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Sequential Dominance

We review the mechanism of sequential right-handed neutrino dominance proposed in the framework of the type I see-saw mechanism to account for bi-large neutrino mixing and a neutrino mass hierarchy in a natural way. We discuss how sequential dominance may also be applied to the right-handed charged leptons, which alternatively allows bi-large lepton mixing from the charged lepton sector. We review how such sequential dominance models may be upgraded to include type II see-saw contributions, resulting in a partially degenerate neutrino mass spectrum with bi-large lepton mixing arising from sequential dominance. We also summarise the model building applications and the phenomenological implications of sequential dominance.


Introduction
Neutrino masses and mixing angles must now be regarded as unavoidable consequences of the firmly established atmospheric and solar neutrino oscillation experiments [1]. A profound consequence of this is that the minimal Standard Model must necessarily be incomplete, and must be extended in some way to account for neutrino masses and mixings. The simplest way to do this appears to be to add right-handed neutrinos to the Standard Model. Since the right-handed neutrinos are gauge singlets electroweak symmetry does not prevent then from having large Majorana masses ranging from a few TeV up to the Planck scale. The right-handed neutrinos may also couple to lefthanded leptons via the usual Higgs doublets. The combination of very large right-handed neutrino Majorana masses, and weak scale Dirac masses from the Higgs couplings leads to suppressed left-handed Majorana neutrino masses which may be identified with the physical neutrino masses responsible for atmospheric and solar neutrino oscillations. This scenario, proposed some time ago in [2], is known as the see-saw mechanism.
Given the simplicity of the see-saw mechanism it has been widely applied to understanding the pattern of neutrino masses and mixings implied by the atmospheric and solar neutrino oscillation data [1]. Although there are alternatives to the see-saw mechanism involving large extra dimensions [3] or R-parity violating supersymmetry [4], we shall consider only the see-saw mechanism here, although we shall later consider a more complicated version of the see-saw mechanism called the type II see-saw mechanism which also involves heavy Higgs triplets (see e.g. [5]). We shall also only consider the case of three active neutrinos, which is the minimal case consistent with the confirmed atmospheric and solar neutrino oscillation data.
Within the framework as described above the goal of the see-saw mechanism is to account for large atmospheric neutrino mixing (θ 23 ∼ 45 • , close to maximal), and large solar neutrino mixing (θ 12 ∼ 30 • , but not maximal) together with the observed atmospheric and solar neutrino mass squared differences [1]. The solar data is consistent with the so called large mixing angle (LMA) MSW solution [6]. The third remaining mixing angle associated with the three active neutrinos is so far unmeasured but must be quite small (θ 13 < ∼ 13 • ) [7]. The neutrino oscillation data does not determine the absolute scale of neutrino masses, nor does it uniquely fix the ordering of neutrino masses, however in a normally ordered hierarchical scheme the neutrino mass values would be roughly given by m 3 ∼ 0.05 eV, m 2 ∼ 0.008 eV, with m 1 ≪ m 2 . However m 1 could in principle be substantially larger, up to the cosmological limit of about 0.23 eV [8].
It has frequently been observed that the simultaneous appearance of hierarchical neutrino masses and two large mixing angles is not natural in the see-saw mechanism. An important exception to this is the sequential dominance mechanism [9,10,11,12,13] (see also [14]) which is the subject of this focus. Sequential dominance is not in itself a model, but is a sub-mechanism within the general framework of the see-saw mechanism, that may be applied to constructing different classes of models. The starting point of sequential dominance is to assume that one of the right-handed neutrinos contributes dominantly in the see-saw mechanism to the heaviest neutrino mass, with the atmospheric mixing angle being determined by a simple ratio of two Yukawa couplings [9,10], which is sometimes referred to as single right-handed neutrino dominance. Sequential dominance corresponds to the further assumption that, together with single right-handed neutrino dominance, a second right-handed neutrino contributes dominantly to the second heaviest neutrino mass, with the large solar mixing angle interpreted as a ratio of Yukawa couplings [11,12]. The third right-handed neutrino is effectively decoupled from the see-saw mechanism, and plays no part in determining the neutrino mass spectrum, although it may play a cosmological role. If the decoupled right-handed neutrino is also the heaviest one then sequential dominance is effectively equivalent to having two right-handed neutrinos [10,11].
We also review how sequential dominance may be generalized to include the righthanded charged leptons [15], which allows bi-large charged lepton mixing consistent with a neutrino mass hierarchy. We then show how such sequential dominance models may be upgraded to include type II see-saw contributions [16], resulting in a partially degenerate neutrino mass spectrum with bi-large lepton mixing arising from sequential dominance.
In section 2 we recall the type I and type II see-saw mechanisms. Section 3 shows how the type I see-saw mechanism can lead to a hierarchical pattern of neutrino masses with bi-large neutrino mixing in a natural way using sequential right-handed neutrino dominance. Section 4 shows how the type I see-saw mechanism can lead to bi-large charged lepton mixing, with naturally small neutrino mixing and hierarchical neutrino masses, using sequential dominance in the right-handed lepton sector. Section 5 shows how a partially degenerate neutrino mass spectrum could originate from the type II seesaw mechanism, with the neutrino mass splittings and mixings controlled by sequential dominance. Section 6 briefly reviews some of the model building applications, while section 7 discusses the phenomenological implications of sequential dominance. Section 8 concludes the review. Our conventions are stated in the Appendix.

