Neutrino properties from Yukawa structure

We discuss the implications for lepton mixing and CP violation of structure in the lepton mass matrices, for the case that neutrino masses are generated by the see-saw mechanism with an hierarchical structure for the Majorana masses. For a particularly interesting case with enhanced symmetry in which the lepton Dirac mass matrices are related to those in the quark sector, the CHOOZ angle is near the present limit and the CP violating phase relevant to thermal leptogenesis and to $\nu 0 \beta \beta $ decay is near maximal.


Introduction
The origin of the structure observed in quark and lepton masses and mixing angles remains one of the most pressing and interesting questions left unanswered by the Standard Model. The continuing improvement in the measurement of the CKM and MNS matrix elements and the neutrino masses has stimulated a renewed theoretical effort to answer these questions.
In the case of quarks one proposed structure going beyond the Standard Model has proved to be robust, giving a quantitatively accurate prediction for the Cabbibo angle (strictly V CKM

12
). It follows from the postulate that the up and down quark mass matrices have a simultaneous "texture" zero in the (1, 1) position 1 and that the magnitude of the matrix elements are symmetric for the first two generations [1]. The measured masses and mixing angles are consistent with additional texture zeros [2], although this may require a departure from the symmetric form of the mass matrices [3]. One reason for the interest in texture zeros is that they may indicate the presence of a new family symmetry which require certain matrix elements be anomalously small. Thus identification of texture zeros may be an important step in unravelling the origin of the fermion masses and mixings.
In this paper we extend the analysis of possible texture zeros to the lepton sector for the case that neutrino masses are given by the see-saw mechanism [4]. In analogy with the quark case we consider the predictions resulting from a symmetric form for the magnitudes of the Dirac mass matrix elements together with texture zeros. Of particular interest is the case of simultaneous zeros in the (1, 1) position. If this proves to be the case it would be a strong indication of a symmetry between the up and the down quarks and the charged lepton and neutrino sectors respectively. For the case that the Majorana mass matrix does not contribute significantly to lepton mixing we obtain predictions for the CHOOZ mixing angle and for the CP violating phases. If the neutrino Majorana mass does contribute significantly to mixing these predictions may be viewed as indicative to the magnitude of these parameters barring what would seem to be an unnatural cancellation between the contribution of the Dirac and Majorana sectors. We also consider the implications further restrictions on the form of the lepton mass matrices. The analysis is done in the context that the mass of one of the Majorana neutrinos is anomalously large [5]. This case includes the possibilities that there is sequential right hand neutrino dominance [6] that offers an attractive way of explaining near bi-maximal neutrino mixing in the case that the quark and neutrino Dirac mass matrices are related [7], [8].
The paper is organised as follows. In Section 2 we review a general parameterisation for the effective light neutrino masses for the case of the see-saw mechanism that is useful in studying the implications of texture zeros. We discuss the constraints on this parameterisation coming from texture zeros, from a symmetric form of the magnitudes of the mass matrix elements and from the case that one of the Majorana neutrinos is anomalously large. In Section 4 we apply this parameterisation to derive general constraints on neutrino mixing and CP violation and consider the implications for leptogenesis. Section 5 summarizes the results.

Parameterisation of the see-saw mechanism
We consider the case of three generations of left-handed SU(2) doublet neutrinos, ν L,i, and three generations of right-handed Standard Model singlet neutrinos, ν R,i . The Lagrangian responsible for lepton masses has the form where Y D ν , Y D l are the matrices of Yukawa couplings which give rise to the neutrino and charged lepton Dirac mass matrices respectively and M M ν is the neutrino Majorana mass matrix. We are interested in studying the implications of simultaneous zeros in Y D ν and Y D l for observable quantities, masses and mixing angles and CP violating phases. For the case of quarks and charged leptons it is easy to do this because the Yukawa couplings are directly related to the mass matrices. For neutrinos, however, the existence of the Majorana masses complicates the connection between the Dirac Yukawa couplings and the neutrino observables. The light neutrino mass matrix, M, is given by the see-saw form Sometimes it is convenient to use an alternative form for the see-saw formula, expressing Y D ν in terms of the neutrino mass eigenvalues, mixing angles and CP violation [9]. In the basis in which the Majorana mass matrix, M M ν , is diagonal the parameterisation has the form where D √ M is the diagonal matrix of the square roots of the eigenvalues of M M ν , D √ m is the diagonal matrix of the roots of the physical masses, m i , of the light neutrinos, W is the neutrino mixing matrix, and R is an orthogonal matrix which parameterises the residual freedom in Y D ν once the other parameters are fixed. It is parameterised by 3 complex "mixing" angles 2 .
