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Renormalisation Group Analysis of Single Right-handed Neutrino Dominance

We perform a renormalisation group (RG) analysis of neutrino masses and mixing angles in the see-saw mechanism in the minimal supersymmetric standard model with three right-handed neutrinos, including the effects of the heavy neutrino thresholds. We focus on the case that one of the right-handed neutrinos provides the dominant contribution to the 23 block of the light Majorana matrix, causing its determinant to approximately vanish and giving an automatic neutrino mass hierarchy, so called single right-handed neutrino dominance which may arise from a U(1) family symmetry. In these models radiative corrections can increase atomospheric and solar neutrino mixing by up to about 10% and 5%, respectively, and may help to achieve bi-maximal mixing. Significantly we find that the radiative corrections over the heavy neutrino threshold region are at least as important as those usually considered from the lightest right-handed neutrino down to low energies.


Introduction
The latest atmospheric neutrino results based on 1117 days of data from Super Kamiokande are still consistent with a standard two neutrino oscillation ν µ → ν τ with a near maximal mixing angle sin 2 2θ 23 > 0.88 and a mass square splitting ∆m 2 23 from 1.5 × 10 −3 to 5 × 10 −3 eV 2 at 90% CL [1]. The sterile neutrino oscillation hypothesis ν µ → ν s is excluded at 99% CL. Super Kamiokande is also beginning to provide important clues concerning the correct solution to the solar neutrino problem. The latest results from 1117 days of data from Super Kamiokande [2] sees a one sigma day-night asymmetry, and a flat energy spectrum, which together disfavour the small mixing angle (SMA) MSW solution [3], the just-so vacuum oscillation hypothesis [4] and the sterile neutrino hypotheses. All three possibilities are now excluded at 95% CL. The results allow much of the large mixing angle (LMA) MSW region, which now looks like the leading candidate for the solution to the solar neutrino problem. For example a typical point in the LMA MSW region is sin 2 2θ 12 ≈ 0.75 and ∆m 2 12 ≈ 2.5 × 10 −5 eV 2 [5].
The see-saw mechanism [6] implies that the three light neutrino masses arise from some large mass scales corresponding to the Majorana masses of some heavy "righthanded neutrinos" N p R M pq RR (p, q = 1, · · · , Z) whose entries take values which extend from ∼ 10 14 GeV down to perhaps several orders of magnitude lower. The presence of electroweak scale Dirac mass terms m ip LR (a 3 × Z matrix) connecting the left-handed neutrinos ν i L (i = 1, . . . 3) to the right-handed neutrinos N p R then results in a very light see-saw suppressed effective 3 × 3 Majorana mass matrix for the left-handed neutrinos ν i L , which are the light physical degrees of freedom observed by experiment. If the neutrino masses arise from the see-saw mechanism then it is natural to assume the existence of a physical neutrino mass hierarchy m ν 1 ≪ m ν 2 ≪ m ν 3 , which implies ∆m 2 23 ≈ m 2 ν 3 , and ∆m 2 12 ≈ m 2 ν 2 , which fixes m ν 3 ≈ 5.6 × 10 −2 eV , and (assuming the LMA MSW solution) m ν 2 ≈ 5.2 × 10 −3 eV , with rather large errors. Thus m ν 2 /m ν 3 ∼ 0.1. In view of such a 23 mass hierarchy the presence of a large 23 mixing angle looks a bit surprising at first sight, especially given our experience with small quark mixing angles. Several explanations have been proposed [7], but the simplest idea is that the contributions to the 23 block of the light effective Majorana matrix come predominantly from a single right-handed neutrino, which causes the 23 subdeterminant to approximately vanish. This mechanism, called single right-handed neutrino dominance (SRHND), was proposed in [8], and developed for bi-maximal mixing in [9].
In this paper we shall be concerned with the effect of radiative corrections in models based on SRHND and U(1) family symmetry [8], and in particular on the case of bi-maximal mixing in these models [9] . We choose these models because analytic estimates suggest that the 23 hierarchy arises from a physical mechanism which accounts for the smallness of the 23 subdeterminant, and so this hierarchy should be stable under radiative corrections. Although the radiative corrections to atmospheric mixing are only a modest 10%, this can nevertheless play an important role in achieving near maximal atmospheric mixing. Our results may be compared to the several RG studies of various models which already exist in the literature [10], [11]. Many of the existing studies do not take proper account of the heavy righthanded neutrino mass thresholds, often only including the effects of running due to the low energy dimension 5 see-saw operator from the see-saw scale down to low energies [10], or running from the high energy scale down to low energies including a single right-handed neutrino mass threshold [11], assuming that all the right-handed neutrinos are degenerate. Our results indicate that the corrections in running from the high energy scale, through the heavy neutrino threshold region, down to the lightest right-handed neutrino mass scale are just as important (and in some cases more so) than the traditionally calculated radiative corrections in running from the lightest right-handed neutrino mass scale down to low energies.
The paper is organised as follows. In section 2 we define the MSSM with Z righthanded neutrinos, where the heavy neutrino mass matrix arises from the vacuum expectation value (VEV) of a singlet field Σ, and write down the renormalisation group equations (RGEs) relevant for running from the unification scale M U ∼ 2 × 10 16 GeV down to low energies. In section 3 we review the analytic discussion of the phenomenological conditions for SRHND and LMA MSW involving the different types of heavy Majorana neutrino textures, and show how a U(1) family symmetry may be used to satisfy them [9]. In section 4 we discuss three explicit examples of this kind, and then perform our numerical RG analysis of these cases. Section 5 concludes the paper.