The See-Saw Mechanism
The see-saw mechanism provides a convincing explanation for the smallness of neutrino masses. In this section, we review its simplest form, the type I see-saw mechanism and its generalization to the type II see-saw mechanism.

Type I See-Saw
Before discussing the see-saw mechanism it is worth first reviewing the different types of neutrino mass that are possible. So far we have been assuming that neutrino masses are Majorana masses of the form where ν L is a left-handed neutrino field and ν C L is the CP conjugate of a left-handed neutrino field, in other words a right-handed antineutrino field. Such Majorana masses are possible to since both the neutrino and the antineutrino are electrically neutral and so Majorana masses are not forbidden by electric charge conservation. For this reason a Majorana mass for the electron would be strictly forbidden. However such Majorana neutrino masses violate lepton number conservation, and in the standard model, assuming only Higgs doublets are present, are forbidden at the renormalizable level by gauge invariance. The idea of the simplest version of the see-saw mechanism is to assume that such terms are zero to begin with, but are generated effectively, after right-handed neutrinos are introduced [2].
If we introduce right-handed neutrino fields then there are two sorts of additional neutrino mass terms that are possible. There are additional Majorana masses of the form where ν R is a right-handed neutrino field and ν C R is the CP conjugate of a right-handed neutrino field, in other words a left-handed antineutrino field. In addition there are Dirac masses of the form Such Dirac mass terms conserve lepton number, and are not forbidden by electric charge conservation even for the charged leptons and quarks. Once this is done then the types of neutrino mass discussed in Eq.2, 3 (but not Eq.1 since we do not assume direct mass terms, e.g. from Higgs triplets, at this stage) are permitted, and we have the mass matrix Figure 1: Diagram illustrating the type I see-saw mechanism.
Since the right-handed neutrinos are electroweak singlets the Majorana masses of the right-handed neutrinos M RR may be orders of magnitude larger than the electroweak scale. In the approximation that M RR ≫ m ν LR the matrix in Eq.4 may be diagonalized to yield effective Majorana masses of the type in Eq.1, The effective left-handed Majorana masses m ν LL are naturally suppressed by the heavy scale M RR . In a one family example, if we take m ν LR = M W and M RR = M GUT , then we find m ν LL ∼ 10 −3 eV which looks good for solar neutrinos. Atmospheric neutrino masses would require a right-handed neutrino with a mass below the GUT scale.
With three left-handed neutrinos and three right-handed neutrinos the Dirac masses m ν LR are a 3 × 3 (complex) matrix and the heavy Majorana masses M RR form a separate 3 × 3 (complex symmetric) matrix. The light effective Majorana masses m ν LL are also a 3 × 3 (complex symmetric) matrix and continue to be given from Eq.5 which is now interpreted as a matrix product. From a model building perspective the fundamental parameters which must be input into the see-saw mechanism are the Dirac mass matrix m ν LR and the heavy right-handed neutrino Majorana mass matrix M RR . The light effective left-handed Majorana mass matrix m ν LL arises as an output according to the see-saw formula in Eq.5.
The version of the see-saw mechanism discussed so far is sometimes called the type I see-saw mechanism. It is the simplest version of the see-saw mechanism, and can be thought of as resulting from integrating out heavy right-handed neutrinos to produce the effective dimension 5 neutrino mass operator where the dot indicates the SU(2) L -invariant product and with Y ν being the neutrino Yukawa couplings and m ν LR = Y ν v u with v u = H u . The type I see-saw mechanism is illustrated diagramatically in Fig. 1.

