Models of neutrino mass, mixing and CP violation

An early suggested pattern of lepton mixing is known as BM mixing with ${s}_{13}^{2}=0$ and ${s}_{12}^{2}={s}_{23}^{2}=1/2$ which could originate from the discrete group S₄ (see later). It has a maximal solar mixing angle [29], and is given by a matrix of the form

$$\begin{eqnarray}{U}_{{\rm{BM}}}=\left(\begin{array}{lll}\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0\\ -\frac{1}{2} & \frac{1}{2} & \frac{1}{\sqrt{2}}\\ \frac{1}{2} & -\frac{1}{2} & \frac{1}{\sqrt{2}}\end{array}\right).\end{eqnarray} \tag{ 7 }$$

A second pattern of lepton mixing which came to dominate the model building community until the measurement of the reactor angle is the TB mixing matrix [30]. This has been associated with models based on the flavour symmetries A₄ and S₄ (see later). Like BM mixing it predicts ${s}_{13}^{2}=0$ and ${s}_{23}^{2}=1/2$ but differs in that it predicts a solar mixing angle given by ${s}_{12}=1/\sqrt{3},$ i.e. θ₁₂ ≈ 35.3°. The mixing matrix is given explicitly by

$$\begin{eqnarray}{U}_{{\rm{TB}}}=\left(\begin{array}{ccc}\sqrt{\frac{2}{3}} & \frac{1}{\sqrt{3}} & 0\\ -\frac{1}{\sqrt{6}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{6}} & -\frac{1}{\sqrt{3}} & \frac{1}{\sqrt{2}}\end{array}\right).\end{eqnarray} \tag{ 8 }$$

Another pattern of lepton mixing which was viable until the reactor angle measurement associates the GR $\varphi =\frac{1+\sqrt{5}}{2}$ with the solar mixing angle. The original GR mixing pattern is related to the flavour symmetry A₅ [31]. As above, it predicts ${s}_{13}^{2}=0$ and ${s}_{23}^{2}=1/2$ but differs by having a solar mixing angle given by ${t}_{12}^{\nu }=1/\varphi ,$ i.e. ${\theta }_{12}^{\nu }\approx 31.7^\circ ,$ resulting in the mixing matrix

$$\begin{eqnarray}{U}_{{\rm{GR}}}=\left(\begin{array}{ccc}\frac{\varphi }{\sqrt{2+\varphi }} & \frac{1}{\sqrt{2+\varphi }} & 0\\ -\frac{1}{\sqrt{4+2\varphi }} & \frac{\varphi }{\sqrt{4+2\varphi }} & \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{4+2\varphi }} & -\frac{\varphi }{\sqrt{4+2\varphi }} & \frac{1}{\sqrt{2}}\end{array}\right).\end{eqnarray} \tag{ 9 }$$

After the measurement of the reactor angle, TB mixing is excluded. However, TB mixing still remains a reasonable approximation to lepton mixing for the solar and atmospheric angles. It therefore makes sense to expand the angles about their TB values [32, 33]:

$$\begin{eqnarray}&&\mathrm{sin}{\theta }_{12}=\displaystyle \frac{1}{\sqrt{3}}(1+s),\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&\mathrm{sin}{\theta }_{23}=\displaystyle \frac{1}{\sqrt{2}}(1+a),\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&\mathrm{sin}{\theta }_{13}=\displaystyle \frac{r}{\sqrt{2}},\end{eqnarray} \tag{ 12 }$$

where s, a, and r are the (s)olar, (a)tmospheric and (r)eactor deviation parameters such that TB mixing [30] is recovered for s = a = r = 0. For example, TBC mixing [26] corresponds to s = a = 0 and r = θ_C, where θ_C is the Cabibbo angle, which is consistent with data at three sigma as shown in figure 3.

Tri-maximal mixing is a variation which preserves either the first or the second column of the TB mixing mixing matrix in equation (8), leading to two versions called TM1 or TM2

$$\begin{eqnarray}{U}_{{\rm{TM1}}}=\left(\begin{array}{ccc}\frac{2}{\sqrt{3}} & - & -\\ -\frac{1}{\sqrt{6}} & - & -\\ \frac{1}{\sqrt{6}} & - & -\end{array}\right),\ \ \ {U}_{{\rm{TM2}}}=\left(\begin{array}{ccc}- & \frac{1}{\sqrt{3}} & -\\ - & \frac{1}{\sqrt{3}} & -\\ - & -\frac{1}{\sqrt{3}} & -\end{array}\right).\end{eqnarray} \tag{ 13 }$$

The dashes indicate that the other elements are undetermined. However these are fixed once the reactor angle is specified (it is a free parameter here). These imply the relations

$$\begin{eqnarray}&&{\rm{TM1}}:\ \ \ \ \left|{U}_{e1}\right|=\sqrt{\displaystyle \frac{2}{3}}\ \ {\rm{and}}\ \ \left|{U}_{\mu 1}\right|=\left|{U}_{\tau 1}\right|=\displaystyle \frac{1}{\sqrt{6}};\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&{\rm{TM2}}:\ \ \ \ \left|{U}_{e2}\right|=\left|{U}_{\mu 2}\right|=\left|{U}_{\tau 2}\right|=\displaystyle \frac{1}{\sqrt{3}}.\end{eqnarray} \tag{ 15 }$$

The atmospheric mixing sum rule

$$\begin{eqnarray}&&a=\lambda r\mathrm{cos}\delta +{\mathcal{O}}({a}^{2},{r}^{2}),\ {\rm{with}}\;{\rm{}}\ \ s={\mathcal{O}}({a}^{2},{r}^{2}),\end{eqnarray} \tag{ 16 }$$

was first derived in [32] by expanding the PMNS matrix to first order in r, s, a. It also follows from a first order expansion of equations (14) and (15), where λ = 1 for TM1 and λ = −1/2 for TM2. The study of correlations of this type, and their application to the discrimination between underlying models, has been shown to be a realistic aim for a next-generation superbeam experiment [34], see for example figure 4.

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Now suppose that neutrino mixing ${U}_{{\rm{TB}}}^{\nu }$ obeys TB exactly so that the PMNS matrix according to equation (6) is given by ${U}_{{\rm{PMNS}}}={U}^{e}{U}_{{\rm{TB}}}^{\nu }$ where ${U}_{{\rm{TB}}}^{\nu }$ is equated to equation (8) while U^e encodes some unknown charged lepton corrections which must be small since U_PMNS is not far from TB mixing. The solar mixing sum rule [35–37] then follows from the assumption that ${\theta }_{23}^{e}={\theta }_{13}^{e}=0.$ If the charged lepton mixing matrix involves a Cabibbo-like mixing, then the PMNS matrix is given by

$$\begin{eqnarray}{U}_{\mathrm{PMNS}}=\left(\begin{array}{ccc}{c}_{12}^{e} & {s}_{12}^{e}{{\rm{e}}}^{-{\rm{i}}{\delta }_{12}^{e}} & 0\\ -{s}_{12}^{e}{{\rm{e}}}^{{\rm{i}}{\delta }_{12}^{e}} & {c}_{12}^{e} & 0\\ 0 & 0 & 1\end{array}\right)\left(\begin{array}{ccc}\sqrt{\frac{2}{3}} & \frac{1}{\sqrt{3}} & 0\\ -\frac{1}{\sqrt{6}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{6}} & -\frac{1}{\sqrt{3}} & \frac{1}{\sqrt{2}}\end{array}\right)=\left(\begin{array}{ccc}\cdots & \cdots & \frac{{s}_{12}^{e}}{\sqrt{2}}{{\rm{e}}}^{-{\rm{i}}{\delta }_{12}^{e}}\\ \cdots & \cdots & \frac{{c}_{12}^{e}}{\sqrt{2}}\\ \frac{1}{\sqrt{6}} & -\frac{1}{\sqrt{3}} & \frac{1}{\sqrt{2}}.\end{array}\right)\end{eqnarray} \tag{ 17 }$$

Comparing to the PMNS parametrization in equation (1) we identify

$$\begin{eqnarray}&&| {U}_{e3}| ={s}_{13}=\displaystyle \frac{{s}_{12}^{e}}{\sqrt{2}},\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&| {U}_{\tau 1}| =| {s}_{23}{s}_{12}-{s}_{13}{c}_{23}{c}_{12}{{\rm{e}}}^{{\rm{i}}\delta }| =\displaystyle \frac{1}{\sqrt{6}},\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}&&| {U}_{\tau 2}| =| -{c}_{12}{s}_{23}-{s}_{12}{s}_{13}{c}_{23}{{\rm{e}}}^{{\rm{i}}\delta }| =\displaystyle \frac{1}{\sqrt{3}},\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&| {U}_{\tau 3}| ={c}_{13}{c}_{23}=\displaystyle \frac{1}{\sqrt{2}}.\end{eqnarray} \tag{ 21 }$$

The first equation predicts a reactor angle θ₁₃ ≈ 9.2° if θ_e ≈ θ_C ≈ 13°. The second and fourth equations allow to eliminate θ₂₃ to give a new relation between the PMNS parameters, θ₁₂, θ₁₃ and δ called a solar sum rule, which may be expanded to first order to give the approximate relations

$$\begin{eqnarray}&&{\theta }_{12}\approx 35.26^\circ +{\theta }_{13}\mathrm{cos}\delta \ \ \ \ {\rm{or}}\ \ \ \ \mathrm{cos}\delta \approx \displaystyle \frac{{\theta }_{12}-35.26^\circ }{{\theta }_{13}}\end{eqnarray} \tag{ 22 }$$

where $35.26^\circ ={\mathrm{sin}}^{-1}\frac{1}{\sqrt{3}},$ which can be recast as [32],

$$\begin{eqnarray}&&s=r\mathrm{cos}\delta +{\mathcal{O}}({a}^{2},{r}^{2}).\end{eqnarray} \tag{ 23 }$$

Recently it has been realized that, keeping ${\theta }_{13}^{e}=0,$ but allowing ${\theta }_{23}^{e}\ne 0,$ the following exact result can be obtained by generalizing equation (17) to allow both ${\theta }_{12}^{e},{\theta }_{23}^{e}\ne 0$ [38]:

$$\begin{eqnarray}&&\displaystyle \frac{\left|{U}_{\tau 1}\right|}{\left|{U}_{\tau 2}\right|}=\displaystyle \frac{| {s}_{12}{s}_{23}-{c}_{12}{s}_{13}{c}_{23}{{\rm{e}}}^{{\rm{i}}\delta }| }{| -{c}_{12}{s}_{23}-{s}_{12}{s}_{13}{c}_{23}{{\rm{e}}}^{{\rm{i}}\delta }| }=\displaystyle \frac{1}{\sqrt{2}},\end{eqnarray} \tag{ 24 }$$

for the previous case ${\theta }_{23}^{e}={\theta }_{13}^{e}=0$ this result follows trivially by taking the ratio of the two equations (19) and (20). Therefore this result applies to that case also. However it turns out that ${\theta }_{23}^{e}$ cancels in this ratio which is therefore a more general sum rule (though not completely general since it still assumes ${\theta }_{13}^{e}=0$). After some algebra, equation (24) leads to an exact prediction for cos δ in terms of the other physical lepton angles

$$\begin{eqnarray}&&\mathrm{cos}\delta =\displaystyle \frac{{t}_{23}{s}_{12}^{2}+{s}_{13}^{2}{c}_{12}^{2}/{t}_{23}-\frac{1}{3}({t}_{23}+{s}_{13}^{2}/{t}_{23})}{\mathrm{sin}2{\theta }_{12}{s}_{13}},\end{eqnarray} \tag{ 25 }$$

as displayed in figure 5.

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When expanded to first order, equation (25) reduces to the leading order sum rule in equation (22). This is not too surprising since the previous sum rule case also satisfies equation (24). The leading order sum rule in equation (22) offers a simple way to understand the results in figure 5. For example from figure 5 it seems that TB neutrino mixing predicts cos δ ≈ 0 if θ₁₂ ≈ 35.26°, which is obvious from² equation (22). This can also be understood from equation (25) where we see that for ${s}_{12}^{2}=1/3$ the leading terms ${t}_{23}{s}_{12}^{2}$ and $\frac{1}{3}{t}_{23}$ in the numerator cancel, leaving $\mathrm{cos}\delta ={s}_{13}/(2\sqrt{2}{t}_{23})\approx 0.05$ which is consistent with the numerical estimates of the error given above for a range of θ₁₂.