The MSSM with Z Right-handed neutrinos
We consider the Yukawa terms with two Higgs doublets augmented by Z right-handed neutrinos, which, are given by where ǫ ab = −ǫ ba , ǫ 12 = 1, and the remaining notation is standard except that the Z right-handed neutrinos N p R have been replaced by their CP conjugates N c p with p, q = 1, . . . , Z and we have introduced a singlet field Σ whose vacuum expectation value (VEV) induces a heavy Majorana matrix M RR =< Σ > Y RR . When the two Higgs doublets get their VEVS < H 2 Replacing CP conjugate fields we can write in a matrix notation where we have assumed that all the masses and Yukawa couplings are real and written Y * = Y . The diagonal mass matrices are given by the following unitary transforma- Below the mass of the lightest right-handed neutrino M R1 the right-handed neutrino masses may be integrated out of the theory, which corresponds to replacing the last two terms in Eq.2 by a dimension 5 operator is simply related to the see-saw mass matrix in Eq.1 when the Higgs fields are replaced by their VEVs Having constructed the light Majorana mass matrix it must then be diagonalised by unitary transformations, The CKM matrix is given by The leptonic analogue of the CKM matrix is the MNS matrix defined as [12] which may be parametrised by a sequence of three rotations about the 1,2 and 3 axes, as in the standard CKM parametrisation, where where s ij = sin θ ij , c ij = cos θ ij , and θ ij refer to lepton mixing angles. Note that we completely ignore CP violating phases in this paper. From the unitarity conditions of the MNS matrix elements in Eq.11, and parametrisation in Eq.12, the mixing angles can be expressed in terms of elements of V M N S as In principle these mixing angles may not be exactly the same as those obtained from two generation analysis of the experimental results.
The renormalisation group equations (RGEs) to one-loop order are: where the wavefunction anomalous dimensions are where t = ln µ (µ is theMS scale) and I is the unit matrix. The RGEs for the gauge couplings are The RGEs above are used to run the Yukawa matrices down from high energies (say the unification or string scale) down to the heaviest right-handed neutrino mass, M RZ , which we assume is equal to the VEV of the Σ field. At this mass scale we perform a rotation of the right-handed neutrino fields to the basis in which M RR is diagonal, according to Eq.5, so that Y ν is replaced by Y ′ ν = Y ν Ω † RR . In this basis one then runs the remaining RGEs (apart from Y RR ) down through the right-handed neutrino thresholds diag(M RZ , . . . , M R1 ), decoupling each right-handed neutrino at its mass threshold from the Y ′ ν contributions which appear on the right-hand side of the RGEs. To be explicit, we replace on the right-hand sides of the RGEs, where θ p = θ(ln µ −ln M Rp ). For the (diagonal) Y RR on the right-hand side we replace it by In the next section we shall see that for the cases of interest it is not necessary to diagonalise M RR in order to implement decoupling of the right-handed neutrinos, and one may remain in the basis defined by the U(1) family charges, since decoupling is facilitated by the simple structures of M RR .
Below the mass of the lightest right-handed neutrino one may use the RGE for κ, the coefficient of the dimension 5 neutrino mass operator, In solving Eq.20, it is convenient to diagonalise Y u and Y e at the scale of the lightest right-handed neutrino, using Eqs.5, then make the approximation of keeping only the largest third family Yukawa eigenvalues. In the diagonal charged lepton basis, κ must be transformed to κ ′ given by then, retaining the third family t and τ Yukawa couplings only, the RGEs for the elements of κ ′ are given by Following from Eq.22 we see that the elements of m ′ where where f = t, τ and i = 1, 2, 3. In running down to m t the charged lepton matrix will remain diagonal to good approximation, so that the low energy MNS matrix is simply given by The mixing matrix V M N S and hence neutrino mixing angles, are the running quantities which can be computed at different energy scales. For example, the running of the neutrino mixing angle relevant to the atmospheric neutrino deficit, θ 23 , can be understood from the evolution equation This equation (see Babu et al in ref. [7]) describes the evolution of the physical 23 mixing angle assuming that we are in the diagonal charged lepton mass basis.