Type II See-Saw
In models with a left-right symmetric particle content like minimal left-right symmetric models, Pati-Salam models or grand unified theories (GUTs) based on SO(10), the type I see-saw mechanism is often generalized to a type II see-saw (see e.g. [5]), where an additional direct mass term m II LL for the light neutrinos is present. With such an additional direct mass term, the general neutrino mass matrix is given by Under the assumption that the mass eigenvalues M Ri of M RR are very large compared to the components of m II LL and m LR , the mass matrix can approximately be diagonalized yielding effective Majorana masses for the light neutrinos. The direct mass term m II LL can also provide a naturally small contribution to the light neutrino masses if it stems e.g. from a see-saw suppressed induced vev. We will refer to the general case, where both possibilities are allowed, as the II see-saw mechanism. Realizing the type II contribution by generating the dimension 5 operator in Eq.6 via the exchange of heavy Higgs triplets of SU(2) L is illustrated diagrammatically in Fig. 2.

Sequential Right-Handed Neutrino Dominance in the Type I See-Saw Mechanism
In this section we discuss an elegant and natural way of accounting for a neutrino mass hierarchy and two large mixing angles, called sequential dominance. The idea of sequential dominance is that one of the right-handed neutrinos contributes dominantly to the see-saw mechanism and determines the atmospheric neutrino mass and mixing. A second right-handed neutrino contributes sub-dominantly and determines the solar neutrino mass and mixing. The third right-handed neutrino is effectively decoupled from the see-saw mechanism.

Single Right-Handed Neutrino Dominance
Consider the case of full neutrino mass hierarchy m 3 ≫ m 2 ≫ m 1 ≈ 0. From Appendix A we see that in the diagonal charged lepton basis, ignoring phases, the neutrino mass matrix is given by: neglecting terms like m 2 θ 13 and setting θ 23 ≈ π/4. Clearly this expression reduces to with m = m 3 in the approximation that m 2 and θ 13 are neglected. However the more exact expression in Eq.11 shows that the required form of m LL should have a very definite detailed structure. The requirement m 2 ≪ m 3 implies that the sub-determinant of the mass matrix m ν LL is small: This requirement in Eq.13 is satisfied by Eq.11, as may be readily seen, and this condition must be reproduced in a natural way (without fine-tuning) by any successful theory. The goal of see-saw model building for hierarchical neutrino masses is therefore to choose input see-saw matrices m ν LR and M RR that will give rise to the form in Eq.11. We now show how the input see-saw matrices can be simply chosen to give this form, with the property of a naturally small sub-determinant in Eq.13 using a mechanism first suggested in [9]. 3 The idea was developed in [10] where it was called single right-handed neutrino dominance (SRHND) . SRHND was first successfully applied to the LMA MSW solution in [11].
To understand the basic idea of dominance, it is instructive to begin by discussing a simple 2 × 2 example, where we have in mind applying this to the atmospheric mixing in the 23 sector: The see-saw formula in Eq.
where the approximation in Eq.15 assumes that the right-handed neutrino of mass Y is sufficiently light that it dominates in the see-saw mechanism: The neutrino mass spectrum from Eq.15 then consists of one neutrino with mass m 3 ≈ (e 2 +f 2 )/Y and one naturally light neutrino m 2 ≪ m 3 , since the determinant of Eq.15 is clearly approximately vanishing, due to the dominance assumption [9]. The atmospheric angle from Eq.15 is tan θ 23 ≈ e/f [9] which can be large or maximal providing e ≈ f , even in the case e, f, b ≪ c that the neutrino Dirac mixing angles arising from Eq.14 are small. Thus two crucial features, namely a neutrino mass hierarchy m 2 3 ≫ m 2 2 and a large neutrino mixing angle tan θ 23 ≈ 1, can arise naturally from the see-saw mechanism assuming the dominance of a single right-handed neutrino. It was also realized that small perturbations from the sub-dominant right-handed neutrinos can then lead to a small solar neutrino mass splitting [9], as we now discuss.