sin θ 2 sin θ 3 cos θ 1 cos θ 3 + sin θ 1 cos θ 2 sin θ 3 sin θ 1 cos θ 3 − cos θ 1 sin θ 3 sin θ 2 cos θ 3 − cos θ 1 sin θ 3 + sin θ 1 cos cos θ 3 − sin θ 1 sin θ 3 − cos cos θ 3 cos θ 2 sin θ 1 sin θ 2 − cos θ 1 sin θ 2 where θ 1 , θ 2 , θ 3 are arbitrary complex angles. These, together with the three Majorana masses, the three light neutrino masses, the three mixing angles and three phases of W make up the eighteen real parameters needed to specify Y D ν . With this form it is straightforward to study the implications of a zero in Y D ν for the physical measureables. In our study of texture zeros we will be interested in simultaneous texture zeros in Y D ν and Y D l . Of course this is basis dependent as a zero in one basis will not in general remain zero after a rotation. In this sense the appearance of simultaneous texture zeros specifies the "texture zero" basis. The idea is that there is some dynamical reason, such as a family symmetry, which generates the texture zero structure. For the case of a family symmetry the "texture zero" basis is just the current quark basis, defined as the one in which the fermion states are eigenstates of the family symmetry group. In the phenomenological analysis of texture zeros this basis choice is taken into account by modifying the parameterisation so that the charged lepton mass matrix is not diagonal. In this case it is the combination U † l W that should be identified with the MNS matrix, where U l is the unitary matrix needed to diagonalise the charged lepton mass matrix, starting from the texture basis.
It is instructive to determine how many free parameters are left in R when Y D ν is constrained in various ways. If any element of Y D ν is zero, there is a reduction of two complex parameters needed to specify Y D ν and a corresponding reduction of the parameters in R. For more than 3 texture zeros there will be relations between measureable quantities 3 . However, depending on the position of the texture zero, there may be predictions for fewer texture zeros.
For the case that Y D ν is symmetric (or hermitian or has off diagonal elements antisymmetric) the number of real parameters needed to specify it are reduced to 12, so in this case R is completely determined. This does not lead to any relations between measurable quantities but if, in addition, there is a texture zero there will be such relations (this is the analogue to the GST relation in the quark sector).
For the case one of the Majorana masses, M M ν,3 , is anomalously heavy the Standard Model singlet component, ν R,3 ,does not play a role in determining the two heaviest of the light neutrino eigenstates. Following from eq(2) we see that in this case the couplings Y D ν 3j , j = 1..3 do not contribute to the light masses and mixing angles. There is also a reduction in the number of parameters needed to specify R.Following from the condition that Y D ν W is finite as M M ν,3 → ∞, we see that in this limit R 3j ∝ 1/M M ν,3 , j = 2, 3 and R ij ≤ O(1), i, j = 1..3 . Inserting these constraints in eq(4) we find the form of R is given by This ± refer to a reflection ambiguity. In practice we can work with the positive sign only and absorb this ambiguity in the unknown phases specified below. The Yukawa couplings Y D ν ij , i = 1, 2, j = 1..3 are thus given in terms of z alone in the limit M M ν,3 → ∞. If we require the (1, 2) block be symmetric, antisymmetric or hermitian, z will be determined and for 1 texture zero there will be relations between measureables. Alternatively more than 1 texture zero will give relations even if the (1, 2) and (2, 1) matrix elements are not related.

