= (2.8±0.1)×10−3 eV or mee = (3.0±0.1)×10−3 eV, (9.52)
depending on the Majorana phase α2. This is also beyond the reach of upcoming exper-
iments. However, we note that neutrinoless double beta decay is an unavoidable conse- quence in the this model.
PART V
CHAPTER 10
Quark Mixing Sum Rules and the
Right Unitarity Triangle
In the last part we have discussed Yukawa couplings in SUSY GUTs for medium and large tanβ. In this part we turn our attention to the case of small tanβ. We start our discussion of that particular case in this chapter with a discussion of quark mixing sum rules and their relation to the right unitarity triangle in the quark sector based on [28].
We assume here that the Yukawa matrices are generated by the vacuum alignment of some family symmetry breaking flavon fields. This point of view defines a preferred basis, which we shall refer to as the flavour basis. We adopt this point of view since in such frameworks, the resulting low energy effective Yukawa matrices are expected to have a correspondingly simple form in the flavour basis associated with the high energy simple flavon vacuum alignment. This suggests that it may be useful to look for simple Yukawa matrix structures in a particular basis, since such patterns may provide a bottom-up route towards a theory of flavour based on a spontaneously broken family symmetry.
Unfortunately, experiment does not tell us directly the structure of the Yukawa matrices and the complexity of the problem, in particular the basis ambiguity from the bottom-up perspective, generally hinders the prospects of deducing even the basic features of the underlying flavour theory from the experimental data. We are left with little alternative but to follow anad hocapproach pioneered some time ago by Fritzsch [22,23] and currently represented by the myriads of proposed effective Yukawa textures, see, e.g. [22–27], whose starting assumption is that in some basis the Yukawa matrices exhibit certain nice features such as symmetries or zeros in specific elements which have become known as texture zeros. For example, in his classic paper, Fritzsch pioneered the idea of having six texture zeros in the 1-1, 2-2, 1-3 entries of the Hermitian up and down quark Yukawa (or mass) matrices [22].
Unfortunately, these six-zero textures are no longer consistent with experimental data, since they imply the bad prediction |Vcb| ∼
p
ms/mb, so texture zerologists have been
forced to retreat to the (at most) four-zero schemes discussed, for example, in [25–27] which give up on the 2-2 texture zeros allowing the good prediction|Vcb| ∼ms/mb.
114 10. Quark Mixing Sum Rules and the Right Unitarity Triangle
tries of both up and down Hermitian mass matrices may also lead to the bad prediction
|Vub|/|Vcb| ∼
p
mu/mc unless |Vcb| results from the cancellation of quite sizeable up- and
down-type quark 2-3 mixing angles, leading to non-negligible induced 1-3 up- and down- type quark mixing [27]. Another possibility is to give up on the 1-3 texture zeros, as well as the 2-2 texture zeros, retaining only two texture zeros in the 1-1 entries of the up and down quark matrices [25]. We reject here both of these options, and instead choose to maintain up to four texture zeros, without invoking cancellations, for example by making the 1-1 element of the up (but not down) quark mass matrix nonzero, while retaining 1-3 texture zeros in both the up and down quark Hermitian matrices, as suggested in [26].
In this chapter we discuss phenomenologically viable textures for hierarchical quark mass matrices which have both 1-3 texture zeros and negligible 1-3 mixing in both the up and down quark mass matrices. We derive quark mixing sum rules applicable to textures of this type, in which Vub is generated from Vcb as a result of 1-2 up-type mixing in Sec. 10.1,
in direct analogy to the lepton sum rules derived in [43, 134–136, 139, 140], and especially discuss how to use the sum rules to show how the right-angled unitarity triangle, i.e.,
α≈90◦, relates to the phases in the up and down quark mass matrices.
In Sec. 10.2 we show how this phase structure can be accounted for by a remarkably simple scheme involving real mass matrices apart from a single element of either the up or down quark mass matrix being purely imaginary. Fritzsch and Xing have previously emphasised how their four-zero scheme with 1-1 and 1-3 texture zeros in the Hermitian up and down quark mass matrices can be used to accommodate right unitarity triangles [27], but since their scheme involves large 2-3 and non-negligible 1-3 up and down quark mixing, our sum rules are not applicable to their case. Therefore, the textures in [25] and [27] do not allow us to explain α ≈ 90◦ by simple structures with a combination of purely real and purely imaginary matrix elements. Recently, it has become increasingly clear that current data is indeed consistent with the hypothesis of a right unitarity triangle, with the best fits giving α = 90.7+4−2..59◦ [142], and this provides additional impetus for our scheme. The phenomenological observation that α ≈ π/2 has also motivated other approaches, see, e.g. [143–145], which are complementary to the approach developed in this chapter. In Sec. 10.2 we discuss also textures with nonzero 1-3 elements in the up sector which, however, turn out to be disfavoured.
We conclude this chapter in Sec. 10.3 with a discussion of the implications of zero 1-3 mixing for the charged lepton and neutrino sectors in the framework of GUTs and show how the quark mixing sum rules may be used to yield an accurate prediction for the reactor mixing angle.
10.1
Quark Mixing Sum Rules
We start our derivation of quark mixing sum rules with the derivation of sum rules for the mixing angles and afterwards we derive a sum rule for the phases. In this whole discussion we always suppose that θd
13 = θ13u = 0. This can be understood as a direct result from a
10.1 Quark Mixing Sum Rules 115
this corresponds only to a convenient choice, it becomes a nontrivial assumption at the level of a specific flavour model.