Sequential Right-Handed Neutrino Dominance
In order to account for the solar and other mixing angles, we must generalize the above discussion to the 3 × 3 case. The SRHND mechanism is most simply described assuming three right-handed neutrinos in the basis where the right-handed neutrino mass matrix is diagonal although it can also be developed in other bases [10,11]. In this basis we write the input see-saw matrices as Each right-handed neutrino in the basis of Eq.17 couples to a particular column of m ν LR in Eq.18. There is no mass ordering of X, Y, Z implied in Eq.17. In [9] it was suggested that one of the right-handed neutrinos may dominate the contribution to m ν LL if it is lighter than the other right-handed neutrinos. The dominance condition was subsequently generalized to include other cases where the right-handed neutrino may be heavier than the other right-handed neutrinos but dominates due to its larger Dirac mass couplings [10]. In any case the dominant right-handed neutrino may be taken to be the one with mass Y without loss of generality.
It was subsequently shown how to account for the LMA MSW solution with a large solar angle [11] by careful consideration of the sub-dominant contributions. Sequential dominance occurs when the right-handed neutrinos dominate sequentially [11], which is the straightforward generalization of Eq.16 where x, y ∈ a, b, c and x ′ , y ′ ∈ p, q, r.
Assuming SRHND with sequential sub-dominance as in Eq.19, then Eq.5, 17, 18 give where the contribution from the right-handed neutrino of mass Z may be neglected according to Eq.19. If the couplings satisfy the sequential dominance condition in Eq.19 then the matrix in Eq.20 resembles the Type IA matrix, and furthermore has a naturally small sub-determinant as in Eq.13. This leads to a full neutrino mass hierarchy and, ignoring phases, the solar angle only depends on the sub-dominant couplings and is given by tan θ 12 ≈ a/(c 23 b − s 23 c) [11]. The simple requirement for large solar angle is then a ∼ b − c [11]. Including phases the neutrino masses are given to leading order in m 2 /m 3 by diagonalizing the mass matrix in Eq.20 using the analytic procedure described in Appendix D of [12]. In the case that d = 0, corresponding to a 11 texture zero in Eq.18, we have [12,13]: where s 12 = sin θ 12 is given below. Note that with SD each neutrino mass is generated by a separate right-handed neutrino, and the sequential dominance condition naturally results in a neutrino mass hierarchy m 1 ≪ m 2 ≪ m 3 . The neutrino mixing angles are given to leading order in m 2 /m 3 by [12,13]: where we have written some (but not all) complex Yukawa couplings as x = |x|e iφx . The phase δ is fixed to give a real angle θ 12 by, The phaseφ is fixed to give a real angle θ 13 bỹ Physically these results show that in sequential dominance the atmospheric neutrino mass m 3 and mixing θ 23 is determined by the couplings of the dominant right-handed neutrino of mass Y . The solar neutrino mass m 2 and mixing θ 12 is determined by the couplings of the sub-dominant right-handed neutrino of mass X. The third right-handed neutrino of mass Z is effectively decoupled from the see-saw mechanism and leads to the vanishingly small mass m 1 ≈ 0.

Types of Sequential Right-Handed Neutrino Dominance
Assuming sequential dominance, there is still an ambiguity regarding the mass ordering of the heavy Majorana right-handed neutrinos. So far we have assumed that the dominant right-handed neutrino of mass Y is dominant because it is the lightest one. We emphasize that this need not be the case. The neutrino of mass Y could be dominant even if it is the heaviest right-handed neutrino, providing its Yukawa couplings are strong enough to overcome its heaviness and satisfy the condition in Eq.19. In hierarchical mass matrix models, it is natural to order the right-handed neutrinos so that the heaviest right-handed neutrino is the third one, the intermediate right-handed neutrino is the second one, and the lightest right-handed neutrino is the first one. It is also natural to assume that the 33 Yukawa coupling is of order unity, due to the large top quark mass. It is therefore possible that the dominant right-handed neutrino is the heaviest (called heavy sequential dominance or HSD), the lightest (called light sequential dominance or LSD), or the intermediate one (called intermediate sequential dominance  or ISD). This leads to the six possible types of sequential dominance corresponding to the six possible mass orderings of the right-handed neutrinos as shown in Table 1. In each case the dominant right-handed neutrino is the one with mass Y , and the leading sub-dominant right-handed neutrino is the one with mass X. The resulting see-saw matrix m ν LL is invariant under re-orderings of the right-handed neutrino columns, but the leading order form of the neutrino Yukawa matrix Y ν is not. It is worth emphasizing that since all the forms above give the same light effective see-saw neutrino matrix m ν LL in Eq.20, under the sequential dominance assumption in Eq.19, this implies that the analytic results for neutrino masses and mixing angles applies to all of these forms. They are distinguished theoretically by different preferred leading order forms of the neutrino Yukawa matrix Y ν shown in the table. These leading order forms follow from the the large mixing angle requirements e ∼ f and a ∼ b − c. 4 Thus we see that LSDa, and ISDa are consistent with a form of Yukawa matrix with small Dirac mixing angles, while HSDa and HSDb correspond to the so called "lop-sided" forms.