Solar sum rules can also be obtained for different types of neutrino mixing such as ${U}_{{\rm{BM}}}^{\nu }$ (which is almost excluded) and ${U}_{{\rm{GR}}}^{\nu }$ (which gives similar results to the case ${U}_{{\rm{TB}}}^{\nu }$ considered here). The general formula given in [38] is

$$\begin{eqnarray}&&\mathrm{cos}\delta =\displaystyle \frac{{t}_{23}{s}_{12}^{2}+{s}_{13}^{2}{c}_{12}^{2}/{t}_{23}-{s}_{12}^{\nu 2}({t}_{23}+{s}_{13}^{2}/{t}_{23})}{\mathrm{sin}2{\theta }_{12}{s}_{13}},\end{eqnarray} \tag{ 26 }$$

where ${s}_{12}^{\nu 2}=\frac{1}{3},\frac{1}{2}$ for ${U}_{{\rm{TB,BM}}}^{\nu }$ and so on. The prospects for studying solar sum rules at JUNO and LBNF is discussed in [38]. A slightly more lengthy but equivalent formula to equation (25) had been previously derived [39] by an alternative method involving an auxiliary phase ϕ without using the elegant result equation (24),

$$\begin{eqnarray}&&\mathrm{cos}\delta =\displaystyle \frac{{t}_{23}}{\mathrm{sin}2{\theta }_{12}{s}_{13}}\left[\mathrm{cos}2{\theta }_{12}^{\nu }+({s}_{12}^{2}-{c}_{12}^{\nu 2})(1-{\mathrm{cot}}^{2}{\theta }_{23}{s}_{13}^{2})\right].\end{eqnarray} \tag{ 27 }$$

We prefer the simpler form in equation (26) which involves ${\theta }_{12}^{\nu }$ in only one place since it exhibits the cancellation between the terms ${t}_{23}{s}_{12}^{2}$ and ${s}_{12}^{\nu 2}{t}_{23}$ when ${s}_{12}^{2}={s}_{12}^{\nu 2}$ responsible for the prediction cos δ ≈ 0 in this case We also advocate the much simpler derivation of equation (26) given in [38].

Finally we give a word of caution that when comparing leading order sum rules to the exact results the ratio ${(\mathrm{cos}\delta )}_{{\rm{exact}}}/{(\mathrm{cos}\delta )}_{{\rm{LO}}}$ used in [40] will lead to misleading results when (cos δ)_LO ≈ 0. In general it is safer to compare them using the experimentally relevant quantity ${\rm{\Delta }}(\mathrm{cos}\delta )={(\mathrm{cos}\delta )}_{{\rm{exact}}}-{(\mathrm{cos}\delta )}_{{\rm{LO}}}$ defined in [38]. For example we find Δ(cosδ) ≲ 0.1 for TB neutrino mixing corrected by charged lepton mixing. It will take experiment a long time to reach this level of precision, so for present purposes the linear approximation in equation (22) is adequate for TB neutrino mixing.

Despite the great progress coming from neutrino oscillation experiments there are still some outstanding questions. Are the lepton mixing angles consistent with TBC mixing? If not then is the atmospheric angle in the first or second octant? What is the leptonic ${\mathcal{C}}{\mathcal{P}}$ violating phase δ? Is the current hint δ ∼ −π/2 going to hold up? Maybe there is no ${\mathcal{C}}{\mathcal{P}}$ violation in the lepton sector? Are neutrino mass squared ordered normally or inverted³ ? What is the lightest neutrino mass? Are neutrino masses Majorana or Dirac in nature? Many neutrino experiments are underway or in the planning stages to address these questions such as T2K, NOνA, Daya Bay, JUNO, RENO, KATRIN, DUNE and many neutrinoless double beta decay experiments running and planned [25].

Everyone can invent her or his personal roadmap of neutrino mass models, one example being that shown in figure 6. The blue boxes contain experimental questions and the red boxes possible theoretical consequences. In this subsection we shall briefly describe the possible theoretical options for producing neutrino mass.

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It is worthwhile to first recall why the observation of non-zero neutrino mass and mixing is evidence for new physics beyond the SM. The most intuitive way to understand why neutrino mass is forbidden in the SM, is to understand that the SM predicts that neutrinos always have a 'left-handed' spin—rather like rifle bullets which spin counter clockwise to the direction of travel. In fact this property was first experimentally measured in 1958, two years after the neutrino was discovered, by Maurice Goldhaber, Lee Grodzins and Andrew Sunyar. More accurately, the 'handedness' of a particle describes the direction of its spin vector along the direction of motion, and the neutrino being 'left-handed' means that its spin vector always points in the opposite direction to its momentum vector. The fact that the neutrino is left-handed, written as ν_L, implies that it must be massless. If the neutrino has mass then, according to special relativity, it can never travel at the speed of light. In principle, a fast moving observer could therefore overtake the spinning massive neutrino and would see it moving in the opposite direction. To the observer, the massive neutrino would therefore appear RH. Since the SM predicts that neutrinos must be strictly left-handed, it follows that neutrinos are massless in the SM. It also follows that the discovery of neutrino mass implies new physics beyond the SM, with profound implications for particle physics and cosmology.

Neutrinos are massless in the SM for three independent reasons:

• There are no RH neutrinos ν_R.
• There are only Higgs doublets (and no Higgs triplets) of SU(2)_L.
• There are only renormalizable terms.

In the SM, the three massless neutrinos ν_e, ν_μ, ν_τ are distinguished by separate lepton numbers L_e, L_μ, L_τ. Neutrinos and antineutrinos are distinguished by total conserved lepton number $L={L}_{e}+{L}_{\mu }+{L}_{\tau }.$ To generate neutrino mass we must relax one or more of the above three conditions. For example, by adding RH neutrinos the Higgs mechanism of the SM can give neutrinos the same type of mass as the Dirac electron mass or other charged lepton and quark masses, which would generally break the separate lepton numbers L_e, L_μ, L_τ, but preserve the total lepton number L. However it is also possible for neutrinos to have a new type of mass of a type first proposed by Majorana, which would also break L. There exists a special case where total lepton number L is broken, but the combination ${L}_{e}-{L}_{\mu }-{L}_{\tau }$ is conserved; such a symmetry would give rise to a neutrino mass matrix with an inverted mass spectrum.

From the theoretical perspective, the main unanswered question is the origin of neutrino mass, and in particular the smallness of neutrino mass. The simplest possibility is that neutrinos have Dirac mass just like the electron mass in the SM, namely due to a term like ${y}_{D}\bar{L}H{\nu }_{{\rm{R}}},$ where L is a lepton doublet containing ν_L, H is a Higgs doublet and ν_R is a RH neutrino. The observed smallness of neutrino masses implies that the Dirac Yukawa coupling y_D must be of order 10⁻¹² to achieve a Dirac neutrino mass of about 0.1 eV. Advocates of Dirac masses point out that the electron mass already requires a Yukawa coupling y_e of about 10⁻⁶, so we are used to such small Yukawa couplings. In this case, all that is required is to add RH neutrinos ν_R to the SM and we are done. Well, almost. It still needs to be explained why the ν_R have zero Majorana mass, after all they are gauge singlets and so nothing prevents them acquiring (large) Majorana mass terms M_RRν_Rν_R where M_RR could be as large as the Planck scale. Moreover, Majorana masses offer a unique (and testable) way to generate neutrino masses (since neutrinos do not carry electric charge) even without RH neutrinos. The simplest way to generate Majorana mass is via y_MΔLL where Δ is a Higgs triplet and y_M is a Yukawa coupling associated with Majorana mass. Alternatively, at the effective level, Majorana neutrino mass can result from some additional dimension five operators which couple two lepton doublets L to two Higgs doublets H first proposed by Weinberg [41],

$$\begin{eqnarray}&&-\displaystyle \frac{1}{2}{{HL}}^{T}\kappa {HL},\end{eqnarray} \tag{ 28 }$$

where κ has dimension ${(\mathrm{mass})}^{-1}.$ This is a non-renormalizable operator, so it violates one of the tenets of the SM. In order to account for a neutrino mass of order 0.1 eV requires κ ∼ 10⁻¹⁴ GeV⁻¹. This suggests a new high energy mass scale M in physics, a small dimensionless coupling associated with κ, or both.

There are basically five different proposals for the origin of neutrino mass:

• The seesaw mechanisms [27, 42, 43], including low scale seesaw mechanisms [44] (Weinberg operator typically from large Majorana mass M = M_R for RH neutrinos ν_R).
• R-parity violating SUSY [45] (Weinberg operator from TeV scale Majorana mass for neutralinos χ).
• TeV scale loop mechanisms [46–48] (Majorana mass from extra Higgs doublets and singlets at the TeV scale).
• Extra dimensions [49] (Dirac mass with small y_D due to RH neutrinos ν_R in the bulk).
• String theory [50, 51] (new mechanisms for generating large Majorana mass for RH neutrinos ν_R from Planck or string scale physics).

These different mechanisms are reviewed in [52]. Returning to the roadmap in figure 6, we see these mechanisms for neutrino mass represented by the red boxes, being related to the experimental question in the blue boxes.

Since no new physics has yet emerged from the LHC, we are led to consider the seesaw mechanism with RH neutrinos. However even in this case both the number of species and the mass spectrum of RH (or sterile) neutrinos is completely unknown [53]. As shown in figure 7 the mass spectrum can cover the whole range with different physical consequences as indicated. It is one of the goals of neutrino physics to determine this spectrum. In this topical review we shall focus on the classic seesaw mechanism [27] with very heavy RH neutrinos, with masses above the TeV scale, which may be incorporated into a theory of flavour, possibly related to string theory.

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In this subsection we consider the high scale (classic) seesaw neutrino model involving just two RH neutrinos ${\nu }_{{\rm{R}}}^{{\rm{sol}}}$ and ${\nu }_{{\rm{R}}}^{{\rm{atm}}}$ with Yukawa couplings [6],⁴

$$\begin{eqnarray}&&({H}_{u}/{v}_{u})(a{\bar{L}}_{e}+b{\bar{L}}_{\mu }+c{\bar{L}}_{\tau }){\nu }_{{\rm{R}}}^{{\rm{sol}}}+({H}_{u}/{v}_{u})(d{\bar{L}}_{e}+e{\bar{L}}_{\mu }+f{\bar{L}}_{\tau }){\nu }_{{\rm{R}}}^{{\rm{atm}}}+{\rm{h.c.}},\end{eqnarray} \tag{ 29 }$$

where H_u is a Higgs doublet and v_u its VEV. The heavy RH Majorana masses are

$$\begin{eqnarray}&&{M}_{{\rm{sol}}}\bar{{\nu }_{{\rm{R}}}^{{\rm{sol}}\;{\rm{}}}}{({\nu }_{{\rm{R}}}^{{\rm{sol}}})}^{c}+{M}_{{\rm{atm}}}\bar{{\nu }_{{\rm{R}}}^{{\rm{atm}}\;{\rm{}}}}{({\nu }_{{\rm{R}}}^{{\rm{atm}}})}^{c}+{\rm{h.c.}}.\end{eqnarray} \tag{ 30 }$$

In the basis, with rows $\left({\bar{\nu }}_{e{\rm{L}}},{\bar{\nu }}_{\mu {\rm{L}}},{\bar{\nu }}_{\tau {\rm{L}}}\right)$ and columns ${\nu }_{{\rm{R}}}^{{\rm{atm}}},{\nu }_{{\rm{R}}}^{{\rm{sol}}},$ the resulting Dirac mass matrix is

$$\begin{eqnarray}{m}^{{\rm{D}}}=\left(\begin{array}{cc}d & a\\ e & b\\ f & c\end{array}\right)\equiv ({m}_{{\rm{atm}}}^{{\rm{D}}}\ \ {m}_{{\rm{sol}}}^{{\rm{D}}}),\ \ \ {m}_{{\rm{atm}}}^{{\rm{D}}}\equiv {(d\ e\ f)}^{T},\ {m}_{{\rm{sol}}}^{{\rm{D}}}\equiv {(a\ b\ c)}^{T}.\end{eqnarray} \tag{ 31 }$$

The (diagonal) RH neutrino heavy Majorana mass matrix M_R with rows ${(\bar{{\nu }_{{\rm{R}}}^{{\rm{atm}}}},\bar{{\nu }_{{\rm{R}}}^{{\rm{sol}}}})}^{T}$ and columns $({\nu }_{{\rm{R}}}^{{\rm{atm}}},{\nu }_{{\rm{R}}}^{{\rm{sol}}})$ is

$$\begin{eqnarray}{M}_{{\rm{R}}}=\left(\begin{array}{cc}{M}_{{\rm{atm}}} & 0\\ 0 & {M}_{{\rm{sol}}}\end{array}\right).\end{eqnarray} \tag{ 32 }$$