Three right-handed neutrinos and SRHND
We now specialise to three right-handed neutrinos and review the conditions for achieving SRHND and the LMA MSW solution [9]. The statement of SRHND is that, of the three right-handed neutrinos, one of them, N R3 , makes the dominant contribution to the 23 block of m LL . This ensures that the 23 sub-determinant approximately vanishes, and a 23 mass hierarchy therefore naturally results.
We first write the neutrino Yukawa matrix in general as There are now three distinct textures for the heavy Majorana neutrino matrix which maintain the isolation of the dominant right-handed neutrino N R3 , namely the diagonal, democratic and off-diagonal textures introduced previously [8,9]. We consider each of them in turn. Note that, assuming SRHND, the contribution to the lepton 23 and 13 mixing angles from the neutrino sector are approximately so that Super-Kamiokande and CHOOZ [13] imply The condition on the 12 mixing angle such that it is relevant for the LMA MSW solution is discussed separately for each case below.

Diagonal Texture
We may invert the heavy Majorana matrix and construct the light Majorana matrix using the see-saw mechanism, The SRHND condition is [9] The 12 mixing angle determines whether we have the LMA MSW or SMA MSW solution, and this depends on the relative magnitude of the sub-dominant entries of m LL , as discussed in [9]. The condition for LMA MSW is [9] max ab X , ac X ,

Off-Diagonal Texture
This is defined by: The off-diagonal case is qualitatively different from the other two cases and gives SRHND is now defined by the conditions

Democratic Texture
The democratic case (assuming the Majorana masses in the upper block are of the same order but are not exactly equal) is defined by: The beauty of SRHND is that it automatically implies m ν2 ≪ m ν3 due to the approximately vanishing 23 subdeterminant, without the need for appeal to cancellations. To understand this simply drop the 1/X terms and observe that the 23 subdeterminant vanishes which implies a massless eigenvalue which is a rather extreme case of a hierarchy! The order of magnitude of m LL is: In this case the SRHND conditions are:

U (1) Family Symmetry
Introducing a U(1) family symmetry [14], [15], [16], [17] provides a convenient way to organise the hierarchies within the various Yukawa matrices. For definiteness we shall focus on a particular class of model based on a single pseudo-anomalous U(1) gauged family symmetry [16]. We assume that the U(1) is broken by the equal VEVs of two singlets θ,θ which have vector-like charges ±1 [16]. The U(1) breaking scale is set by < θ >=<θ > where the VEVs arise from a Green-Schwartz mechanism [18] with computable Fayet-Illiopoulos D-term which determines these VEVs to be one or two orders of magnitude below M U . Additional exotic vector matter with mass M V allows the Wolfenstein parameter [19] to be generated by the ratio [16] The idea is that at tree-level the U(1) family symmetry only permits third family The conditions which ensure that the third dominant neutrino is isolated require that the elements λ |n 1 +n 3 +σ| , λ |n 2 +n 3 +σ| be sufficiently small. The diagonal, off-diagonal and democratic textures then emerge as approximate cases [8,9]. The neutrino Yukawa matrix is explicitly which may be compared to the notation in Eq.28. The requirement of large 23 mixing and small 13 mixing expressed in Eq.30 becomes The remaining conditions for the U(1) charges depend on the specific heavy Majorana texture under consideration [9].
The charged lepton Yukawa matrix is given by where e i are the U(1) charges of the charged lepton singlet fields.
For the quarks we shall assume a common form for the textures of Y u and Y d