Sequential Right-Handed Lepton Dominance in the Type I See-Saw Mechanism
In this section we show how bi-large mixing could originate from the charged lepton sector using a generalization of sequential right-handed neutrino dominance [11,12] to all right-handed leptons [15]. We write the mass matrices for the charged leptons m E as In our notation, each right-handed charged lepton couples to a column in m E . For the charged leptons, the sequential dominance conditions are [15]: They imply the desired hierarchy for the charged lepton masses m τ ≫ m µ ≫ m e and small right-handed mixing of U e R . We assume zero mixing from the neutrino sector which corresponds to the MNS matrix being given by U MNS = U e L · diag (1, e iβ ν 2 , e iβ ν 3 ) in the conventions in Appendix A. A natural possibility for obtaining a small θ 13 is [15] |d ′ |, |e ′ | ≪ |f ′ | .
In leading order in |d ′ |/|f ′ | and |e ′ |/|f ′ |, for the mixing angles θ 12 , θ 23 and θ 13 , we obtain where the Dirac CP phase δ is determined such that θ 13 is real, which requires Given tan(δ), δ has to be chosen such that tan(θ 13 ) ≥ 0 in order to match with the usual convention θ 13 ≥ 0. The phases β e 2 and β e 3 from the charge lepton sector are given by Note that in the case that the neutrino sector induces Majorana phases, the total Majorana phases β 2 and β 3 of the MNS matrix are given by θ 13 only depends on d ′ /f ′ and e ′ /f ′ from the Yukawa couplings to the sub-dominant right-handed muon and on θ 12 . We find that in the limit |d ′ |, |e ′ | ≪ |f ′ |, the two large mixing angles θ 12 and θ 23 approximately depend only on a ′ /c ′ and b ′ /c ′ from the righthanded tau Yukawa couplings. Both mixing angles are large if a ′ , b ′ and c ′ are of the same order.
In addition to achieving bi-large mixing from the charged lepton sector, we also require now small mixing from the neutrino sector. Usually, sequential RHND [11,12] is viewed as a framework for generating large solar mixing θ 12 and large atmospheric mixing θ 23 in the neutrino mass matrix. However, given sequential dominance in the neutrino sector in Eq.19 which guarantees a neutrino mass hierarchy, one can easily find the conditions for small mixing from the neutrinos as well from Eq.25,26. Using the notation of subsection 3.2, we need d, e ≪ f and a ≪ b, c. Small mixing from the neutrino sector thus requires three small entries in m ν LR . As shown in [16], three zero entries in Y ν might stem from a spontaneously broken SO(3) flavour symmetry and real vacuum alignment. Other realizations might by found via Abelian or discrete symmetries.

Type II See-Saw Upgrade
In type I see-saw models, it seems to be difficult to obtain a partially degenerate or quasi-degenerate neutrino mass spectrum in a natural way, whereas hierarchical masses seem to be natural. The direct mass term in type II models on the other hand has the potential to provide a natural way for generating neutrino masses with a partial degeneracy. In this section we show that it is possible to obtain a partially degenerate neutrino mass spectrum by essentially adding a type II direct neutrino mass contribution proportional to the unit matrix. In this case the neutrino mass scale is controlled by the type II direct mass term, while the neutrino mass splittings (which are generally now much smaller) and mixings continue to be determined by the type I see-saw matrix using sequential dominance as described earlier.
Thus we shall consider a type II upgrade [16], where the mass matrix of the light neutrinos in Eq.9 has the particular form in the basis where the mass matrix M RR of the heavy right-handed neutrinos is diagonal.
To understand the effect of the type II contribution m II LL , we consider the diagonalization of m I LL by a unitary transformation (m I LL ) diag = V m I LL V T . If we assume for the moment that the type I see-saw mass matrix m I LL is real, which implies that V is an orthogonal matrix, we obtain The additional direct mass term leaves the predictions for the mixings from the type I see-saw contribution unchanged in this case. This allows to transform many type I see-saw models for hierarchical neutrino masses into type II see-saw models for partially degenerate or quasi-degenerate neutrino masses while maintaining the predictions for the mixing angles. Obviously, in the general complex case, it is no longer that simple since for a unitary matrix V V T = ½ and the phases will have impact on the predictions for the mixings. However, as we will see below, with sequential right-handed neutrino dominance [11,12] for the type I contribution to the neutrino mass matrix, and a particular phase structure, the known techniques and mechanisms for explaining the bi-large lepton mixings can be directly applied also in the presence of CP phases.