The charged lepton mass matrix
The MNS matrix is given by U † l W and has a contribution coming from the matrix U l which diagonalises the charged lepton mass matrix. The latter has to reproduce the hierarchical structure of lepton masses and this may place constraints on the magnitude of the charged lepton mixing angles. Let us consider the case the lepton mass matrix is symmetric and that, like the quarks, the hierarchy of lepton masses is due to an hierarchical structure in the matrix elements and not due to a cancellation between different contributions. This is what is expected if there is an underlying Grand Unified symmetry relating leptons to quarks. Moreover a cancellation between different contributions to lepton masses seems very difficult to reconcile with an underlying family symmetry as it requires non-trivial relations between different matrix elements which are difficult to arrange even in the context of non-Abelian family symmetry. With this constraint it is easy to limit (U l ) 23 , because (M l ) 2 23 ≤ m µ m τ , giving Similarly one obtains a bound on (U l ) 12 from the constraint that (M l ) 2 12 < m e m µ which follows from taking the deteminant of the mass matrix. This in turn implies with equality occurring if there is a texture zero in the (1, 1) position. The constraint on (M l ) 2 12 also leads to the constraint |(U l ) 13 , which occurs when there is a texture zero in the (2, 2) position, we have the bound |(U l ) 13 | ≤ me mτ . If, however, |(U l ) 23 | ≪ mµ mτ we have (M l ) 22 = m µ and then from the determinant we have (M l ) 2 13 ≤ m e m τ which again gives In practice the magnitudes of (U l ) 23 and (U l ) 13 are so small that they do not affect the mixing coming from the neutrino sector. However (U l ) 12 close to the upper bound given in eq (6) does give a significant contribution to the CHOOZ angle. Its effect is considered below. The discussion above relies on a symmetric structure relating the magnitudes of the charged lepton mass matrix elements. If we relax this condition there is no constraint on the magnitude of the matrix elements of U l . In this case the contributions to the MNS matrix coming from the neutrino sector should be considered as an indication of the lower bound on the MNS matrix elements, assuming there is no delicate cancellation between the contributions of U l and W.
We turn now to a determination of the relations that follow for various form of the Yukawa couplings. We first consider in detail how the analysis proceeds for the case the texture zero is in the (1, 1) position and both Y D ν and Y D l are symmetric. In the analogous case in the quark sector a (1, 1) texture zero leads to the remarkably successful GST relation [1], so this case is particularly interesting for, if it leads to a phenomenologically realistic prediction, it may indicate a connection betweeen quarks and leptons.
As discussed above we are interested in the case M 1,2 /M 3 ≪ m 2 /m 3 and the Majorana mass matrix, M M ν is diagonal and real. We include the CP violating phases in U M N S , i.e.we write it in the form U = V.diag(e −iφ/2 , e −iφ ′ /2 , 1) where φ and φ ′ are the CP violating phases and V has the form of the CKM matrix. In this case a symmetric structure in the Dirac neutrino mass matrices and a texture zero will lead to a relation between measurable parameters. Following from eq(3) the condition Y D ν 11 = 0 gives 4 where W is the matrix acting on the left-handed neutrino states needed to diagonalise the Dirac neutrino mass matrix. To express this in terms of U M N S we use the constraints of eqs (6,7,8) to determine W. There is a residual phase ambiguity because the basis in which the MNS matrix has the standard form can be different from the "symmetry" basis in which the texture zero appears. This corresponds to the simultaneous redefininition of the phase of the left-and righthanded states such that the Dirac structure is invariant (the change in the Majorana matrix is absorbed in a redefinition of φ and φ ′ in eq (9)). With this we have W = U l P U M N S where P = diag(e iα 1 , e iα 2 , e iα 3 ). From the symmetric constraint Y D ν 12 = Y D ν 21 one obtains Substituting for tan z leads to the relation where det W =e iβ . We choose the phases of the right handed charged leptons such that U l is real in the (1, 2) block. Then in leading order we have W ij ≃ e iα i U ij except for where we have written U M N S = U. In eq(12) we have dropped terms involving the roots of ratios of lepton masses relative to unity. Using eq(12) in eq (11) gives For the case of a (1, 1) texture zero in Y D l 11 we have (U l ) 12 = me mµ . For the case of a texture zero in Y D l 12 , (U l ) 12 = 0. Other possibilities for a lepton texture zero or no texture zero at all give (U l ) 12 ≤ me mµ . The implications of eq(13) for the CHOOZ angle are shown in Fig(1) for the case (U l ) 12 = 0 and me mµ respectively 5 . In these plots we have chosen a random distribution of the unknown phases β, α i . One may see there is a clustering of values within a small range with the CHOOZ angle near the current bound, sin θ 13 < 0.24 at 3σ. This implies that, barring an unnatural cancellation between terms, we expect a large CHOOZ angle, in the range that would make the long baseline neutrino factory searches for CP violation feasible. To quantify this we have determined the range of the CHOOZ angle which includes 95% of the points, giving sin θ 13 > 0.1 over the whole range of M 1 /M 2 .