The seesaw formula is [27],

$$\begin{eqnarray}&&{m}^{\nu }=-{m}^{{\rm{D}}}{M}_{{\rm{R}}}^{-1}{({m}^{{\rm{D}}})}^{T},\end{eqnarray} \tag{ 33 }$$

where m^ν is the the light effective left-handed Majorana neutrino mass matrix (i.e. the physical neutrino mass matrix), m^D is the Dirac mass matrix in LR convention and M_R is the (heavy) Majorana mass matrix. Using the seesaw formula dropping the overall minus sign which is physically irrelevant, the light effective left-handed Majorana neutrino mass matrix m^ν (i.e. the physical neutrino mass matrix) is, by multiplying the matrices

$$\begin{eqnarray}{m}^{\nu }={m}^{{\rm{D}}}{M}_{{\rm{R}}}^{-1}{({m}^{{\rm{D}}})}^{T}=\left(\begin{array}{ccc}\frac{{a}^{2}}{{M}_{{\rm{sol}}}}+\frac{{d}^{2}}{{M}_{{\rm{atm}}}} & \frac{{ab}}{{M}_{{\rm{sol}}}}+\frac{{de}}{{M}_{{\rm{atm}}}} & \frac{{ac}}{{M}_{{\rm{sol}}}}+\frac{{df}}{{M}_{{\rm{atm}}}}\\ \frac{{ab}}{{M}_{{\rm{sol}}}}+\frac{{de}}{{M}_{{\rm{atm}}}} & \frac{{b}^{2}}{{M}_{{\rm{sol}}}}+\frac{{e}^{2}}{{M}_{{\rm{atm}}}} & \frac{{bc}}{{M}_{{\rm{sol}}}}+\frac{{ef}}{{M}_{{\rm{atm}}}}\\ \frac{{ac}}{{M}_{{\rm{sol}}}}+\frac{{df}}{{M}_{{\rm{atm}}}} & \frac{{bc}}{{M}_{{\rm{sol}}}}+\frac{{ef}}{{M}_{{\rm{atm}}}} & \frac{{c}^{2}}{{M}_{{\rm{sol}}}}+\frac{{f}^{2}}{{M}_{{\rm{atm}}}}\end{array}\right).\end{eqnarray} \tag{ 34 }$$

The SD [6] assumptions are that $d\ll e,f$ and

$$\begin{eqnarray}&&\displaystyle \frac{{(e,f)}^{2}}{{M}_{{\rm{atm}}}}\gg \displaystyle \frac{{(a,b,c)}^{2}}{{M}_{{\rm{sol}}}}.\end{eqnarray} \tag{ 35 }$$

By explicit calculation, one can check that $\mathrm{det}{m}^{\nu }=0.$ Since the determinant of a Hermitian matrix is the product of mass eigenvalues

$$\begin{eqnarray*}&&\mathrm{det}({m}^{\nu }{m}^{\nu \dagger })={m}_{1}^{2}{m}_{2}^{2}{m}_{3}^{2},\end{eqnarray*}$$

one may deduce that one of the mass eigenvalues of the complex symmetric matrix above is zero, which under the SD assumption is the lightest one m₁ = 0 with ${m}_{3}\gg {m}_{2}$ since the model approximates to a single RH neutrino model [6]. Hence we see that SD implies a normal neutrino mass hierarchy. Including the solar RH neutrino as a perturbation, it can be shown that, for d = 0, together with the assumption of a dominant atmospheric RH neutrino in equation (35), leads to the approximate results for the solar and atmospheric angles [6],

$$\begin{eqnarray}&&\mathrm{tan}{\theta }_{23}\sim \displaystyle \frac{e}{f},\ \ \ \mathrm{tan}{\theta }_{12}\sim \displaystyle \frac{\sqrt{2}a}{b-c}.\end{eqnarray} \tag{ 36 }$$

Under the above SD assumption, each of the RH neutrinos contributes uniquely to a particular physical neutrino mass. The SD framework above with d = 0 leads to the relations in equation (36) together with the reactor angle bound [7],

$$\begin{eqnarray}&&{\theta }_{13}\lesssim {m}_{2}/{m}_{3}.\end{eqnarray} \tag{ 37 }$$

This result shows that SD allows for large values of the reactor angle, consistent with the measured value. Indeed the measured reactor angle, observed a decade after this theoretical bound was derived, approximately saturates the upper limit. In order to understand why this is so, we must go beyond the SD assumptions stated so far, and enter the realms of CSD.

Let us return to equation (31) and set d = 0 and e = f, with b = a and c = −a [35]. The motivation is that from equation (36) one then approximately expects the good phenomenological relations t₂₃ ∼ 1 and ${t}_{12}\sim 1/\sqrt{2},$ although the value of the reactor angle bounded by equation (37) remains to be seen. With the above assumption, equation (34) becomes

$$\begin{eqnarray}{m}^{\nu }=\left(\begin{array}{ccc}\frac{{a}^{2}}{{M}_{{\rm{sol}}}} & \frac{{a}^{2}}{{M}_{{\rm{sol}}}} & \frac{-{a}^{2}}{{M}_{{\rm{sol}}}}\\ \frac{{a}^{2}}{{M}_{{\rm{sol}}}} & \frac{{a}^{2}}{{M}_{{\rm{sol}}}}+\frac{{e}^{2}}{{M}_{{\rm{atm}}}} & \frac{-{a}^{2}}{{M}_{{\rm{sol}}}}+\frac{{e}^{2}}{{M}_{{\rm{atm}}}}\\ \frac{-{a}^{2}}{{M}_{{\rm{sol}}}} & \frac{-{a}^{2}}{{M}_{{\rm{sol}}}}+\frac{{e}^{2}}{{M}_{{\rm{atm}}}} & \frac{{a}^{2}}{{M}_{{\rm{sol}}}}+\frac{{e}^{2}}{{M}_{{\rm{atm}}}}\end{array}\right).\end{eqnarray} \tag{ 38 }$$

By explicit calculation one then finds that the neutrino mass matrix is exactly diagonalized by the TB mixing matrix in equation (8),

$$\begin{eqnarray}{U}_{\mathrm{TB}}^{T}{m}^{\nu }{U}_{\mathrm{TB}}=\left(\begin{array}{ccc}0 & 0 & 0\\ 0 & \frac{3{a}^{2}}{{M}_{{\rm{sol}}}} & 0\\ 0 & 0 & \frac{2{e}^{2}}{{M}_{{\rm{atm}}}}\end{array}\right).\end{eqnarray} \tag{ 39 }$$

If the charged lepton mass matrix is diagonal, the interpretation is that these constrained couplings d = 0, e = f with b = a and c = −a lead to TB mixing, with the lightest neutrino mass m₁ = 0, the second lightest neutrino identified as the solar neutrino with mass ${m}_{2}=\frac{3{a}^{2}}{{M}_{{\rm{sol}}}}$ and the heaviest neutrino identified as the atmospheric neutrino with mass ${m}_{3}=\frac{2{a}^{2}}{{M}_{{\rm{atm}}}}.$ While TB mixing accurately gives the good relations t₂₃ = 1 and ${t}_{12}=1/\sqrt{2},$ unfortunately it also gives θ₁₃ = 0. This is known as CSD [35].

We can generalize the original idea of CSD to other examples of Dirac mass matrix with (in the notation of equation (31)) d = 0 and e = f as before, but now with b = na and $c=(n-2)a,$ for any positive integer n, which we refer to as CSD(n). The original CSD in equation (38) with b = a and c = −a is identified as the special case CSD(n = 1). The motivation for CSD(n) is that for any n equation (36) implies t₂₃ ∼ 1 and ${t}_{12}\sim 1/\sqrt{2},$ although these results are strongly dependent on the relative phase between the first and second column of the Dirac mass matrix. CSD(n) then corresponds to the following pattern of couplings in the Dirac mass matrix in equation (31):

• CSD(1): ${({m}_{{\rm{atm}}}^{{\rm{D}}})}^{T}=(0,e,e),\;$ ${({m}_{{\rm{sol}}}^{{\rm{D}}})}^{T}=(a,a,-a)$ [35].
• CSD(2): ${({m}_{{\rm{atm}}}^{{\rm{D}}})}^{T}=(0,e,e),\;$ ${({m}_{{\rm{sol}}}^{{\rm{D}}})}^{T}=(a,2a,0)$ [56].
• CSD(3): ${({m}_{{\rm{atm}}}^{{\rm{D}}})}^{T}=(0,e,e),\;$ ${({m}_{{\rm{sol}}}^{{\rm{D}}})}^{T}=(a,3a,a)$ [57].
• CSD(4): ${({m}_{{\rm{atm}}}^{{\rm{D}}})}^{T}=(0,e,e),\;$ ${({m}_{{\rm{sol}}}^{{\rm{D}}})}^{T}=(a,4a,2a)$ [57–59].
• CSD(n): ${({m}_{{\rm{atm}}}^{{\rm{D}}})}^{T}=(0,e,e),\;$ ${({m}_{{\rm{sol}}}^{{\rm{D}}})}^{T}=(a,{na},(n-2)a)$ [60].

For the general case of CSD(n) the Dirac mass matrix is then

$$\begin{eqnarray}{m}^{{\rm{D}}}={Y}^{\nu }{v}_{u}=\left(\begin{array}{cc}0 & a\\ e & {na}\\ e & (n-2)a\end{array}\right).\end{eqnarray} \tag{ 40 }$$

The constrained couplings will be justified later with the help of discrete family symmetry. For now we simply assume these couplings motivated by the desire to obtain an approximately maximal atmospheric angle tan θ₂₃ ∼ e/f ∼ 1 and trimaximal solar angle $\mathrm{tan}{\theta }_{12}\sim \sqrt{2}a/(b-c)\sim 1/\sqrt{2}.$ Since experiment indicates that the bound θ₁₃ ≲ m₂/m₃ is almost saturated, these schemes require certain phase choices $\mathrm{arg}(a/e)$ in order to achieve the desired reactor angle, leading to predictions for the ${\mathcal{C}}{\mathcal{P}}$-violating phase δ_CP, discussed below.

In a CSD(n) framework [60], the low energy effective Majorana neutrino mass matrix in equation (34) in the two RH neutrino case may be written as

$$\begin{eqnarray}{m}_{(n)}^{\nu }={m}_{a}\left(\begin{array}{ccc}0 & 0 & 0\\ 0 & 1 & 1\\ 0 & 1 & 1\end{array}\right)+{m}_{b}{{\rm{e}}}^{{\rm{i}}\eta }\left(\begin{array}{ccc}1 & n & n-2\\ n & {n}^{2} & n(n-2)\\ n-2 & n(n-2) & {(n-2)}^{2}\end{array}\right),\end{eqnarray} \tag{ 41 }$$

where η is the only physically important phase, which depends on the relative phase between the first and second column of the Dirac mass matrix, $\mathrm{arg}(a/e).$ By comparing equations (38) and (41) for n = 1 we identify ${m}_{a}=\frac{{e}^{2}}{{M}_{{\rm{atm}}}}$ and ${m}_{b}=\frac{{a}^{2}}{{M}_{{\rm{sol}}}},$ which hold for any value of n. This can be thought of as the minimal (two RH neutrino) predictive seesaw model since, for a given n, only three parameters m_a, m_b, η describe the entire neutrino sector (three neutrino masses and the PMNS matrix). CSD(n) with two RH neutrinos always predicts the lightest physical neutrino mass to be zero, m₁ = 0. It also immediately predicts TM1 mixing since

$$\begin{eqnarray}&&{m}_{(n)}^{\nu }\left(\begin{array}{c}2\\ -1\\ 1\end{array}\right)=\left(\begin{array}{c}0\\ 0\\ 0\end{array}\right).\end{eqnarray} \tag{ 42 }$$

In other words the column vector (2, −1, 1)^T is an eigenvector of ${m}_{(n)}^{\nu }$ with a zero eigenvalue, i.e. it is the first column of the PMNS mixing matrix, corresponding to m₁ = 0, which means TM1 mixing in equation (13).

For a given choice of the positive integer n, there are three real input parameters m_a, m_b and η from which two light physical neutrino masses m₂, m₃, three lepton mixing angles, the ${\mathcal{C}}{\mathcal{P}}$-violating phase δ_CP and two Majorana phases are derived; a total of nine physical parameters (including the prediction m₁ = 0) from three input parameters, i.e. six predictions for each value of n. As the Majorana phases are not known and δ_CP is only tentatively constrained by experiment, this leaves five presently measured observables, namely the two neutrino mass squared differences and the three lepton mixing angles, from only three input parameters. Essentially the input parameters m_a and m_b are fixed by the two physical neutrino mass squared differences, which implies that the entire PMNS mixing matrix is determined by only a single parameter, namely the phase η. The resulting best fit predictions for CSD(n) [60] are shown in figure 8 as a function of n. As can be seen, CSD(2) gives a reactor angle which is too small⁵ , while CSD(n ≥ 5) gives a reactor angle which is too large. CSD(3) and CSD(4) allow a reactor angle in the desired Goldilocks (experimentally preferred) region θ₁₃ ∼ 8.5°. This value occurs for special choices of phase η ∼ 2π/3 for CSD(3) and η ∼ 4π/5 for CSD(4) where positive values of these phases yield negative values of δ_CP ∼ −90° for CSD(3) and δ_CP ∼ −120° for CSD(4), with the mixing angles being independent of the sign of the phase.

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It is straightforward to extend the previous ideas of SD to the case of three RH neutrinos [6]. The starting assumption is that of the SM supplemented by three RH neutrinos with masses in the classic seesaw range TeV to M_GUT in figure 7. Then the light neutrino mass matrix emerges from the seesaw formula in equation (33). The basic idea of SD was given in the framework of the two-right handed neutrino model above, but can now be extended to a third, almost decoupled, RH neutrino, as follows.