Type II Upgrade of a ISD Model
As an example of a type II model where the bi-large lepton mixing stems from the neutrino mass matrix, we now consider explicitly the model A1 of table 4 in [16] with sequential right-handed neutrino dominance [11,12] for the type I part m I ν of the neutrino mass matrix. The leading order Dirac mass matrices are where here a, b, c, e, f, r and a ′ , b ′ , c ′ , e ′ , f ′ , r ′ are real. M RR and the type II contribution m II LL are given by denoting the mass of the dominant right-handed neutrino by Y and the mass of the sub-dominant one by X. The sequential RHND condition we impose is then The leading order type II neutrino mass matrix is given by the type II see-saw formula of Eq.9. The masses of the charged leptons are given by m τ = r ′ , m µ = e ′ and m e = a ′ . In addition we note that the mixings θ e 12 , θ e 13 and θ e 23 , which stem from U e L and could contribute to the MNS matrix, are very small. Furthermore, in leading order each column of M e has a common complex phase, which can be absorbed by U e R . Therefore, the charged leptons do not influence the leptonic CP phases in this approximation.
Using the analytical methods for diagonalizing neutrino mass matrices with small θ 13 derived in [12], from m ν LL = m II LL + m I LL we find for the mixing angles and with δ defined by Given tan( δ), δ has to be chosen such that sin(θ 23 )|b| + cos(θ 23 )|c| sign (b c e f ) This does not effect θ 13 , which we have defined to be ≥ 0, however it is relevant for extracting the Dirac CP phase δ, given by with P being defined by The mass eigenvalues of the complete type II neutrino mass matrix are given by and, for m II = 0, the Majorana phases β 2 and β 3 can be extracted by In the classes of type II see-saw models with sequential right-handed neutrino dominance for the type I contribution to the neutrino mass matrix and real vacuum alignment leading to the phase structure of the Yukawa matrices as in Eq.40, the solar and the atmospheric neutrino mixings θ 12 and θ 23 are independent of the type II mass scale m II and of the complex phases of the neutrino Yukawa matrix. We can thus upgrade these types of models continously from hierarchical neutrino mass spectra to partially degenerate ones, while maintaining the predictions for the two large lepton mixings.

Model Building Applications of Sequential Dominance
We have seen that sequential dominance is not a model, but is a general sub-mechanism within the see-saw mechanism. Sequential dominance may be used to obtain hierarchical type I neutrino masses, together with bi-large lepton mixing, in a completely natural way, overcoming the usual naturalness objection to the see-saw mechanism in this case.
We have also seen that sequential dominance may be extended to the case where the lepton mixing arises from the charged lepton sector. Furthermore we have seen that the sequential dominance mechanism is also useful within the framework of the type II see-saw mechanism in the case that the additional type II mass contributions are proportional to the unit matrix. Despite the successes of sequential dominance, the conditions on which it is based have just been stated without any explanation. It also remains to be seen how the mechanism of sequential dominance can be used to construct realistic unified models of flavour.
In this section we discuss some of the model building applications of sequential dominance. We shall see that the use of sequential dominance is ideally suited to GUTs and family symmetry models, has already been used in quite a number of works of this nature. Sequential dominance also makes contact with studies based on two righthanded neutrinos. Finally there have been some interesting cosmological applications that have recently been proposed. Given the simplicity and naturalness of sequential dominance, it is reasonable to expect that it will continue to be exploited increasingly in the future.

Effective Two Right-Handed Neutrino Models
In sequential dominance we have seen that one of the right-handed neutrinos effectively decouples from the see-saw mechanism. Without loss of generality we have denoted the mass of this decoupled right-handed neutrino as Z. From Table 1 we see that the decoupled right-handed neutrino of mass Z may be the lightest, the heaviest of the intermediate mass right-handed neutrino. If it is the lightest or the second lightest then it could in principle play an important part in leptogenesis or inflation and so have cosmological relevance even though it is decoupled from the see-saw mechanism. However if it is the heaviest right-handed neutrino, as in LSDa or ISDa in Table 1, then it would be expected to play no part in phenomenology. In these cases, the heaviest neutrino of mass Z is completely decoupled from physics, and sequential dominance reduces to effectively two right-handed neutrino models, as pointed out in [11,13]. Recently there have been several studies based on the "minimal see-saw" involving two right-handed neutrinos [17], and it is worth bearing in mind that such models could naturally arise as the limiting case of sequential dominance.

GUT and Family Symmetry Models
There are many models in the literature based on single right-handed neutrino dominance or sequential dominance. For example explicit realisations of the small determinant condition of Altarelli and Feruglio implicitly involve single right-handed neutrino dominance, or sequential dominance, together with U(1) family symmetry and SU (5) GUTs [18]. An example of sequential dominance of the HSD type in Pati-Salam models with U(1) family symmetry was considered in [19]. Single right-handed neutrino dominance has also been applied to SO(10) GUT models involving a U(2) family symmetry [20]. Sequential dominance of the LSD type with SU(3) family symmetry and SO (10) GUTs has been considered in [21]. Type II up-gradable models based on sequential dominance of the ISD type with SO(3) family symmetry have been considered in [15,16]. This list is not exhaustive, but represents a subset of models based on single or sequential right-handed neutrino dominance. The main point is that sequential dominance can readily be included in a wide range GUT and family symmetry models, and it enhances the naturalness of such models.