Finally we determine the implications of our results for thermal leptogenesis, assuming that the lightest Majorana state dominates [12]. In this case the asymmetry is given by Note that ǫ depends only on tan z. The dependence of tan z on low energy phases may be read from eq (10) showing which combination is relevant for leptogenesis. The magnitude of ǫ/ǫ max is plotted in Fig(3). Note that, if we ignore the charged lepton contribution coming from a nontrivial U l , a (1, 1) texture zero with an hierarchical Majorana mass spectrum gives the same value for the CP violating phase in double beta decay as the CP violating phase determining the lepton asymmetry in leptogenesis [14]. This explains the correlation seen between the plots of Figs(3), although note that in Figs(3b) a significant charged lepton contribution has been added.  Whether this asymmetry can lead to the observed baryon asymmetry depends on the subsequent washout. This is characterised by the parameter m 1 [15]. It is given by For the case of a (1,1) texture zero the value of m 1 is given in Fig(4). In the whole region of parameter space m 1 ≫ m 2 and so the washout will reduce the baryon asymmetry below the observed value unless M 1 is very large [16]. In the case of SUGRA this implies a reheat temperature above the gravitino abundance bound implying that in this case thermal leptogenesis cannot work. However in other supersymmetry breaking mediation scenarios, such as gauge mediation, the gravitino is much lighter and a heavier M 1 is consistent with the gravitino bound.   It is straightforward to apply the analysis just discussed to the other possible positions for a single texture zero in the Dirac neutrino matrix. The results are presented in Table 1. Note that, unlike the case for a (1, 1) texture zero, the prediction for tan z is in terms of the measured large MNS matrix elements. As a result one obtains a definite prediction for leptogenesis which is also given in the Table. For the case of (1, 2) and (1, 3) texture zeros we see that tan z is suppressed by m 2 m 3 which leads to a near maximal form for ǫ ǫmax . The bound on m 1 is only mildly stronger than the absolute bound m 1 ≥ m 2 , so the washout effects are expected to be less efficient than in the (1, 1) texture zero case. For the case of the (2, 2) and (2, 3) texture zeros tan z is enhanced by m 3 m 2 which leads to a m 2 m 3 suppression in ǫ ǫmax . The bound on m 1 in this case is comparable to the one for a (1, 1) texture zero but is independent of M 1 /M 2 . As a result baryogenesis through thermal leptogenesis will not proceed in these cases either.
For the case of the (1, 2) texture zero the prediction for the CHOOZ angle depends only on unknown phases with the distribution is shown in Fig(5). For a (1,3) texture zero the CHOOZ angle also depends on the ratio M 1 /M 2 as in the previous cases. This is plotted in Fig(6). In both cases θ 13 is predicted to be large, although the 95% lower range is smaller than that found for the (1, 1) texture zero case.  Table 1: The constraints following from a symmetric mass matrix and a single texture zero . χ is 1 for a (1,1) texture zero in the charged lepton sector and 0 for a (1,2) texture zero. If there is no lepton texture zero χ lies between these limiting cases. c 12 is cos(θ 12 ).
For the case of the (2, 2) and (2, 3) texture zeros one obtains a relation between the large elements of the MNS matrix. From this one may extract a relation between the phases and a prediction for M 1 /M 2 . Unfortunately these do not lead to a relation between measureable parameters, although the constraint that M 1 /M 2 ≃ m 2 /m 3 may be of interest in model building.

The case of two texture zeros
For two texture zeros one obtains a prediction even without imposing the symmetric constraint. There are fifteen ways of assigning two texture zeros to the first two rows of the Dirac neutrino mass matrix (the third row plays no role in the case the third Majorana neutrino is anomalously heavy). All but five lead to inconsistent results; below we discuss only the viable choices.