Extending the preceding example of two RH neutrinos, it is possible to implement the seesaw mechanism with three RH neutrinos using the SD mechanism [6]. The SD assumption can be made precise as follows. In the basis, M_R = diag(M_atm, M_sol, M_dec) where the Dirac mass matrix is constructed by extending equation (31) to the three columns ${m}^{{\rm{D}}}=({m}_{{\rm{atm}}}^{{\rm{D}}},{m}_{{\rm{sol}}}^{{\rm{D}}},{m}_{{\rm{dec}}}^{{\rm{D}}}),$ where ${m}_{{\rm{atm}}}^{{\rm{D}}}={(d,e,f)}^{T},\;$ ${m}_{{\rm{sol}}}^{{\rm{D}}}={(a,b,c)}^{T},$ etc, the SD assumption in equation (35) generalizes to:

$$\begin{eqnarray}&&\displaystyle \frac{{({m}_{{\rm{atm}}}^{{\rm{D}}})}^{\dagger }{m}_{{\rm{atm}}}^{{\rm{D}}}}{{M}_{{\rm{atm}}}}\gg \displaystyle \frac{{({m}_{{\rm{sol}}}^{{\rm{D}}})}^{\dagger }{m}_{{\rm{sol}}}^{{\rm{D}}}}{{M}_{{\rm{sol}}}}\gg \displaystyle \frac{{({m}_{{\rm{dec}}}^{{\rm{D}}})}^{\dagger }{m}_{{\rm{dec}}}^{{\rm{D}}}}{{M}_{{\rm{dec}}}}.\end{eqnarray} \tag{ 43 }$$

Equation (43) immediately predicts a normal neutrino mass hierarchy:

$$\begin{eqnarray}&&{m}_{3}\gg {m}_{2}\gg {m}_{1}\end{eqnarray} \tag{ 44 }$$

which is the main consequence of SD. The lightest physical neutrino mass m₁ is much smaller than the others since the corresponding RH neutrino ${\nu }_{{\rm{R}}}^{{\rm{dec}}}$ being approximately decoupled from the seesaw mechanism. The heaviest physical neutrino has mass m₃ much larger than m₂ since the atmospheric RH neutrino makes the dominant contribution to the seesaw mechanism. The model approximates to the two RH neutrino case (as considered previously) where m₁ = 0. The SM with three such RH neutrinos is depicted in figure 9.

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One of the exciting things about the discovery of neutrino masses and mixing angles is that this provides additional information about the flavour problem—the problem of understanding the origin of three families of quarks and leptons and their masses and mixing angles. In the framework of the seesaw mechanism, new physics beyond the SM is required to violate lepton number and generate RH neutrino masses which may be as large as the GUT scale. This is also exciting since it implies that the origin of neutrino masses is also related to some GUT symmetry group G_GUT, which unifies the fermions within each family. Some possible candidate unified gauge groups are shown in figure 10.

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Let us take G_GUT = SU(5) as an example [61]. Each family of quarks (with colour r, b, g) and leptons fits nicely into SU(5) representations of left-handed (L) fermions, $F=\bar{5}$ and T = 10

$$\begin{eqnarray}&&F={\left(\begin{array}{c}{d}_{r}^{c}\\ {d}_{b}^{c}\\ {d}_{g}^{c}\\ {e}^{-}\\ -{\nu }_{e}\end{array}\right)}_{{\rm{L}}},\qquad T={\left(\begin{array}{ccccc}0 & {u}_{g}^{c} & -{u}_{b}^{c} & {u}_{r} & {d}_{r}\\ . & 0 & {u}_{r}^{c} & {u}_{b} & {d}_{b}\\ . & . & 0 & {u}_{g} & {d}_{g}\\ . & . & . & 0 & {e}^{c}\\ . & . & . & . & 0\end{array}\right)}_{{\rm{L}}},\end{eqnarray} \tag{ 45 }$$

where c denotes ${\mathcal{C}}{\mathcal{P}}$ conjugated fermions. The SU(5) representations $F=\bar{5}$ and T = 10 decompose into multiplets of the SM gauge group ${SU}{(3)}_{{\rm{C}}}\times {SU}{(2)}_{{\rm{L}}}\times U{(1)}_{Y}$ as F = (d^c, L), corresponding to

$$\begin{eqnarray}&&\bar{5}=(\bar{3},1,1/3)\oplus (1,\bar{2},-1/2),\end{eqnarray} \tag{ 46 }$$

and T = (u^c, Q, e^c), corresponding to

$$\begin{eqnarray}&&10=(\bar{3},1,-2/3)\oplus (3,2,1/6)\oplus (1,1,1).\end{eqnarray} \tag{ 47 }$$

Thus a complete quark and lepton SM family (Q, u^c, d^c, L, e^c) is accommodated in the $F=\bar{5}$ and T = 10 representations, with RH neutrinos, whose ${\mathcal{C}}{\mathcal{P}}$ conjugates are denoted as ν^c, being singlets of SU(5), ν^c = 1. The Higgs doublets H_u and H_d which break electroweak symmetry in a two Higgs doublet model are contained in the SU(5) multiplets H₅ and ${H}_{\bar{5}}.$

The Yukawa couplings for one family of quarks and leptons are given by

$$\begin{eqnarray}&&{y}_{u}{H}_{5i}{T}_{{jk}}{T}_{{lm}}{\epsilon }^{{ijklm}}+{y}_{\nu }{H}_{5i}{F}^{i}{\nu }^{c}+{y}_{d}{H}_{\bar{5}}^{i}{T}_{{ij}}{F}^{j},\end{eqnarray} \tag{ 48 }$$

where ^ijklm is the totally antisymmetric tensor of SU(5) with i, j, j, k, l = 1, ..., 5, which decompose into the SM Yukawa couplings

$$\begin{eqnarray}&&{y}_{u}{H}_{u}{{Qu}}^{c}+{y}_{\nu }{H}_{u}L{\nu }^{c}+{y}_{d}({H}_{d}{{Qd}}^{c}+{H}_{d}{e}^{c}L).\end{eqnarray} \tag{ 49 }$$

Notice that the Yukawa couplings for down quarks and charged leptons are equal at the GUT scale. Generalizing this relation to all three families we find the SU(5) prediction for Yukawa matrices

$$\begin{eqnarray}&&{Y}_{d}={Y}_{e}^{T},\end{eqnarray} \tag{ 50 }$$

which is successful for the third family, but fails badly for the first and second families. Georgi and Jarlskog [62] suggested to include a higher Higgs representation ${H}_{\bar{45}}$ which is responsible for the 2-2 entry of the down and charged lepton Yukawa matrices. Dropping SU(5) indices for clarity

$$\begin{eqnarray}&&{({Y}_{d})}_{22}{H}_{\bar{45}}{T}_{2}{F}_{2},\end{eqnarray} \tag{ 51 }$$

decomposes into the second family SM Yukawa couplings

$$\begin{eqnarray}&&{({Y}_{d})}_{22}({H}_{d}{Q}_{2}{d}_{2}^{c}-3{H}_{d}{e}_{2}^{c}{L}_{2}),\end{eqnarray} \tag{ 52 }$$

where the factor of −3 is an SU(5) Clebsch–Gordan coefficient⁶ . Assuming a hierarchical Yukawa matrix with a zero Yukawa element (texture) in the 1-1 position, results in the GUT scale Yukawa relations

$$\begin{eqnarray}&&{y}_{b}={y}_{\tau },\quad {y}_{s}=\displaystyle \frac{{y}_{\mu }}{3},\quad {y}_{d}=3{y}_{e},\end{eqnarray} \tag{ 53 }$$

which, after renormalization group running effects are taken into account, are consistent with the low energy masses. The precise viability of these relations has been widely discussed in the light of recent progress in lattice theory which enable more precise values of quark masses to be determined, especially the strange quark mass (see, e.g., [63]). In SUSY theories with low values of the ratio of Higgs VEVs, the relation for the third generation y_b = y_τ at the GUT scale remains viable, but a viable GUT scale ratio of y_μ/y_s is more accurately achieved within SUSY SU(5) GUTs using a Clebsch factor of 9/2, as proposed in [64], which is 50% higher than the Georgi–Jarlskog prediction of 3. For other Clebsch relations see [65].

As already remarked, it is a remarkable fact that the smallest leptonic mixing angle, the reactor angle, is of a similar magnitude to the largest quark mixing angle, the Cabibbo angle, indeed they may even be equal to each other up to a factor of $\sqrt{2}.$ Such relationships may be a hint of a connection between leptonic mixing and quark mixing, where such a connection might be achieved using GUTs [66, 67]. For example, the Georgi–Jarlskog relations discussed above already lead to the left-handed charged lepton mixing angle having a simple relation with the RH down-type quark mixing angle ${\theta }_{12}^{{e}_{{\rm{L}}}}\approx {\theta }_{12}^{{d}_{{\rm{R}}}}/3$ where the approximation assumes hierarchical Yukawa matrices, with the 1-1 elements being approximately zero. If the upper 2 × 2 Yukawa matrices are symmetric (as motivated by the successful Gatto–Sartori–Tonin (GST) relation [68] which relates the 12 mixing ${\theta }_{12}^{{d}_{{\rm{L}},{\rm{R}}}}$ to the down and strange mass by ${\theta }_{12}^{{d}_{{\rm{L}},{\rm{R}}}}\approx \sqrt{{m}_{d}/{m}_{s}}$) then we may drop the L, R subscripts and this relation simply becomes ${\theta }_{12}^{e}={\theta }_{12}^{d}/3.$ In large classes of models, the quark mixing originates predominantly from the down-type quark sector, in which case this relation becomes ${\theta }_{12}^{e}={\theta }_{{\rm{C}}}/3.$ If one starts from TB mixing in the neutrino sector, resulting from some discrete family symmetry, then, using the results in section 4.3 such a charged lepton correction results in a reactor angle in the lepton sector of ${\theta }_{13}\approx {\theta }_{{\rm{C}}}/(3\sqrt{2}).$ This is a factor of 3 too small to account for the observed reactor angle, but it illustrates how the reactor angle could possibly be related to the Cabibbo angle using GUTs. Indeed it has been suggested that perhaps the charged lepton mixing angle is exactly equal to the Cabibbo angle in some GUT model, leading to ${\theta }_{13}\approx {\theta }_{{\rm{C}}}/\sqrt{2}$ [26, 69]. However it is non-trivial to reconcile such large charged lepton mixing with the successful relationships between charged lepton and down-type quark masses, and it seems more likely that charged lepton mixing is not entirely responsible for the reactor angle.

The above discussion provides an additional motivation for combining GUTs with discrete family symmetry in order to account for the reactor angle. Putting these two ideas together we are suggestively led to a framework of new physics beyond the SM based on commuting GUT and family symmetry groups

$$\begin{eqnarray}&&{G}_{{\rm{GUT}}}\times {G}_{{\rm{FAM}}}.\end{eqnarray} \tag{ 54 }$$

The spectrum of quark and lepton masses may also provide some motivation for considering a family symmetry as well as a grand unified symmetry, acting in different directions, as illustrated in figure 11. The (scaled) heights of the towers representing the fermion masses, show vast hierarchies which are completely mysterious in the SM. Some popular family symmetries which admit triplet representations are shown in figure 12. The mathematics of these and other groups has recently been reviewed in [11, 14, 15] to which we refer the interested reader for more details.

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Here we just mention the family symmetry A₄ as it is the smallest non-Abelian finite group with an irreducible triplet representation. A₄ is the symmetry group of the tetrahedron. There are 12 independent transformations of the tetrahedron and hence 12 group elements as follows:

• Four rotations by 120 ° clockwise (seen from a vertex) which are T-type.
• Four rotations by 120 ° anti-clockwise(seen from a vertex) which are T-type.
• Three rotations by 180 ° which are S-type.
• One unit operator ${\mathcal{I}}$.

The generators of the A₄ group, can be written as S and T with ${S}^{2}={T}^{3}={({ST})}^{3}={\mathcal{I}}.$ All 12 group elements can be formed by multiplying together these two generators in all possible ways.

