The LMA solution from bimaximal lepton mixing at the GUT scale by renormalization group running

We show that in see-saw models with bimaximal lepton mixing at the GUT scale and with zero CP phases, the solar mixing angle θ 12 generically evolves towards sizably smaller values due to renormalization group effects, whereas the evolution of θ 13 and θ 23 is comparatively small. The currently favored LMA solution of the solar neutrino problem can thus be obtained in a natural way from bimaximal mixing at the GUT scale. We present numerical examples for the evolution of the leptonic mixing angles in the Standard Model and the MSSM, in which the current best-ﬁt values of the LMA mixing angles are produced. These include a case where the mass eigenstates corresponding to the solar mass squared difference have opposite CP parity.


Introduction
Recent experimental evidence strongly favors the LMA solution of the solar neutrino problem with a large but non-maximal value of the solar mixing angle θ 12 [1][2][3][4]. An overview of the current allowed regions for the mixing angles and the mass squared differences is given in Table 1.
A big problem for model builders is to explain the deviation of θ 12 from maximal mixing, while keeping θ 23 maximal and θ 13 small at the same time. The renormalization group (RG) evolution is a possible candidate for accomplishing this. Therefore, it is interesting to investigate the evolution of the mixing angles from the GUT scale to the electroweak (EW) or SUSY-breaking scale. A number of studies with three neutrinos considered the possibility of increasing a small mixing angle via RG evolution [7][8][9][10]. Others focused on the case of nearly degenerate neutrinos [11][12][13][14][15][16], on the existence of fixed points [17], or on the effect of non-zero Majorana phases on the stability of the RG evolution [18]. Table 1 Experimental data for the neutrino mixing angles and mass squared differences. For the solar angle θ 12 and the solar mass squared difference, the LMA solution has been assumed. The results stem from the analysis of the recent SNO data [3], the Super-Kamiokande atmospheric data [5] and the CHOOZ experiment [6] Best-fit value Range (for θ ∈ [0 • , 45 • ]) C . L .
θ 12  We consider the see-saw scenario, i.e., the Standard Model (SM) or MSSM extended by 3 heavy neutrinos that are singlets under the SM gauge groups and have large explicit (Majorana) masses with a non-degenerate spectrum. Due to this non-degeneracy, one has to use several effective theories, with the singlets partly integrated out, when studying the evolution of the effective mass matrix of the light neutrinos [19,20]. Below the lowest mass threshold, the neutrino mass matrix is given by the effective dimension 5 neutrino mass operator in the SM or MSSM. The relevant RGE's were derived in [20][21][22][23][24][25].
In this Letter, we assume bimaximal mixing at the GUT scale with vanishing CP phases and positive mass eigenvalues. We calculate the RG running numerically in order to obtain the mixing angles at low energy and to compare them with the experimentally favored values. We include the regions above and between the see-saw scales in our study, which have not been considered in most of the previous works. We find that the solar mixing angle changes considerably, while the evolution of the other angles is comparatively small, so that values compatible with the LMA solution can be obtained. We present analytic approximations that help to understand this behavior and show that it is rather generic.

Bimaximal mixing at the GUT scale
At the GUT scale, we assume bimaximal mixing in the lepton sector. We restrict ourselves to the case of positive mass eigenvalues and real parameters, so that there is no CP violation. In the basis where the charged lepton Yukawa matrix is diagonal, up to phase conventions the general parametrization of the effective Majorana mass matrix of the light neutrinos is then From Eq. (3a) we see that a > 0. Eqs. (4) imply that the solar mass squared difference m 2 sol = m 2 2 − m 2 1 is related to c, while the atmospheric one, m 2 atm = m 2 3 − m 2 1 , is controlled by b. Thus, a > |b| > |c|. For b > 0 we obtain a normal mass hierarchy, while for b < 0 the mass hierarchy is inverted, as illustrated in Fig. 1. For positive c, m 1 < m 2 , otherwise m 1 > m 2 . Hence, m 2 sol is positive only if c is. If a |b|, |c|, the spectrum is called degenerate. We use the convention that the mass label 2 is attached in such a way that 0 θ 12 45 • . This can always be accomplished by the replacement c ↔ −c.
In our see-saw scenario, the effective mass matrix of the light neutrinos is ≈ 174 GeV. Obviously, the singlet Yukawa and mass matrices Y ν and M cannot be determined uniquely from this relation, i.e., there is a set of {Y ν , M} configurations that yield bimaximal mixing. After choosing an initial condition for Y ν , M (and thus the see-saw scales) is fixed by the see-saw formula (5).