From Table 1 we may readily solve the constraint following from equating the two forms for tan z that follow from (1, 1) and (2, 2) texture zeros. This gives the prediction for U 13 given in Table 2. One may see it is identical to the prediction (c.f. Figure 5) obtained for a single texture zero in the (1, 2) position with the symmetric condition imposed although in this case we have not imposed this condition. If one further imposes the condition that the matrix is symmetrical one also obtains the prediction for U 13 given in eq (13). Equating these results fixes one combination of the phases (which does not lead to new relations between measurable phases) and fixes the ratio M 1 /M 2 ≃ m 2 /m 3 . The prediction for ǫ ǫmax is as given in Table 1 for the (2, 2) texture zero case.
The remaining possibilities are given in Table 2. The prediction for the CHOOZ angle is approximately the same for the (1, 1) and (2,3) or the (1, 3) and (2, 1) cases and is shown in Figure 7(a). The remaining case with a (1, 1) and a (2, 1) texture zero is shown in Figure 7(b).
For the case of (1, 1) and (2, 3) texture zeros one again needs M 1 /M 2 ≃ m 2 /m 3 if one requires the Dirac mass matrix be symmetric. For the last two cases there is no solution if one additionally  1) and (2, 1). The plot is for the χ = 1 case and is plotted againt the relative phase between the two terms appearing in Table 2.
Texture zero U 13 (1, 1) and (2, 2) ± m 2 m 3 Table 2: The constraints following from two texture zeros. Only those cases shown are consistent apart from the (1,2), (2,1) case which has already been discussed when considering symmetric textures. Also shown are the additional constraints following from imposing a symmetric structure for the two cases this is consistent. χ is 0 for a (1,1) texture zero in the charged lepton sector and 0 for a (1,2) texture zero. For no lepton texture zero χ is between these limiting cases.
imposes the condition the Dirac mass matrix be symmetric. In all cases the prediction for ǫ ǫmax is as given in Table 1 for the appropriate texture zero. This follows because the prediction comes from the constraint on tan z only and does not require the symmetric condition.

Summary and Conclusions
The combination of the see-saw mechanism, an hierarchical structure for the Majorana mass matrix and a combination of texture zeros and/or a symmetrical form for the moduli of the mass matrix elements leads to relations amongst observable properties of neutrinos. In this paper we have determined these predictions in a model independent way.
The case of a (1, 1) texture zero is of particular interest because, in the quark sector, it leads to a relation in excellent agreement with experiment. In the neutrino case the equivalent (1, 1) texture zero leads to a prediction for the CHOOZ angle that is close to the present limit and a near maximal CP violating phase relevant to thermal leptogenesis and to ν0ββ. For the (1, 1) texture zero, thermal leptogenesis cannot give rise to acceptable baryogenesis while satisfying the gravitino bounds on the reheat temperature. Therefore, an acceptable range of baryogenesis is only possible if the gravitino constraints are relaxed, for example in theories in which the supersymmetry breaking occurs at a lower scale.
In the case that the texture zero appears in the (1, 2) or (1, 3) positions the CHOOZ angle is still predicted to be large, encouraging for long baseline CP violation studies. Furthermore, in these cases washout effects after thermal leptogenesis are not too efficient and could allow for adequate baryogenesis. The case of (2, 2) and (2, 3) texture zeros does not lead to phenomenologically interesting relations. However there are five viable cases in which two texture zeros can be present. In these cases a large CHOOZ angle is again predicted.
The determination of the parameters involved in the see-saw mechanism is an illdefined problem due to the large number of parameters relative to measureables. The best hope is that the system has a high degree of symmetry, reducing the number of parameters. Our analysis has explored a particularly promising possibility suggested by the structure observed in the quark sector in which the Dirac masses have one (or more) texture zero(s) and the magnitude of the mass matrix elements may be symmetric. In addition we have assumed an hierarchical structure for the Majorana matrix, motivated by the fact this can readily explain the large neutrino mixing angles while having a relation between quark and lepton Dirac masses. Such a structure for the Dirac and Majorana masses can be derived from an underlying family symmetry [17] and, if the resultant predictions for neutrino properties should be confirmed, it would provide strong support for such an underlying symmetry organising the fermion mass matrices.