A₄ has four irreducible representations, three singlets 1, 1' and 1'' and one triplet. The products of singlets are:

$$\begin{eqnarray}1\otimes 1=1, & {1}^{\prime }\otimes 1^{\prime\prime} =1, & {1}^{\prime }\otimes {1}^{\prime }={1}^{\prime\prime } & {1}^{\prime\prime }\otimes {1}^{\prime\prime }={1}^{\prime }.\end{eqnarray} \tag{ 55 }$$

Later we shall sometimes work in the real basis of the triplet representation [70],

$$\begin{eqnarray}S=\left(\begin{array}{ccc}1 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & -1\end{array}\right),\ \ \ T=\left(\begin{array}{ccc}0 & 1 & 0\\ 0 & 0 & 1\\ 1 & 0 & 0\end{array}\right)\end{eqnarray} \tag{ 56 }$$

which generate 12 real 3 × 3 matrix group elements after multiplying these two matrices together in all possible ways [70]. In this basis one has the following Clebsch rules for the multiplication of two triplets, $3\times 3=1+{1}^{\prime }+{1}^{\prime\prime }+{3}_{1}+{3}_{2},$ with

$$\begin{eqnarray}{({ab})}_{1} & = & {a}_{1}{b}_{1}+{a}_{2}{b}_{2}+{a}_{3}{b}_{3};\\ {({ab})}_{{1}^{\prime }} & = & {a}_{1}{b}_{1}+{\omega }^{2}{a}_{2}{b}_{2}+\omega {a}_{3}{b}_{3};\\ {({ab})}_{{1}^{\prime\prime }} & = & {a}_{1}{b}_{1}+\omega {a}_{2}{b}_{2}+{\omega }^{2}{a}_{3}{b}_{3};\\ {({ab})}_{{3}_{1}} & = & ({a}_{2}{b}_{3},{a}_{3}{b}_{1},{a}_{1}{b}_{2});\\ {({ab})}_{{3}_{2}} & = & ({a}_{3}{b}_{2},{a}_{1}{b}_{3},{a}_{2}{b}_{1}),\end{eqnarray} \tag{ 57 }$$

where a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃) are the two triplets and ${\omega }^{3}=1.$

The starting point for family symmetry models is to consider the Klein symmetry of the neutrino mass matrix. First consider the phase symmetry of the diagonal charged lepton mass matrix M_e,

$$\begin{eqnarray}&&{T}^{\dagger }({M}_{e}^{\dagger }{M}_{e})T={M}_{e}^{\dagger }{M}_{e},\end{eqnarray} \tag{ 58 }$$

where T = diag(1, ω , ω²) and⁷ ω = e^2πi/n. For example for n = 3 clearly T generates the group ${Z}_{3}^{T}.$ In any case, the Klein symmetry of the neutrino mass matrix, in this basis, is given by

$$\begin{eqnarray}&&{m}^{\nu }={S}^{T}{m}^{\nu }S,\ \ \ {m}^{\nu }={U}^{T}{m}^{\nu }U,\end{eqnarray} \tag{ 59 }$$

where [71]

$$\begin{eqnarray}&&S={U}_{{\rm{PMNS}}}^{*}\ {\rm{diag}}(+1,-1,-1)\ {U}_{{\rm{PMNS}}}^{T},\end{eqnarray} \tag{ 60 }$$

$$\begin{eqnarray}&&U={U}_{{\rm{PMNS}}}^{*}\ {\rm{diag}}(-1,+1,-1)\ {U}_{{\rm{PMNS}}}^{T},\end{eqnarray} \tag{ 61 }$$

$$\begin{eqnarray}&&{SU}={U}_{{\rm{PMNS}}}^{*}\ {\rm{diag}}(-1,-1,+1)\ {U}_{{\rm{PMNS}}}^{T}\end{eqnarray} \tag{ 62 }$$

and

$$\begin{eqnarray}&&{\mathcal{K}}=\{1,S,U,{SU}\}\end{eqnarray} \tag{ 63 }$$

is called the Klein symmetry ${Z}_{2}^{S}\times {Z}_{2}^{U}.$

The idea of direct models is that the three generators S, T, U introduced above are embedded into a discrete family symmetry G which is broken by new Higgs fields called 'flavons' of two types: ϕ^l whose VEVs preserve T and ϕ^ν whose VEVs preserve S, U. These flavons are segregated such that ϕ^l only appears in the charged lepton sector and ϕ^ν only appears in the neutrino sector as depicted in figure 13, thereby enforcing the symmetries of the mass matrices. Note that the full Klein symmetry ${Z}_{2}^{S}\times {Z}_{2}^{U}$ of the neutrino mass matrix is enforced by symmetry in the direct approach.

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Following the measurement of the reactor angle, it has emerged that the only viable direct models are those based on Δ (6N²) [72–74]. Unfortunately large N is required in order to achieve the desired reactor angle. Moreover such models generally predict the ${\mathcal{C}}{\mathcal{P}}$ phase δ = 0, π resulting in the atmospheric sum rule [73],

$$\begin{eqnarray}&&{\theta }_{23}=45^\circ \mp {\theta }_{13}/\sqrt{2}.\end{eqnarray} \tag{ 64 }$$

which follows since the PMNS matrix has the TM2 form shown in equation (13).

The inclusion of discrete family symmetry and GUTs into a theory of flavour offers the possibility of having spontaneously broken ${\mathcal{C}}{\mathcal{P}}$ symmetry. The idea is that the high energy theory respects ${\mathcal{C}}{\mathcal{P}}$ but it becomes spontaneously broken along with the discrete family symmetry and GUT symmetry. As with all types of spontaneous symmetry breaking, this offers the possibility of understanding the origin of ${\mathcal{C}}{\mathcal{P}}$ violation, and relating ${\mathcal{C}}{\mathcal{P}}$ violation in the quark and lepton sectors. For example, it is possible that the ${\mathcal{C}}{\mathcal{P}}$ violating phases in the quark and lepton sectors may be predicted within this kind of approach.

The direct approach can be generalized to the case of a conserved ${\mathcal{C}}{\mathcal{P}}$ (see [75] and references therein) which is spontaneously broken as shown in figure 14. This approach has been studied for Δ (6N²) in [76]. However since we already know that δ = 0, π in this case it only fixes the Majorana phases.

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The generalized ${\mathcal{C}}{\mathcal{P}}$ approach has also been used in the semi-direct approach (defined below) where the phase δ is undetermined without ${\mathcal{C}}{\mathcal{P}}$. Here the results are more interesting since for the smaller groups like A₄ and S₄ one generally predicts a discrete choice including δ = ±π/2 [77]. However for larger groups in the series Δ(6N²) and Δ(3N²), broken in a semi-direct way, the discrete predictions for δ proliferate [78]. Motivated by the the good experimental prospects for measuring leptonic ${\mathcal{C}}{\mathcal{P}}$ violation, there has been considerable theoretical interest in ${\mathcal{C}}{\mathcal{P}}$ symmetry in different approaches to family symmetry models [79].

Recently an invariant approach ${\mathcal{C}}{\mathcal{P}}$ symmetry in family symmetry models has been discussed [80]. It is worthwhile to first recap how the invariant approach works for any theory where the Lagrangian is specified. Following [81], to study ${\mathcal{C}}{\mathcal{P}}$ symmetry in any model one divides a given Lagrangian as follows ${\mathcal{L}}={{\mathcal{L}}}_{{\mathcal{C}}{\mathcal{P}}}+{{\mathcal{L}}}_{\mathrm{rem}}$ where ${{\mathcal{L}}}_{{\mathcal{C}}{\mathcal{P}}}$ is the part that automatically conserves ${\mathcal{C}}{\mathcal{P}}$(like the kinetic terms and gauge interactions) while ${{\mathcal{L}}}_{\mathrm{rem}}$ includes the ${\mathcal{C}}{\mathcal{P}}$ violating non-gauge interactions such as the Yukawa couplings. Then one considers the most general ${\mathcal{C}}{\mathcal{P}}$ transformation that leaves ${{\mathcal{L}}}_{{\mathcal{C}}{\mathcal{P}}}$ invariant and check if invariance under ${\mathcal{C}}{\mathcal{P}}$ restricts ${{\mathcal{L}}}_{\mathrm{rem}}$—only if this is the case can ${\mathcal{L}}$ violate ${\mathcal{C}}{\mathcal{P}}$.

In the presence of a family symmetry G, one may check if a given vacuum leads to spontaneous ${\mathcal{C}}{\mathcal{P}}$ violation, as follows. Consider a Lagrangian invariant under G and ${\mathcal{C}}{\mathcal{P}}$, containing a series of scalars which under ${\mathcal{C}}{\mathcal{P}}$ transform as $({\mathcal{C}}{\mathcal{P}}){\phi }_{i}{({\mathcal{C}}{\mathcal{P}})}^{-1}={U}_{{ij}}{\phi }_{j}^{*}.$ In order for the vacuum to be ${\mathcal{C}}{\mathcal{P}}$ invariant, the following relation has to be satisfied: $\langle 0| {\phi }_{i}| 0\rangle ={U}_{{ij}}\langle 0| {\phi }_{j}^{*}| 0\rangle$ [82]. The presence of G usually allows for many choices for U. If (and only if) no choice of U exists which satisfies the previous condition, will the vacuum violate ${\mathcal{C}}{\mathcal{P}}$, leading to spontaneous ${\mathcal{C}}{\mathcal{P}}$ violation. In order to prove that no choice of U exists one can construct ${\mathcal{C}}{\mathcal{P}}$-odd invariants.

As a brief review of how to derive ${\mathcal{C}}{\mathcal{P}}$-odd invariants, consider the Lagrangian of the leptonic part of the SM extended by Majorana neutrino masses. After electroweak breaking at low energies, the most general mass terms are as in equation (3) which we rewrite in matrix form as

$$\begin{eqnarray}&&{{\mathcal{L}}}^{{\rm{lepton}}}=-{\bar{e}}_{{\rm{L}}}{m}_{l}{e}_{{\rm{R}}}-\frac{1}{2}{\bar{\nu }}_{{\rm{L}}}{m}_{\nu }{\nu }_{{\rm{L}}}^{c}+{\rm{h.c.}},\end{eqnarray} \tag{ 65 }$$

due to the SU(2)_L structure, the most general ${\mathcal{C}}{\mathcal{P}}$ transformation which leaves the leptonic gauge interactions invariant are (ignoring spin)

$$\begin{eqnarray}&&L(x)\to {{UL}}^{*}({x}_{P}),\ {e}_{{\rm{R}}}(x)\to {{Ve}}_{{\rm{R}}}^{*}({x}_{P}),\end{eqnarray} \tag{ 66 }$$

where L = (ν_L, e_L ) are the left-handed neutrino and charged lepton fields in a weak basis, and x_P are the parity (three-space) inverted coordinates.

In order for ${{\mathcal{L}}}^{{\rm{lepton}}}$ to be ${\mathcal{C}}{\mathcal{P}}$ invariant under equation (66), the terms shown in the equation (65) go into their respective h.c. terms and vice versa:

$$\begin{eqnarray}&&{U}^{\dagger }{m}_{\nu }{U}^{*}={m}_{\nu }^{*},\ \ \ {U}^{\dagger }{m}_{l}V={m}_{l}^{*}.\end{eqnarray} \tag{ 67 }$$

From equation (67) one can infer how to build combinations of the mass matrices that will result in equations where U and V cancel entirely. The condition for ${\mathcal{C}}{\mathcal{P}}$ to be conserved is [81]:

$$\begin{eqnarray}&&{I}_{1}\equiv \mathrm{Tr}\;\left({\left[{H}_{\nu },{H}_{l}\right]}^{3}\right)=\mathrm{Tr}\;\left({\left[{H}_{\nu }{H}_{l}-{H}_{l}{H}_{\nu }\right]}^{3}\right)=0,\end{eqnarray} \tag{ 68 }$$

where ${H}_{\nu }\equiv {m}_{\nu }{m}_{\nu }^{\dagger }$ and ${H}_{l}\equiv {m}_{l}{m}_{l}^{\dagger }.$ This equation is a necessary and sufficient condition for Dirac ${\mathcal{C}}{\mathcal{P}}$ invariance, since it follows from the existence of ${\mathcal{C}}{\mathcal{P}}$ transformations in equation (67). If the mass matrices are chosen such that I₁ = 0 then Dirac type ${\mathcal{C}}{\mathcal{P}}$ is conserved while if ${I}_{1}\ne 0$ then Dirac type ${\mathcal{C}}{\mathcal{P}}$ is violated⁸ .

As pointed out in [80], once a Lagrangian is specified, which is invariant under a family symmetry G and some ${\mathcal{C}}{\mathcal{P}}$ transformation, then the consistency relations [75, 77] are automatically satisfied. In order to prove this it is sufficient to consider some generic Lagrangian invariant under a family symmetry transformation, involving some mass term m (Dirac or Majorana), then define H = mm^†. Under some G transformation, ρ(g), the mass term remains unchanged implying:

$$\begin{eqnarray}&&\rho {(g)}^{\dagger }H\rho (g)=H.\end{eqnarray} \tag{ 69 }$$

Invariance of the Lagrangian under ${\mathcal{C}}{\mathcal{P}}$ transformation U requires the mass term to swap with its h.c., hence:

$$\begin{eqnarray}&&{U}^{\dagger }{HU}={H}^{*}.\end{eqnarray} \tag{ 70 }$$

Taking the complex conjugate of equation (69) we find

$$\begin{eqnarray}&&{(\rho {(g)}^{\dagger })}^{*}{H}^{*}\rho {(g)}^{*}={H}^{*}={U}^{\dagger }{HU},\end{eqnarray} \tag{ 71 }$$

using equation (70) for the last equality. Using equation (70) again:

$$\begin{eqnarray}&&{(\rho {(g)}^{\dagger })}^{*}{U}^{\dagger }{HU}\rho {(g)}^{*}={U}^{\dagger }{HU}.\end{eqnarray} \tag{ 72 }$$

Hence by using once more equation (69) for a g', we find

$$\begin{eqnarray}&&U{(\rho {(g)}^{\dagger })}^{*}{U}^{\dagger }{HU}\rho {(g)}^{*}{U}^{\dagger }=H=\rho {({g}^{\prime })}^{\dagger }H\rho ({g}^{\prime }).\end{eqnarray} \tag{ 73 }$$

Comparing both sides of equation (73) we see

$$\begin{eqnarray}&&U\rho {(g)}^{*}{U}^{\dagger }=\rho ({g}^{\prime })\end{eqnarray} \tag{ 74 }$$

which is just the consistency relation [75, 77]. In other words, if we consider equations (69) and (70) we do not need to consider the consistency condition separately since it always follows.