Sneutrino Inflation Models
Sequential dominance has recently also been applied to sneutrino inflation [22]. Requiring a low reheat temperature after inflation, in order to solve the gravitino problem, forces the sneutrino inflaton to couple very weakly to ordinary matter and its superpartner almost to decouple from the see-saw mechanism. This decoupling of a right-handed neutrino from the see-saw mechanism is a characteristic of sequential dominance.

Phenomenological Implications of Sequential Dominance
We now review phenomenological consequences of type I see-saw models with sequential dominance and their type II upgrades for the low energy neutrino parameters and highenergy mechanisms as leptogenesis and minimal lepton flavour violation (LFV). In order to compare the predictions of see-saw models based on sequential dominance with the experimental data obtained at low energy, the renormalization group (RG) running of the effective neutrino mass matrix has to be taken into account.

Renormalization Group Corrections
For type I models with sequential dominance, the running of the mixing angles is generically small [23] since the mass scheme is strongly hierarchical. When the neutrino mass scale is lifted, e.g. via a type II upgrade, a careful treatment of the RG running of the neutrino parameters, including the energy ranges between and above the see-saw scale [23,24], is required. For convenient estimates of the running below the see-saw scales, the approximate analytical formulae for the running of the parameters [25] can be used. Dependent on tan β in the MSSM, on the size of the neutrino Yukawa couplings and on the neutrino mass scale, the RG effects can be sizable or cause only small corrections. 5

Dirac and Majorana CP Phases and Neutrinoless Double Beta Decay
At present, the CP phases in the lepton sector are unconstraint by experiment. In type I see-saw models based on sequential dominance, there is no restriction on them from a theoretical point of view. The type-II-upgrade scenario however predicts that all observable CP phases, i.e. the Dirac CP phase δ relevant for neutrino oscillations and the Majorana CP phases β 2 and β 3 , become small as the neutrino mass scale increases.
The key process for measuring the neutrino mass scale could be neutrinoless double beta decay. The decay rates depend on an effective Majorana mass defined by m ν = | i (U MNS ) 2 1i m i |. Future experiments which are under consideration at present might increase the sensitivity to m ν by more than an order of magnitude. For type I models with sequential dominance, which have a hierarchical mass scheme, m ν can be very small, below the accessible sensitivity.
For models where the neutrino mass scale is lifted via a type II upgrade [16], there is a close relation between the neutrino mass scale, i.e. the mass of the lightest neutrino and m ν . Since the CP phases are small, there can be no significant cancellations in m ν . This implies that the effective mass for neutrinoless double beta decay is approximately equal to the neutrino mass scale m ν ≈ m II and therefore neutrinoless double beta decay will be observable in the next round of experiments if the neutrino mass spectrum is partially degenerate.

Theoretical expectations for the Mixing Angles
In order to discriminate between models, precision measurements of the neutrino mixing angles have the potential to play an important role.
One important parameter is the value of the mixing angle θ 13 , which is at present only bounded from above to be smaller than approximately 13 • . In the type I sequential dominance case, the mixing angle θ 13 is typically of the order O(m I 2 /m I 3 ). In the type-IIupgrade scenario this ratio decreases with increasing neutrino mass scale and is smaller than ≈ 5 • for partially degenerate neutrinos even if it was quite large in the type I limit. Sizable RG corrections, which are usually expected for partially degenerate neutrinos, are suppressed in the type-II-upgrade scenario due to small CP phases β 2 , β 3 and δ [25].
Another important parameter is θ 23 . Its present best-fit value is close to 45 • , however comparably large deviations are experimentally allowed as well. With sequential dominance, we expect minimal deviations of θ 23 from 45 • of the order O(m I 2 /m I 3 ), which could be observed by future long-baseline experiments in the type I see-saw case. 6 In the type II upgraded version, the corrections can be significantly smaller since the ratio m I 2 /m I 3 decreases with increasing neutrino mass scale [16]. For large tan β in the MSSM, the major source for the corrections can be RG effects [25], which are un-suppressed for small CP phases.