Solving the RGE's
To study the RG running of the leptonic mixing angles and neutrino masses, all parameters of the theory have to be evolved from the GUT scale to the EW or SUSY-breaking scale, respectively. Since the heavy singlets have to be integrated out at their mass thresholds, which are non-degenerate in general, a series of effective theories has to be used. The derivation of the RGE's and the method for dealing with these effective theories are given in [20]. Starting at the GUT scale, the strategy is to successively solve the systems of coupled differential equations of the form for all the parameters M, . . .} of the theory in the energy ranges corresponding to the effective theories denoted by (n). At each see-saw scale, tree-level matching is performed. Due to the complicated structure of the set of differential equations, the exact solution can only be obtained numerically. However, to understand certain features of the RG evolution, an analytic approximation at the GUT scale will be derived in Section 5.

Examples for the running of the mixing angles
Figs. 2 and 3 show typical numerical examples for the running of the mixing angles from the GUT scale to the EW or SUSY-breaking scale. They contain an important effect that appears for most choices of the initial parameters: The solar angle θ 12 changes drastically, while the changes in θ 13 and θ 23 are comparatively small. This agrees remarkably well with the experimentally favored scenario.   3. Example for the RG evolution of the mixing angles in the SM extended by heavy singlets from the GUT scale to the EW scale for a normal mass hierarchy and Y ν = X diag(1, ε, ε 2 ) with ε = 0.65, a = 0.0655 eV and X = 1. In this example, the lightest neutrino has a mass of 0.024 eV.

Analytic approximation for the running of the mixing angles at the GUT scale
In order to understand the effect found numerically in the previous section, we now derive an analytic approximation for the RG evolution of the mixing angles at the GUT scale. It is only affected by the part of the RGE that is not proportional to the unit matrix, which is given by with C e = −3/2, C ν = 1/2 in the SM and C e = C ν = 1 in the MSSM. Analogously to Eq. (1), m ν is parametrized by where µ is the renormalization scale, t := ln µ µ 0 , and m diag := diag(m 1 , m 2 , m 3 ). In general, the real Y ν can be written as However, the effective mixing matrix m ν is invariant under the transformations Y ν → V T Y ν and M → V T MV , which correspond to a change of basis for the heavy sterile neutrinos. Thus, V (ξ 12 , ξ 13 , ξ 32 ) in Eq. (9) can be absorbed into M, leading to the simpler parametrization Furthermore, we use the approximation that the effect of the charged lepton Yukawa matrices Y e can be neglected compared to that of the neutrino Yukawa matrix. Note that in the MSSM a large tan β can yield a relatively large Y e , which can also have sizable effects that are neglected in this approximation.

Parameters at the GUT scale
The considerable change of the solar mixing angle found in the previous sections raises the question whether the parameter region of the LMA solution might be reached by RG evolution, if one starts with bimaximal mixing at high energy. We will investigate this possibility by further numerical calculations in the following. To reduce the parameter space for the numerical analysis, we choose a specific neutrino Yukawa coupling Y ν at the GUT scale. We assume that it is diagonal and of the form Y ν and M are now determined by the parameters {ε, X, a, b, c}. Moreover, we fix the GUT scale values of b and c by the requirement that the solar and atmospheric mass squared differences obtained at the EW scale after the RG evolution be compatible with the allowed experimental regions. Thus, we are left with the free parameters X, ε and a. The parameter ε controls the hierarchy of the entries in Y ν and thus the degeneracy of the see-saw scales, while a determines the mass of the lightest neutrino. The dependence of physical quantities on ε and a is shown in Fig. 4. The effect of changing the scale X of the neutrino Yukawa coupling will be discussed in Section 6.3. As mentioned above, we work in the basis where the Yukawa matrix of the charged leptons is diagonal.