The above considerations about whether ${\mathcal{C}}{\mathcal{P}}$ is conserved or violated apply separately both to the original theory (defined by some high energy Lagrangian, above the scale of symmetry breaking) and to the spontaneously broken theory (defined by some low energy effective Lagrangian, below the scale of symmetry breaking). We are mainly interested in theories which respect ${\mathcal{C}}{\mathcal{P}}$ at high energy, but where ${\mathcal{C}}{\mathcal{P}}$ is spontaneously broken, since these allow for the possibility of being able to predict the amount of ${\mathcal{C}}{\mathcal{P}}$ violation (e.g. the physical ${\mathcal{C}}{\mathcal{P}}$ violating phases in some basis).

Taking a less constrained approach to model building one may suppose that we start from only smaller discrete family groups such as S₄, which leads to either TB or BM mixing at leading order, or A₅ which leads to GR mixing at leading order as shown in figure 15. Then we suppose that at higher order, one or more of the generators S, T, U is broken, which is necessary in this approach since the resulting BM, TB and GR mixing patterns discussed in equations (7)–(9) are excluded. There are two interesting possibilities depicted in figure 15 as follows:

• (1)  
  If the T generator protecting the charged lepton mass matrix is broken, then we can expect charged lepton corrections leading to the solar sum rules discussed in section 4.3.
• (2)  
  If the U generator is broken then this leads to either TM1 or TM2 mixing depending on whether SU or S is preserved, leading to atmospheric sum rules as discussed in section 4.2.

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The semi-direct approach was first used in [84, 85] for A₄ where there is no U generator to start with and also S₄ which is broken to A₄ at higher order [85]. It was subsequently generalized to von Dyck groups in [86]. In all cases the reactor angle is not predicted but described by a free parameter. This is a retreat from the original goal of predicting lepton mixing angles using symmetry.

The final logical possibility is that the family symmetry is completely broken in both the neutrino and charged lepton sectors as shown in figure 16 for the example of the smallest family symmetry group that admits triplet representations, namely A₄. In this approach, we allow the flavons ϕ^l and ϕ^ν to have not only symmetry preserving vacuum alignments, but also new alignments which are orthogonal to them and break the symmetry. In the following, ϕ^l refers to ϕ_e,μ,τ which only enter the charged lepton sector and are responsible for a diagonal charged lepton mass matrix, while ϕ^ν refers to ${\phi }_{\mathrm{atm}},\;$ ${\phi }_{\mathrm{sol}},$ and ${\phi }_{\mathrm{dec}}$ which only enter the neutrino sector and are responsible for a Dirac mass matrix of the CSD(n) form.

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The advantages of the indirect approach over the previous approaches are:

• (1)  
  It can involve a small family symmetry group such as A₄ (unlike the direct approach which involves large family symmetry groups).
• (2)  
  It is highly predictive since it can yield CSD(n) where the entire PMNS matrix is predicted in terms of one input parameter (unlike the semi-direct approach which only predicts sum rules).

The basic starting point is to consider some small family symmetry such as A₄ which admits triplet representations. The family symmetry is broken by triplet flavons ϕ_i whose vacuum alignment will control the structure of the Yukawa couplings. To illustrate how this works, we sketch a model, where the relevant operators responsible for the Yukawa structure in the neutrino sector are

$$\begin{eqnarray}&&\displaystyle \frac{1}{{\rm{\Lambda }}}{H}_{u}(L\cdot {\phi }_{\mathrm{atm}}){\nu }_{{\rm{atm}}}^{c}+\displaystyle \frac{1}{{\rm{\Lambda }}}{H}_{u}(L\cdot {\phi }_{\mathrm{sol}}){\nu }_{{\rm{sol}}}^{c}+\displaystyle \frac{1}{{\rm{\Lambda }}}{H}_{u}(L\cdot {\phi }_{\mathrm{dec}}){\nu }_{{\rm{dec}}}^{c},\end{eqnarray} \tag{ 75 }$$

where L is the SU(2) lepton doublet, assumed to transform as a triplet under the family symmetry, while ${\nu }_{{\rm{atm}}}^{c},{\nu }_{{\rm{sol}}}^{c},{\nu }_{{\rm{dec}}}^{c}$ are ${\mathcal{C}}{\mathcal{P}}$ conjugates of the RH neutrinos and H_u is the electroweak scale up-type Higgs field, the latter being family symmetry singlets but distinguished by some additional quantum numbers. In the charged-lepton sector, we consider the operators

$$\begin{eqnarray}&&\displaystyle \frac{1}{{\rm{\Lambda }}}{H}_{d}(L\cdot {\phi }_{e}){e}^{c}+\displaystyle \frac{1}{{\rm{\Lambda }}}{H}_{d}(L\cdot {\phi }_{\mu }){\mu }^{c}+\displaystyle \frac{1}{{\rm{\Lambda }}}{H}_{d}(L\cdot {\phi }_{\tau }){\tau }^{c},\end{eqnarray} \tag{ 76 }$$

where ${e}^{c},{\mu }^{c},{\tau }^{c}$ are the ${\mathcal{C}}{\mathcal{P}}$ conjugated RH electron, muon and tau respectively. The RH neutrino Majorana superpotential is typically chosen to give a diagonal mass matrix

$$\begin{eqnarray}&&{M}_{{\rm{R}}}=\mathrm{diag}({M}_{{\rm{atm}}},{M}_{{\rm{sol}}},{M}_{{\rm{dec}}}).\end{eqnarray} \tag{ 77 }$$

Details of the origin of these operators (e.g. in terms of Majoron fields), the relative values of ${M}_{{\rm{atm}}},{M}_{{\rm{sol}}},{M}_{{\rm{dec}}}$ as well as the inclusion of any off-diagonal terms in M_R will all depend on the additional specifications of the model beyond our sketchy model here (we shall consider real models shortly in the next section 7).

The idea is that CSD(n) discussed in section 5.4 emerges from flavon vacuum alignments in the effective operators involving three flavon fields ${\phi }_{\mathrm{atm}},\;$ ${\phi }_{\mathrm{sol}},$ and ${\phi }_{\mathrm{dec}}$ which are triplets under the flavour symmetry and acquire VEVs that break the family symmetry completely in both the neutrino and charged lepton sectors. The subscripts are chosen by noting that ${\phi }_{\mathrm{atm}}$ correlates with the atmospheric neutrino mass m₃, ${\phi }_{\mathrm{sol}}$ with the solar neutrino mass m₂, and ${\phi }_{\mathrm{dec}}$ with the lightest neutrino mass m₁, which in CSD is light enough that the associated third RH neutrino can, to good approximation, be thought of as decoupled from the theory [6]. CSD(n) corresponds to the choice of vacuum alignments

$$\begin{eqnarray}&&\langle {\phi }_{\mathrm{atm}}\rangle ={v}_{\mathrm{atm}}\left(\begin{array}{c}0\\ 1\\ 1\end{array}\right),\qquad \langle {\phi }_{\mathrm{sol}}\rangle ={v}_{\mathrm{sol}}\left(\begin{array}{c}1\\ \mathrm{in}\\ n-2\end{array}\right),\qquad \langle {\phi }_{\mathrm{dec}}\rangle ={v}_{\mathrm{dec}}\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right),\end{eqnarray} \tag{ 78 }$$

where n is a positive integer, and the only phases allowed are in the overall proportionality constants. Such vacuum alignments arise from symmetry preserving alignments together with orthogonality conditions [57, 58], as discussed below.

The starting point for understanding the alignments in equation (78) are the symmetry preserving vacuum alignments of A₄, namely:

$$\begin{eqnarray}&&\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right),\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right),\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right),\left(\begin{array}{c}\pm 1\\ \pm 1\\ \pm 1\end{array}\right),\end{eqnarray} \tag{ 79 }$$

which each preserve some subgroup of A₄ in a basis where the 12 group elements in the triplet representation are real as in equation (56) [70] (i.e. each alignment in equation (79) is an eigenvector of at least one non-trivial group element with eigenvalue +1.) The first alignment in equation (78), which completely breaks the A₄ symmetry, arises from the orthogonality conditions

$$\begin{eqnarray}&&\left(\begin{array}{c}0\\ 1\\ 1\end{array}\right)\;\perp \;\left(\begin{array}{c}1\\ 1\\ -1\end{array}\right),\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right)\end{eqnarray} \tag{ 80 }$$

involving two symmetry preserving alignments selected from equation (79). The following symmetry breaking alignment may be obtained which is orthogonal to the alignment in equation (80) and one of the symmetry preserving alignments

$$\begin{eqnarray}&&\left(\begin{array}{c}2\\ -1\\ 1\end{array}\right)\;\perp \;\left(\begin{array}{c}1\\ 1\\ -1\end{array}\right),\left(\begin{array}{c}0\\ 1\\ 1\end{array}\right).\end{eqnarray} \tag{ 81 }$$

The CSD(n) alignment in equation (78) is orthogonal to the above alignment in equation (81),

$$\begin{eqnarray}&&\left(\begin{array}{c}1\\ n\\ n-2\end{array}\right)\;\perp \;\left(\begin{array}{c}2\\ -1\\ 1\end{array}\right),\end{eqnarray} \tag{ 82 }$$

where the orthogonality in equation (82) is maintained for any value of n (not necessarily integer). To pin down the value of n and show that it is a particular integer requires a further orthogonality condition.

For example, for n = 3, the desired alignment is obtained from the two orthogonality conditions

$$\begin{eqnarray}&&\left(\begin{array}{c}1\\ 3\\ 1\end{array}\right)\;\perp \;\left(\begin{array}{c}2\\ -1\\ 1\end{array}\right),\left(\begin{array}{c}1\\ 0\\ -1\end{array}\right),\end{eqnarray} \tag{ 83 }$$

where the first condition above is a particular case of equation (82) and the second condition involves a new alignment, obtained from two of the symmetry preserving alignments in equation (79),

$$\begin{eqnarray}&&\left(\begin{array}{c}1\\ 0\\ -1\end{array}\right)\perp \left(\begin{array}{c}1\\ 1\\ 1\end{array}\right),\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right).\end{eqnarray} \tag{ 84 }$$

Using equation (75), the vacuum alignments in equation (78) make up the columns of the Dirac neutrino Yukawa matrix ${Y}^{\nu }\propto (\langle {\phi }_{\mathrm{atm}}\rangle ,\langle {\phi }_{\mathrm{sol}}\rangle ,\langle {\phi }_{\mathrm{dec}}\rangle ),$ giving a Dirac mass matrix

$$\begin{eqnarray}{m}^{{\rm{D}}}={Y}^{\nu }{v}_{u}=\left(\begin{array}{ccc}0 & a & 0\\ e & {na} & 0\\ e & (n-2)a & c\end{array}\right),\end{eqnarray} \tag{ 85 }$$

which is an extension of equation (40) to include a third (decoupled) RH neutrino, where ${m}^{{\rm{D}}}=({m}_{{\rm{atm}}}^{{\rm{D}}},{m}_{{\rm{sol}}}^{{\rm{D}}},{m}_{{\rm{dec}}}^{{\rm{D}}})$ and the coefficients e, a, and c are generally complex. The charged-lepton Yukawa matrix is chosen to be diagonal (up to model-dependent corrections, assumed small), corresponding to the existence of three flavons ϕ_e, ϕ_μ and ϕ_τ in the charged-lepton sector which acquire VEVs with alignments [57, 58]

$$\begin{eqnarray}&&\langle {\phi }_{e}\rangle ={v}_{e}\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right),\qquad \langle {\phi }_{\mu }\rangle ={v}_{\mu }\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right),\qquad \langle {\phi }_{\tau }\rangle ={v}_{\tau }\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right).\end{eqnarray} \tag{ 86 }$$

Given this choice, it is clear that Y^e is diagonal, hence ${U}_{{e}_{{\rm{L}}}}$ is the identity matrix up to diagonal phase rotations, and that ${U}_{\mathrm{PMNS}}={U}_{{\nu }_{{\rm{L}}}}^{\dagger },$ i.e. simply the matrix that diagonalizes the neutrino mass matrix, up to charged lepton phase rotations.

As an example of an 'indirect' model, an 'A to Z of flavour with PS' based on the PS gauge group has been proposed [87] as sketched in figure 17. The PS symmetry leads to ${Y}^{u}={Y}^{\nu },$ where the columns of the Yukawa matrices are determined as in equation (75) with the flavon alignments as in equation (78) for the case n = 4. The first column is proportional to the alignment (0, e, e) the second column proportional to the CSD(4) orthogonal alignment, (a, 4a, 2a) and the third column is proportional to the alignment (0, 0, c), where $e\ll a\ll c$ gives the hierarchy ${m}_{u}\ll {m}_{c}\ll {m}_{t}.$ This structure predicts a Cabibbo angle θ_C ≈ 1/4 in the diagonal Y^d ∼ Y^e basis enforced by the first three alignments in equation (79). It also predicts a normal neutrino mass hierarchy with ${\theta }_{13}\approx 9^\circ ,$ θ₂₃ ≈ 45° and δ ≈ 260° [87].