Minimal Lepton Flavour Violation
At leading order in a mass insertion approximation the branching fractions of LFV processes are given by 7 where l 1 = e, l 2 = µ, l 3 = τ , and where the off-diagonal slepton doublet mass squared is given in the leading log approximation (LLA) by With sequential dominance, using the notation of Eqs.17,18, the leading log coefficients relevant for µ → eγ and τ → µγ are given approximately as From Table 1 and Eq.53 it can be seen which types of SD will lead to large rates for µ → eγ and τ → µγ. For example the results for HSD show a large rate for τ → µγ which is the characteristic expectation of lop-sided models in general [27] and HSD in particular. A global analysis of LFV has been performed in the constrained minimal supersymmetric standard model (CMSSM) for the case of sequential dominance, focussing on the two cases of HSDa and LSDa [28]. The results in [28] are based on an exact calculation, and the error incurred compared to the LLA study [29] can be as much as 100%. For LSDa τ → µγ is well below observable values. Therefore τ → µγ provides a good discriminator between the HSDa and LSDa types of dominance. In [28] it is shown that the rate for µ → eγ may determine the order of the sub-dominant neutrino Yukawa couplings in the flavour basis.

Leptogenesis
Leptogenesis and lepton flavour violation are important indicators which can help to resolve the ambiguity of right-handed neutrino masses in Table 1. In the LSD and HSD cases of sequential dominance leptogenesis has been studied with some interesting results [30]. In general successful leptogenesis for such models requires the mass of the lightest right-handed neutrino to be quite high, and generally to exceed the gravitino constraints if supersymmetry is assumed. However, putting this to one side for the moment, interesting links between the phase relevant for leptogenesis and the phase δ measurable in neutrino oscillation experiments have been made. The precise link depends on how many "texture" zeroes are assumed to be present in the neutrino Dirac mass matrix. For example if two texture zeroes are assumed then there is a direct link between δ and the leptogenesis phase, with the sign of δ being predicted from the fact that we are made of matter rather than antimatter. On the other hand if only the physically motivated texture zero in the 11 entry of the Dirac mass matrix is assumed, then the link is more indirect [13]. More generally in three right-handed neutrino models with sequential dominance, if the dominant right-handed neutrino is the lightest one (LSD) then the washout parameterm 1 ∼ O(m 3 ), which is rather too large compared to the optimal value of around 10 −3 eV, while if the dominant right-handed neutrino is either the intermediate one or the heaviest one then one findsm 1 ∼ O(m 2 ) or arbitrarym 1 , which can be closer to the desired value [30].

Discussion and Conclusions
Neutrino masses and mixings are now established experimental phenomena which must be included in some extended version of the Standard Model. The simplest mechanism for describing small neutrino masses is the see-saw mechanism, however the simultaneous appearance of hierarchical neutrino masses and two large mixing angles is not natural in the see-saw mechanism. The simplest solution to this difficulty is to assume sequential dominance which has been the subject of this review.
We have reviewed the mechanism of sequential right-handed neutrino dominance which was proposed in the framework of the type I see-saw mechanism to account for bi-large neutrino mixing and a neutrino mass hierarchy in a natural way. We have discussed how sequential dominance may also be applied to the right-handed charged leptons, which alternatively allows bi-large lepton mixing in the charged lepton sector. We reviewed how such sequential dominance models may be upgraded to include type II see-saw contributions, resulting in a partially degenerate neutrino mass spectrum with bi-large lepton mixing arising from sequential dominance. We also saw that the use of sequential dominance is ideally suited to GUTs and family symmetry models, and mentioned some examples of such models. We also pointed out the interesting case where sequential dominance reduces effectively to the case of two right-handed neutrinos, and mentioned some interesting cosmological applications that have recently been proposed such as sneutrino inflation.
We also reviewed some phenomenological consequences of type I see-saw models with sequential dominance and their type II upgrades for the low energy neutrino parameters and high-energy mechanisms as leptogenesis and minimal lepton flavour violation, both of which can be probes of different types of sequential dominance. While RG effects are expected to be quite small for type I sequential dominance, they become increasingly important for the type II upgrade sequential dominance as the neutrino mass scale increases. We noted that neutrinoless double beta decay is practically unobservable in type I sequential dominance, but may well be observed in the next round of experiments in the type II upgrade sequential models if the neutrino masses are partially degenerate. We saw that both θ 13 and the correction to θ 23 are controlled by the ratio m I 2 /m I 3 which decreases with increasing neutrino mass scale, with interesting consequences.
Given the simplicity and naturalness of sequential dominance, we expect it to continue to be used and exploited ubiquitously in the future.

Acknowledgements
We acknowledge support from the PPARC grant PPA/G/O/2002/00468.

A Our Conventions
The MNS matrix is then given by We use the parameterization U MNS = R 23 U 13 R 12 P 0 with R 23 , U 13 , R 12 and P 0 being defined as and where s ij and c ij stand for sin(θ ij ) and cos(θ ij ), respectively. The matrix P 0 contains the possible Majorana phases β 2 and β 3 . δ is the Dirac CP phase relevant for neutrino oscillations.