Renormalisation Group Analysis of SRHND
In [9] we tabulated the simplest charges which satisfy all the conditions given above, and so provide a natural account of the atmospheric and solar neutrinos via the LMA MSW effect. In Table 1  In case A, before rotating to the diagonal charged lepton mass basis, we may easily estimate the order of the entries in m LL in Eq.32 from the matrices Y ν and M RR in Table 1, and hence verify the SRHND conditions Eq.33, and the LMA MSW condition Eq.34. Using Eq.32 we find  Eq.36 from the matrices Y ν and M RR in Table 1 as, where the first entry in each element corresponds to the 1/Y contributions coming from the right-handed neutrino N 3R , and clearly dominates the 23 block by a factor of λ 2 . In this case it does not dominate the other elements outside the 23 block. The 13 element from N 3R is suppressed relative to the 23 block elements by a factor of λ 2 , leading to a CHOOZ angle of this order. The subdominant entries in the 12,13,22,23 elements are again the same order, leading to a large 12 angle suitable for LMA MSW.
Finally in case C, we verify that the SRHND condition Eq.41 and LMA MSW condition Eq.42 are satisfied, by constructing the entries in m LL in Eq.40 from the matrices Y ν and M RR in Table 1, where the first entry in each element corresponds to the 1/Y contributions coming from the dominant right-handed neutrino, which dominates in the 23 block by a factor of λ 2 and in the 12 and 13 elements by a factor of λ. The 13 element is smaller than the 23 elements by a factor of λ leading to a CHOOZ angle of this order. The subdominant 12,13,22,23 elements are all the same order leading to LMA MSW.
We now turn to a numerical treatment of these three cases. In integrating the RGEs down to low energies the heavy thresholds for the diagonal texture in Eq.31 are dealt with exactly as described in the previous section, except that no rotation is required to get to the diagonal M RR basis since we already begin with that form.
For the democratic texture in Eq.39 there are essentially only two mass thresholds to consider X and Y since N R1 and N R2 are approximately degenerate with mass X and we can ignore any small mass difference between them to leading order. Similarly for the off-diagonal texture in Eq.35 we also have only two mass thresholds to consider X and Y since N R1 and N R2 are now exactly degenerate with mass X. Thus for both democratic and off-diagonal textures we can replace on the right-hand sides of the RGEs by where θ 1,2 = θ(ln µ − ln X), θ 3 = θ(ln µ − ln Y ). We replace Y RR on the right-hand side of the RGEs by  Tables 2,3,4 is always positive. This means that the overall sign of the RGE is negative, which implies an increasing S at as the energy scale is reduced. Note that the largest contribution to the atmospheric mixing angle arises from the charged lepton sector, for this choice of parameters, as is clear from examining V eL in Tables 2,3,4. The solar mixing quantity S sol also increases, but the increases of 1 − 5% are mild in comparison to S at . In cases A,C S sol approaches maximal mixing, while in case B its low energy value is about 0.71, which are well near the best fit sin 2 2θ 12 ≈ 0.76.
In Figures 1-3 we display the RG running of the mixing angles S sol and S at and the ratio of neutrino masses m ν2 /m ν3 as a function of t = ln µ, for the three cases A,B,C. We prefer to show the variation in the neutrino mass ratio, rather than the absolute values of the two neutrino mass eigenvalues since they will depend to some extent on the running vacuum expectation value, although the effect on the present analysis is not large since we assume high tan β. We also do not consider the ratio m ν1 /m ν2 since this is not experimentally measurable in the hierarchical case.
In Figure 1(a) we show the variations of S sol and S at with energy scale for case A. As already stated, the atmospheric mixing angle runs more rapidly than the solar mixing angle, and although S at starts out smaller then S sol at M U , it quickly grows larger. Note the effect of the heavy right-handed neutrino mass thresholds which change the slope of the curves, which grow steeply over the heavy threshold region.
In Figure 1(b) we give the variation of the neutrino mass ratio m ν2 /m ν3 with t = ln µ for case A. Here the effects of the three heavy right-handed neutrino thresholds is clearly seen, with again a steep rise in this mass ratio over the heavy threshold region.
In Figure 2(a) we show the variations of S sol and S at with energy scale for case B. Here the atmospheric angle starts out larger than the solar angle, but still grows more rapidly. In Figure 2(b) we give the variation of the neutrino mass ratio m ν2 /m ν3 with t = ln µ for case B. The qualitative shape of this curve is similar to Figure 1 but there are only two heavy neutrino mass thresholds in this case, and also the mass ratio is larger throughout.
In Figure 3(a) we show the variations of S sol and S at with energy scale for case B.
Here the solar angle starts out much larger than the atmospheric angle, and as before the atmospheric angle grows more rapidly and approaches the solar angle. In Figure   3(b) we give the variation of the neutrino mass ratio m ν2 /m ν3 with t = ln µ for case B. Because of the choice of c 12 in Table 1, the two heavy right-handed neutrino mass thresholds are very close together in this case.    It is important to emphasise that more than 50% of the radiative corrections to all the physical quantities m ν 2 /m ν 3 , S at , S sol arises from the RG evolution over the range from the GUT scale M U = 2.0 × 10 16 GeV through the heavy right-handed neutrino threshold region down to the lightest right-handed neutrino mass scale M R 1 ∼ 10 14 GeV. Therefore even though this range covers only two orders of magnitude in energy, it is just as important as the RG evolution effects from M R 1 ∼ 10 14 GeV down to low energies which covers over 12 orders of magnitude in energy, and is the region commonly considered in the literature.