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The model is based on the PS gauge group, with A₄ × Z₅ (A to Z) family symmetry

$$\begin{eqnarray}&&{SU}{(4)}_{{\rm{C}}}\times {SU}{(2)}_{{\rm{L}}}\times {SU}{(2)}_{{\rm{R}}}\times {A}_{4}\times {Z}_{5}.\end{eqnarray} \tag{ 87 }$$

The quarks and leptons are unified in the PS representations as follows

$$\begin{eqnarray}{F}_{i} & = & {(4,2,1)}_{i}={\left(\begin{array}{cccc}u & u & u & \nu \\ d & d & d & e\end{array}\right)}_{i}\to ({Q}_{i},{L}_{i}),\\ {F}_{i}^{c} & = & {(\bar{4},1,2)}_{i}={\left(\begin{array}{cccc}{u}^{c} & {u}^{c} & {u}^{c} & {\nu }^{c}\\ {d}^{c} & {d}^{c} & {d}^{c} & {e}^{c}\end{array}\right)}_{i}\to ({u}_{i}^{c},{d}_{i}^{c},{\nu }_{i}^{c},{e}_{i}^{c}),\end{eqnarray} \tag{ 88 }$$

where the SM multiplets ${Q}_{i},{L}_{i},{u}_{i}^{c},{d}_{i}^{c},{\nu }_{i}^{c},{e}_{i}^{c}$ resulting from PS breaking are also shown and the subscript i ( = 1, 2, 3) denotes the family index. The left-handed quarks and leptons form an A₄ triplet F, while the three (${\mathcal{C}}{\mathcal{P}}$ conjugated) RH fields F^c_i are A₄ singlets, distinguished by Z₅ charges α, α³, 1, for i = 1, 2, 3, respectively. Clearly the PS model cannot be embedded into an SO(10) GUT since different components of the 16-dimensional representation of SO(10) would have to transform differently under A₄ × Z₅, which is impossible. On the other hand, the PS gauge group and A₄ could emerge directly from string theory.

The PS gauge group is broken at the GUT scale to the SM

$$\begin{eqnarray}&&{SU}{(4)}_{{\rm{C}}}\times {SU}{(2)}_{{\rm{L}}}\times {SU}{(2)}_{{\rm{R}}}\to {SU}{(3)}_{{\rm{C}}}\times {SU}{(2)}_{{\rm{L}}}\times U{(1)}_{Y},\end{eqnarray} \tag{ 89 }$$

by PS Higgs, H^c and $\bar{{H}^{c}},$

$$\begin{eqnarray}{H}^{c} & = & (\bar{4},1,2)=({u}_{H}^{c},{d}_{H}^{c},{\nu }_{H}^{c},{e}_{H}^{c}),\\ \bar{{H}^{c}} & = & (4,1,2)=({\bar{u}}_{H}^{c},{\bar{d}}_{H}^{c},{\bar{\nu }}_{H}^{c},{\bar{e}}_{H}^{c}).\end{eqnarray} \tag{ 90 }$$

These acquire VEVs in the 'RH neutrino' directions, with equal VEVs close to the GUT scale 2 × 10¹⁶ GeV,

$$\begin{eqnarray}&&\langle {H}^{c}\rangle =\langle {\nu }_{H}^{c}\rangle =\langle \bar{{H}^{c}}\rangle =\langle {\bar{\nu }}_{H}^{c}\rangle \sim 2\times {10}^{16}\ {\rm{GeV}},\end{eqnarray} \tag{ 91 }$$

so as to maintain SUSY gauge coupling unification.

Our starting point is to assume that the high energy theory, above the PS breaking scale, conserves ${\mathcal{C}}{\mathcal{P}}$ symmetry. Under a ${\mathcal{C}}{\mathcal{P}}$ transformation, the A₄ singlet fields ξ, Σ_u, Σ_d transform into their complex conjugates

$$\begin{eqnarray}&&\xi \to {\xi }^{*},\ {{\rm{\Sigma }}}_{u}\to {{\rm{\Sigma }}}_{u}^{*},\ {{\rm{\Sigma }}}_{d}\to {{\rm{\Sigma }}}_{d}^{*},\end{eqnarray} \tag{ 92 }$$

where the complex conjugate fields transform in the complex conjugate representations under A₄ × Z₅. For example if ξ ∼ α⁴, under Z₅, then ξ* ∼ α. Similarly if Σ_u ∼ 1', Σ_d ∼ 1'', under A₄, then ${{\rm{\Sigma }}}_{u}^{*}\sim {1}^{\prime\prime },$ ${{\rm{\Sigma }}}_{d}^{*}\sim {1}^{\prime }.$ On the other hand, in a particular basis, for A₄ triplets ϕ ∼ (ϕ₁, ϕ₂, ϕ₃), a consistent definition of ${\mathcal{C}}{\mathcal{P}}$ symmetry requires the second and third triplet components to swap under ${\mathcal{C}}{\mathcal{P}}$,

$$\begin{eqnarray}&&\phi \to ({\phi }_{1}^{*},{\phi }_{3}^{*},{\phi }_{2}^{*}).\end{eqnarray} \tag{ 93 }$$

With the above definition of ${\mathcal{C}}{\mathcal{P}}$, all coupling constants g and explicit masses m are real due to CP conservation and the only source of phases can be the VEVs of fields which break A₄ × Z₅. In the model of interest, all the physically interesting ${\mathcal{C}}{\mathcal{P}}$ phases will arise from Z₅ breaking.

Let us now consider the A₄ triplet fields ϕ which also carry Z₅ charges. In the full model there are four such triplet fields, or 'flavons', denoted as ${\phi }_{1}^{u},\;$ ${\phi }_{2}^{u},\;$ ${\phi }_{1}^{d},\;$ ${\phi }_{2}^{d}.$ The idea is that ${\phi }_{i}^{u}$ are responsible for up-type quark flavour, while ${\phi }_{i}^{d}$ are responsible for down-type quark flavour.

The structure of the Yukawa matrices depends on the so-called CSD(4) vacuum alignments of these flavons, with the overall phases quantized due to Z₅,

$$\begin{eqnarray}&&\langle {\phi }_{1}^{u}\rangle =\displaystyle \frac{{V}_{1}^{u}}{\sqrt{2}}{{\rm{e}}}^{{\rm{i}}m\pi /5}\left(\begin{array}{c}0\\ 1\\ 1\end{array}\right),\qquad \langle {\phi }_{2}^{u}\rangle =\displaystyle \frac{{V}_{2}^{u}}{\sqrt{21}}{{\rm{e}}}^{{\rm{i}}m\pi /5}\left(\begin{array}{c}1\\ 4\\ 2\end{array}\right),\end{eqnarray} \tag{ 94 }$$

and

$$\begin{eqnarray}&&\langle {\phi }_{1}^{d}\rangle ={V}_{1}^{d}{{\rm{e}}}^{{\rm{i}}n\pi /5}\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right),\qquad \langle {\phi }_{2}^{d}\rangle ={V}_{2}^{d}{{\rm{e}}}^{{\rm{i}}n\pi /5}\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right).\end{eqnarray} \tag{ 95 }$$

We note here that the vacuum alignments in equation (95) and the first alignment in equation (94) are fairly 'standard' alignments that are encountered in TB mixing models, while the second alignment in equation (94) is obtained using orthogonality arguments, as discussed in section 6.7. In particular we are using the vacuum alignments in equation (78) for the case n = 4, where we identify ${\phi }_{\mathrm{atm}}$ and ${\phi }_{\mathrm{sol}}$ with ${\phi }_{1}^{u}$ and ${\phi }_{2}^{u}.$ We also use the alignments in equation (86) where we identify ϕ_e and ϕ_μ with ${\phi }_{1}^{d}$ and ${\phi }_{2}^{d}.$

The model will involve Higgs bi-doublets of two kinds, h_u which lead to up-type quark and neutrino Yukawa couplings and h_d which lead to down-type quark and charged lepton Yukawa couplings. In addition a Higgs bidoublet h₃, which is also an A₄ triplet, is used to give the third family Yukawa couplings.

After the PS and A₄ breaking, most of these Higgs bi-doublets will get high scale masses and will not appear in the low energy spectrum. In fact only two light Higgs doublets will survive down to the TeV scale, namely H_u and H_d. The basic idea is that the light Higgs doublet H_u with hypercharge Y = +1/2, which couples to up-type quarks and neutrinos, is a linear combination of components of the Higgs bi-doublets of the kind h_u and h₃, while the light Higgs doublet H_d with hypercharge Y = −1/2, which couples to down-type quarks and charged leptons, is a linear combination of components of Higgs bi-doublets of the kind h_d and h₃,

$$\begin{eqnarray}&&{h}_{u},{h}_{3}\to {H}_{u},\ \ \ {h}_{d},{h}_{3}\to {H}_{d}.\end{eqnarray} \tag{ 96 }$$

The renormalizable Yukawa operators, which respect PS and A₄ symmetries, have the following form, leading to the third family Yukawa couplings shown, using equations (88), (96),

$$\begin{eqnarray}&&F.{h}_{3}{F}_{3}^{c}\to {Q}_{3}{H}_{u}{u}_{3}^{c}+{Q}_{3}{H}_{d}{d}_{3}^{c}+{L}_{3}{H}_{u}{\nu }_{3}^{c}+{L}_{3}{H}_{d}{e}_{3}^{c},\end{eqnarray} \tag{ 97 }$$

where we have used equations (88), (96). The non-renormalizable operators, which respect PS and A₄ symmetries, have the following form

$$\begin{eqnarray}&&F.{\phi }_{i}^{u}{h}_{u}{F}_{i}^{c}\to Q.\langle {\phi }_{i}^{u}\rangle {H}_{u}{u}_{i}^{c}+L.\langle {\phi }_{i}^{u}\rangle {H}_{u}{\nu }_{i}^{c},\end{eqnarray} \tag{ 98 }$$

$$\begin{eqnarray}&&F.{\phi }_{i}^{d}{h}_{d}{F}_{i}^{c}\to Q.\langle {\phi }_{i}^{d}\rangle {H}_{d}{d}_{i}^{c}+L.\langle {\phi }_{i}^{d}\rangle {H}_{d}{e}_{i}^{c},\end{eqnarray} \tag{ 99 }$$

where i = 1 gives the first column of each Yukawa matrix, while i = 2 gives the second column and we have used equations (88), (96). Thus the third family masses are naturally larger since they correspond to renormalizable operators, while the hierarchy between first and second families arises from a hierarchy of flavon VEVs. The lepton operators in equations (98), (99) may be compared to the operators in equations (75), (76).

Inserting the vacuum alignments in equations (94) and (95) into equations (98) and (99), together with the renormalizable third family couplings in equation (97), gives the Yukawa matrices of the form

$$\begin{eqnarray}{Y}^{u}{v}_{u}={Y}^{\nu }{v}_{u}=\left(\begin{array}{ccc}0 & a & 0\\ e & 4a & 0\\ e & 2a & c\end{array}\right),\ \ \ {Y}^{d}\sim {Y}^{e}\sim \left(\begin{array}{ccc}{y}_{d}^{0} & 0 & 0\\ 0 & {y}_{s}^{0} & 0\\ 0 & 0 & {y}_{b}^{0}\end{array}\right).\end{eqnarray} \tag{ 100 }$$

The PS unification predicts the equality of Yukawa matrices Y^u = Y^ν and Y^d ∼ Y^e, while the A₄ vacuum alignment predicts the structure of each Yukawa matrix, essentially identifying the first two columns with the vacuum alignments in equations (94) and (95). With a diagonal RH Majorana mass matrix, Y^ν leads to a successful prediction of the PMNS mixing parameters. Also the Cabibbo angle is given by θ_C ≈ 1/4 [59]. Thus equation (100) is a good starting point for a theory of quark and lepton masses and mixing, although the other quark mixing angles and the quark ${\mathcal{C}}{\mathcal{P}}$ phase are approximately zero. However the above discussion ignores the effect of Clebsch factors which will alter the relationship between elements of Y^d and Y^e, which also include off-diagonal elements responsible for small quark mixing angles in the full model discussed in [87].

In realistic unified models involving an SO(10)-inspired pattern of Dirac and heavy RH neutrino masses, assuming the type I seesaw, the lightest RH neutrino N₁ is too light to yield successful thermal leptogenesis, barring highly fine-tuned solutions, while the second heaviest RH neutrino N₂ is typically in the correct mass range. In [88] we discussed N₂ dominated leptogenesis in the A to Z model, where N₁ is identified with ${\nu }_{1}^{c}={\nu }_{{\rm{atm}}}^{c},$ while N₂ is identified with ${\nu }_{2}^{c}={\nu }_{{\rm{sol}}}^{c}$ and N₃ is identified with ${\nu }_{3}^{c}={\nu }_{{\rm{dec}}}^{c},$ as depicted in figure 17. In the A to Z model the neutrino Dirac mass matrix is equal to the up-type quark mass matrix and has the particular constrained structure in equation (100). We showed that flavour coupling effects in the Boltzmann equations are crucial to the success of such N₂ dominated leptogenesis in this model, by helping to ensure that the flavour asymmetries produced at the N₂ scale survive N₁ washout. The numerical results, supported by analytical insight, showed that in order to achieve successful N₂ leptogenesis, consistent with neutrino phenomenology, requires a 'flavour swap scenario' whereby the asymmetry generated in the tauon flavour emerges as a surviving asymmetry dominantly in the muon flavour. However successful leptogenesis requires a less hierarchical pattern of RH neutrino masses than naively expected, at the expense of some mild fine-tuning involving significant off-diagonal elements of the heavy RH Majorana mass matrix. This leads to large deviations from the CSD(4) predictions, including a NO neutrino spectrum with an atmospheric neutrino mixing angle well into the second octant and a Dirac phase δ_CP ≈ 20°, a set of predictions that will be tested soon in neutrino oscillation experiments, as discussed in [88].