Allowed parameter space regions
The parameter space regions in which the RG evolution produces low-energy values compatible with the LMA solution are shown in Fig. 5 for the SM and the MSSM (tan β = 5) with a normal mass hierarchy. We find that for the form of Y ν under consideration, hierarchical and degenerate neutrino mass schemes as well as degenerate and non-degenerate see-saw scales are possible. For inverted neutrino mass spectra, allowed parameter space regions exist as well.
We would like to stress that the shape of the allowed parameter space regions strongly depends on the choice of the initial value of Y ν at the GUT scale. One also has to ensure that the sign of m 2 sol is positive, as the LMA solution requires this if the convention is used that the solar mixing angle is smaller than 45 • . With bimaximal mixing at the GUT scale, the sign of m 2 sol is not defined by the initial conditions. Using the analytic approximation  of Section 5, the sign just below the GUT scale can be calculated. We find m 2 sol > 0 for F 1 < 0 and vice versa. However, in order to predict the sign of m 2 sol at low energy, the numerical RG evolution has to be used. This excludes some of the possible choices for the neutrino Yukawa coupling Y ν at the GUT scale. For example, among the possibilities with diagonal Y ν it excludes Y ν = diag(ε 2 , ε, 1).

Dependence on the scale X of the neutrino Yukawa coupling
For small values of X, the contribution from Y ν to the evolution of the mixing angles above the largest see-saw scale is suppressed by a factor of X 2 . Nevertheless, the evolution to the LMA solution is still possible, as can be seen from the example in Fig. 6. Here the large change of θ 12 also seems to be generic but takes place between the seesaw scales, which shows the importance of carefully studying the RG behavior in these intermediate regions [20]. Note that in this case the analytic approximation of Section 5 cannot be applied, since it is only valid at the GUT scale.

Effect of neutrino CP parities
An example for the running of the mixing angles to the LMA solution with a negative CP parity for the state with mass m 2 is shown in Fig. 7. For this we have chosen a different diagonal structure for Y ν , at the GUT scale. Here, the evolution to the LMA solution is possible due to running between the see-saw scales. A more detailed study of the effect of CP phases will be given in a forthcoming paper [27].
The large RG effects in this case seem surprising at first sight, since previous studies, e.g. [18,26], found that opposite CP parities for m 1 and m 2 prevent a sizable change of the solar mixing angle by RG evolution. However, these works did not consider the energy region between the see-saw scales, where the largest change occurs in our example. This fact explains the apparent discrepancy.

Low scale values of θ 13 and θ 23
The mixing angles θ 13 and θ 23 are affected by the RG evolution as well, i.e., they do not stay at their initial values θ 13 = 0 • and θ 23 = 45 • . However, lower bounds on their changes cannot be given unless a specific model is chosen. As one can see from the previous examples, the changes can be tiny. For instance, the evolution of Fig. 2 gives θ 13 = 0.02 • , which corresponds to sin 2 (2θ 13 ) = 5 × 10 −7 , and θ 23 = 0.28 • . On the other hand, other choices of Y ν at the GUT scale produce θ 13 and θ 23 that come close to the experimental bounds. This can make it possible to discriminate between models with different initial values Y ν (M GUT ).

Summary and conclusions
We have shown that in see-saw scenarios the experimentally favored neutrino mass parameters with the LMA solution of the solar neutrino problem can be obtained in a rather generic way from bimaximal mixing at the GUT scale by renormalization group running. We have concentrated on the case of vanishing CP phases, which implies positive mass eigenvalues. In an example where the mass eigenstates corresponding to the solar mass squared difference have opposite CP parity, we have demonstrated that an evolution towards the LMA solution is possible in this case as well. The general case of arbitrary CP phases is beyond the scope of this Letter and will be studied elsewhere [27]. The mixing angles evolved down to the electroweak scale show a strong dependence on the mass scale of the lightest neutrino, on the degeneracy of the see-saw scales, and on the form of the neutrino Yukawa coupling. A generic feature of the renormalization group evolution is that the solar mixing angle θ 12 evolves towards sizably smaller values, whereas the change of θ 13 and θ 23 is comparatively small. In the SM and MSSM, we find extensive regions in parameter space which are compatible with the LMA solution for normal and inverted neutrino mass hierarchies and for large and small absolute scales of the neutrino Yukawa couplings. Thus, RG running may provide a natural explanation for the observed deviation of the LMA mixing angles from bimaximality.