Conclusion
We have studied the effects of radiative corrections on neutrino masses and mixing angles, in evolving the theory defined at the GUT scale down to low energies. Our approach is to run down all the Yukawa matrices from the GUT scale down through the heavy right-handed neutrino mass thesholds to low energy, replacing the neutrino Yukawa matrices by the dimension 5 neutrino mass operator at the lowest righthanded neutrino mass threshold. We have found that in the realistic cases considered, the atmospheric and solar neutrino mixing parameters receive radiative corrections of order 10% and 5%, respectively, while the ratios of the neutrino masses change by up about 50% in going from M U to low energy. Importantly, more than 50% of the overall radiative corrections arises from running through the threshold region, even though it only accounts for two orders of magnitude in energy. Thus many of the existing analyses in the literature which ignore this threshold region could endanger a significant error.
We have considered a realistic class of models known as SRHND [8,9], in which the contribution to the 23 block of the light effective Majorana matrix m LL is dominated by a single right-handed neutrino. The small neutrino mass hierarchy m ν 2 /m ν 3 ∼ 0.1 originates from the smallness of the 23 subdeterminant of m LL (which is zero in the limit that a single right-handed neutrino is the only contribution). We have focussed on cases with two large mixing angles θ 23 and θ 12 , with the CHOOZ angle θ 13 being small -the so-called bi-maximal mixing scenario. The couplings of the dominant right-hand neutrino controls the 23 and 13 mixing angles, and the subdominant right-handed neutrino couplings control the 12 angle. A U(1) family symmetry is used to generate a controlled expansion of all the Yukawa couplings in powers of the Wolfenstein parameter λ, and with a suitable choice of U(1) charges SRHND may be achieved with subdominant right-handed neutrino contributions being of the correct order of magnitude to generate the desired neutrino mixing angles and mass hierarchy.
The U(1) charges also control the charged lepton Yukawa matrix, resulting in large contributions to the physical lepton mixing angles from the charged lepton sector. In general one would expect the two contributions to the 23 mixing angle (from the charged leptons and the neutrinos) not to cancel exactly due to order unity coefficients in the Yukawa matrices which are not predicted, and our numerical results include these effects. Similarly it would be surprising if the physical mixing angles turned out to be exactly maximal at the GUT scale. Therefore we have considered examples in which the 23 and 12 angles start out large but not maximal, and we have also assumed large tan β, where the τ Yukawa coupling is large and the effect of radiative corrections is maximised. Although the effect of radiative corrections on the mixing angles is always ≤ 10%, showing that the models are quite stable, we have shown that the effects may play an important role in driving the initially large (but not maximal) mixing angle towards its maximal value. This is true both of the atmospheric mixing angle and the solar mixing angle, although in the latter case the effects are milder, which helps to explain why the atmospheric angle is larger than the solar angle. In principle, a different choice of parameters could have caused the neutrino angles to have grown smaller at low energies. However it is significant that for the cases considered both mixing angles become magnified showing that the low energy approximate bi-maximal scenario could partly result from radiative corrections in SRHND models.