In this section we describe a fairly complete A₄ × SU(5) SUSY GUT model which implements CSD(3) with two RH neutrinos [89]. This model has the following virtues:

• It is fully renormalizable at the GUT scale, with an explicit SU(5) breaking sector and a spontaneously broken ${\mathcal{C}}{\mathcal{P}}$ symmetry.
• The MSSM is reproduced with R-parity emerging from a discrete ${{\mathbb{Z}}}_{4}^{R}.$
• Doublet-triplet splitting is achieved through the missing partner mechanism [90].
• A μ term is generated at the correct scale.
• Proton decay is sufficiently suppressed.
• It solves the strong ${\mathcal{C}}{\mathcal{P}}$ problem through the Nelson–Barr mechanism [91, 92].
• It explains the hierarchies in the quark sector, and successfully fits all of the quark masses, mixing angles and the ${\mathcal{C}}{\mathcal{P}}$ phase, using only ${\mathcal{O}}(1)$ parameters.
• It justifies the CSD(3) alignment which accurately predicts the leptonic mixing angles, as well as a normal neutrino mass hierarchy.
• It involves two RH neutrinos with the lighter one dominantly responsible for the atmospheric neutrino mass.
• There is only one physical phase in the model, called η, which is responsible for ${\mathcal{C}}{\mathcal{P}}$ violation in both leptogenesis and neutrino oscillations.
• A ${{\mathbb{Z}}}_{9}$ flavour symmetry fixes the phase η to be one of ninth roots of unity [93].

Apart from A₄ × SU(5) the model also involves the discrete symmetries ${{\mathbb{Z}}}_{9}\times {{\mathbb{Z}}}_{6}\times {{\mathbb{Z}}}_{4}^{R}.$ It is renormalizable at the GUT scale, but many effects, including most fermion masses, come from non-renormalizable terms that arise when heavy messenger fields are integrated out. Unwanted or potentially dangerous terms are forbidden by the symmetries and the prescribed messenger sector, including any terms that would generate proton decay or strong ${\mathcal{C}}{\mathcal{P}}$ violation. Such terms may arise from Planck scale suppressed terms, but prove to be sufficiently small. Due to the completeness of the model, the field content is too big to be listed here, but the superfields relevant for quarks, leptons and Higgs, including flavons, are shown in table 1.

Table 1.  Superfields containing SM fermions, the Higgses and relevant flavons.

Field	Representation
	A4	SU(5)	${{\mathbb{Z}}}_{9}$	${{\mathbb{Z}}}_{6}$	${{\mathbb{Z}}}_{4}^{R}$
F	3	$\bar{5}$	0	0	1
T1	1	10	5	0	1
T2	1	10	7	0	1
T3	1	10	0	0	1
${N}_{{\rm{atm}}}^{c}$	1	1	7	3	1
${N}_{{\rm{sol}}}^{c}$	1	1	8	3	1
Γ	1	1	0	3	1
H5	1	5	0	0	0
${H}_{\bar{5}}$	1	$\bar{5}$	2	0	0
H45	1	45	4	0	2
${H}_{\bar{45}}$	1	$\bar{45}$	5	0	0
ξ	1	1	2	0	0
${\theta }_{2}$	1	1	1	4	0
ϕatm	3	1	3	1	0
ϕsol	3	1	2	1	0

The SM fermions are contained within superfields F and T_i. The MSSM Higgs doublet H_u originates from a combination of H₅ and H₄₅, and H_d from a combination of ${H}_{\bar{5}}$ and ${H}_{\bar{4}5}.$ Having the Higgs doublets inside these different representations generates the correct relations between down-type quarks and charged leptons. Doublet-triplet splitting is achieved by the missing partner mechanism [90].

The field ξ which gains a VEV ${v}_{\xi }\sim 0.06{M}_{\mathrm{GUT}}$ generates a hierarchical fermion mass structure in the up-type quark sector through terms like ${v}_{u}{T}_{i}{T}_{j}{({v}_{\xi }/M)}^{6-i-j},$ where v_u is the VEV of H_u. It also partially contributes to the mass hierarchy for down-type quarks and charged leptons and provides the mass scales for the RH neutrinos as discussed later. It further produces a highly suppressed μ term $\sim {({v}_{\xi }/M)}^{8}{M}_{\mathrm{GUT}}.$ The resulting symmetric Yukawa matrix for up-type quarks is

$$\begin{eqnarray}{Y}_{{ij}}^{u}={u}_{{ij}}{\left(\displaystyle \frac{\langle \xi \rangle }{M}\right)}^{{n}_{{ij}}}\sim \left(\begin{array}{ccc}{\tilde{\xi }}^{4} & {\tilde{\xi }}^{3} & {\tilde{\xi }}^{2}\\ & {\tilde{\xi }}^{2} & \tilde{\xi }\\ & & 1\end{array}\right),\end{eqnarray} \tag{ 101 }$$

where $\tilde{\xi }=\langle \xi \rangle /M\sim 0.1.$ The up-type Yukawa matrix Y^u is highly nondiagonal while the down-type and charged lepton Yukawa matrices Y^d ∼ Y^e, derived from terms like FϕTH, are nearly diagonal

$$\begin{eqnarray}{Y}_{\mathrm{LR}}^{d}\sim {Y}_{\mathrm{RL}}^{e}\sim \left(\begin{array}{ccc}\frac{\langle \xi \rangle {v}_{e}}{{v}_{{{\rm{\Lambda }}}_{24}}^{2}} & \frac{\langle \xi \rangle {v}_{\mu }}{{v}_{{{\rm{\Lambda }}}_{24}}{v}_{{H}_{24}}} & 0\\ 0 & \frac{{v}_{{H}_{24}}{v}_{\mu }}{{M}^{2}} & 0\\ 0 & 0 & \frac{{v}_{\tau }}{M}\end{array}\right),\end{eqnarray} \tag{ 102 }$$

where v_e,μ,τ are charged lepton flavon VEVs as in equation (86), while ${v}_{{{\rm{\Lambda }}}_{24}}$ and ${v}_{{H}_{24}}$ are the respective VEVs of heavy Higgs Λ₂₄ and H₂₄, and we include the subscripts LR to emphasize the role of the off-diagonal term to left-handed mixing from Y^d. The off-diagonal term in Y^e also provides a tiny contribution to left-handed charged lepton mixing ${\theta }_{12}^{e}\sim {m}_{e}/{m}_{\mu }$ which may safely be neglected. It also introduces ${\mathcal{C}}{\mathcal{P}}$ violation to the CKM matrix via the phase of $\langle \xi \rangle .$

The relevant terms in the superpotential giving neutrino masses are

$$\begin{eqnarray}&&{W}_{\nu }={y}_{1}{H}_{5}F\displaystyle \frac{{\phi }_{\mathrm{atm}}}{\langle {\theta }_{2}\rangle }{N}_{\mathrm{atm}}^{c}+{y}_{2}{H}_{5}F\displaystyle \frac{{\phi }_{\mathrm{sol}}}{\langle {\theta }_{2}\rangle }{N}_{\mathrm{sol}}^{c}+{y}_{3}\displaystyle \frac{{\xi }^{2}}{{M}_{{\rm{\Gamma }}}}{N}_{\mathrm{atm}}^{c}{N}_{\mathrm{atm}}^{c}+{y}_{4}\xi {N}_{\mathrm{sol}}^{c}{N}_{\mathrm{sol}}^{c},\end{eqnarray} \tag{ 103 }$$

where the y_i are dimensionless couplings, expected to be ${\mathcal{O}}(1).$ The alignment of the flavon vacuum is fixed by the form of the superpotential, with ϕ_atm and ϕ_sol gaining VEVs according to CSD(3) in equation (78):

$$\begin{eqnarray}&&\langle {\phi }_{\mathrm{atm}}\rangle ={v}_{\mathrm{atm}}\left(\begin{array}{c}0\\ 1\\ 1\end{array}\right),\qquad \qquad \langle {\phi }_{\mathrm{sol}}\rangle ={v}_{\mathrm{sol}}\left(\begin{array}{c}1\\ 3\\ 1\end{array}\right).\end{eqnarray} \tag{ 104 }$$

This results in a low energy effective Majorana mass matrix of the form in equation (41) for n = 3, namely

$$\begin{eqnarray}{m}^{\nu }={m}_{a}\left(\begin{array}{ccc}0 & 0 & 0\\ 0 & 1 & 1\\ 0 & 1 & 1\end{array}\right)+{m}_{b}{{\rm{e}}}^{{\rm{i}}\eta }\left(\begin{array}{ccc}1 & 3 & 1\\ 3 & 9 & 3\\ 1 & 3 & 1\end{array}\right),\end{eqnarray} \tag{ 105 }$$

where η is the only physically important phase, which depends on the relative phase between the first and second column of the Dirac mass matrix in the flavour basis. The phase η is responsible for ${\mathcal{C}}{\mathcal{P}}$ violation in both leptogenesis and neutrino oscillations. We identify

$$\begin{eqnarray}&&{m}_{a}=\left|\displaystyle \frac{{y}_{1}^{2}{v}_{u}^{2}{v}_{\mathrm{atm}}^{2}{M}_{{\rm{\Gamma }}}}{{y}_{3}{\langle {\theta }_{2}\rangle }^{2}{v}_{\xi }^{2}}\right|,\qquad {m}_{b}=\left|\displaystyle \frac{{y}_{2}^{2}{v}_{u}^{2}{v}_{\mathrm{sol}}^{2}}{{y}_{4}{\langle {\theta }_{2}\rangle }^{2}{v}_{\xi }}\right|.\end{eqnarray} \tag{ 106 }$$

The Abelian flavour symmetry ${{\mathbb{Z}}}_{9}$ fixes the phase η to be one of the ninth roots of unity, through a variant of the mechanism used in [93]. The particular choice η = 2π/3 can give the neutrino mixing angles with great accuracy. Furthermore, this phase corresponds to δ_CP ≈ −π/2, consistent with hints from experimental data.

The relevant best fit parameters from our model are given in table 2, along with the model predictions for the leptonic mixing angles and neutrino masses, for tan β = 5 .

Table 2.  Best fit parameters and predictions for an A₄ × SU(5) SUSY GUT with CSD(3) and a fixed phase η = 2π/3, as described in [89]. The spectrum is NO with lightest neutrino mass m₁ = 0 and hence the remaining Majorana phase (predicted but not indicated) will be practically impossible to measure.

n	ma	mb	η	θ12	θ13	θ23	δCP	m2	m3
	(meV)	(meV)	(rad)	(°)	(°)	(°)	(°)	(meV)	(meV)
3	26.57	2.684	$\frac{2\pi }{3}$	34.3	8.67	45.8	−86.7	8.59	49.8

Using the above estimates, in [94] we estimated the baryon asymmetry of the Universe (BAU) for this model resulting from N₁ leptogenesis:

$$\begin{eqnarray}&&{Y}_{B}\approx 2.5\times {10}^{-11}\mathrm{sin}\eta \left[\displaystyle \frac{{M}_{1}}{{10}^{10}\;\mathrm{GeV}}\right].\end{eqnarray} \tag{ 107 }$$

Using η = 2π/3 and the observed value of Y_B fixes the lightest RH neutrino mass:

$$\begin{eqnarray}&&{M}_{1}\approx 3.9\times {10}^{10}\;\mathrm{GeV}.\end{eqnarray} \tag{ 108 }$$

Note that the phase η controls the BAU via leptogenesis in equation (107). The phase η also controls the entire PMNS matrix, including all the lepton mixing angles as well as all low energy ${\mathcal{C}}{\mathcal{P}}$ violation. The single phase η is the therefore the source of all ${\mathcal{C}}{\mathcal{P}}$ violation in this model, including both ${\mathcal{C}}{\mathcal{P}}$ violation in neutrino oscillations and in leptogenesis, providing a direct link between these two phenomena in this model. We not only have a correlation between the sign of the BAU and the sign of low energy leptonic ${\mathcal{C}}{\mathcal{P}}$ violation, but we actually know the value of the leptogenesis phase: it is η = 2π/3 which leads to the observed excess of matter over antimatter for M₁ ≈ 4.10¹⁰ GeV together with an observable neutrino oscillation phase ${\delta }_{\mathrm{CP}}\approx -\pi /2.$