If neutrinos are to play a relevant cosmological rôle, they must be essentially degenerate with a mass matrix of the bimaximal mixing type. We study this scenario in the MSSM framework, finding that if neutrino masses are produced by a seesaw mechanism, the radiative corrections give rise to mass splittings and mixing angles that can accommodate the atmospheric and the (large angle MSW) solar neutrino oscillations. This provides a natural origin for the hierarchy. On the other hand, the vacuum oscillation solution to the solar neutrino problem is always excluded. We discuss also in the SUSY scenario other possible effects of radiative corrections involving the new neutrino Yukawa couplings, including implications for triviality limits on the Majorana mass, the infrared fixed point value of the top Yukawa coupling, and gauge coupling and bottomtau unification.
CERNTH/99142
May 1999
IEMFT193/99
CERNTH/99142
hepph/9905381
1 Introduction
If neutrinos are to play a relevant cosmological role, their masses should be (eV). In that case, since atmospheric and solar neutrino anomalies [1] indicate that masssquared splittings are at most , neutrinos must be almost degenerate [2][5]. On the other hand, supersymmetry (SUSY) is a key ingredient in most of the extensions of the Standard Model (SM) which are candidate for a more fundamental theory. In this paper we will analyze, within the supersymmetric framework, under which circumstances the “observed” mass splittings between quasidegenerate neutrinos arise naturally (or not), as a radiative effect, in agreement with all the available experimental data.
This problem has been also addressed in a recent paper by Ellis and Lola [6], in which they treat the neutrino mass matrix, , as an effective operator, emerging at some scale, , with the bimaximal mixing form. Then the renormalization group (RG) analysis shows that the splittings and mixings at low energy are not in agreement with observations. Here we take a more general point of view. Besides exploring the effective operator scenario, we focus our attention in the (well motivated) case in which this operator is produced by a seesaw mechanism^{2}^{2}2An alternative possibility to generate small nonzero neutrino masses in the MSSM involves parity breaking [7]. See Ref. [8] for a discussion on the interplay between the two possibilities.. This introduces crucial differences in the analysis. In particular, the form of is modified by a first stage of RG running of the neutrino DiracYukawa matrix and the righthanded neutrino mass matrix from the high energy scale (say or ) to , which is identified with the Majorana mass scale. As we will see, this modification allows in many cases to reconcile the scenario with experiment, providing also a natural origin for the “observed” solaratmospheric hierarchy of splittings, .
In a recent paper [9], we performed a similar analysis in the SM framework (in which the only particles added to the SM are three righthanded neutrinos), also with positive results. There are similarities and differences between the SUSY and the SM cases. First, SUSY introduces additional unknowns in the scenario, particularly the supersymmetric mass spectrum. In the analysis, the only rôle of this spectrum is to give the threshold scale(s) below which the effective theory is just the SM. In this sense, the combination of (negative) experimental data and naturalness requires TeV. Of course, there may be appreciable differences between the masses of various supersymmetric particles (squarks, gluino, charginos, etc.). Still, the variation in the results of the RG analysis are not important. Hence, we will take a unique threshold at TeV throughout the paper. Let us mention here that, apart from three righthanded neutrinos, we will assume a minimal spectrum of particles; in other words we will work within the minimal supersymmetric standard model (MSSM). Second (and more important), in the supersymmetric regime, the chargedlepton Yukawa couplings are multiplied by a factor with respect to their SM value. These couplings (together with the neutrino Yukawa couplings) play a major rôle in the radiative modification of the form of . Thus, the results are going to present a strong dependence on (we recall that is defined as the ratio of the expectation values of the two supersymmetric Higgs doublets, ). Finally, the renormalization group equations (RGEs) themselves are different in the SUSY and in the SM cases. The difference is not just quantitative (i.e. differences in the size of the various coefficients), but also qualitative. In particular, the modification of the texture due to the contribution from the chargedlepton Yukawa couplings has opposite signs in the two cases (while the contribution from the neutrino Yukawa couplings themselves remains with the same sign).
Let us briefly review the current relevant experimental constraints on neutrino masses and mixing angles (a more detailed account is given in ref. [9]). Observations of atmospheric neutrinos are well described by oscillations driven by a mass splitting and a mixing angle in the range [10]
(1) 
Concerning the solar neutrino problem, as has been shown in ref. [2] the small angle MSW solution is unplausible in a scenario of nearly degenerate neutrinos, so we are left with the large angle MSW (LAMSW) and the vacuum oscillation (VO) solutions, which require mass splittings and mixing angles in the following ranges
LAMSW solution:
(2) 
VO solution:
(3) 
From the previous equations, it is apparent the hierarchy of mass splittings between the different species of neutrinos, , which should be reproduced by any natural explanation of those splittings. Let us also remark that it is not clear at the moment the value of the upper bound on [see eq.(2)]. As we will see, an upper limit like or even greater, may disallow the scenario examined in this paper. For the moment, we will not consider any upper bound on (see ref.[9] for a more detailed discussion). On the other hand, according to the most recent combined analysis of SK CHOOZ data (last paper of ref. [10]) the third independent angle, say (the one mixing the electron with the most split mass eigenstate), is constrained to have low values, at 90% (99%) C.L.
Other relevant experimental information concerns the nonobservation of neutrinoless double decay, which requires the element of the matrix to be bounded as [11]
(4) 
In addition, Tritium decay experiments indicate eV for any mass eigenstate with a significant component [12]. Finally, concerning the cosmological relevance of neutrinos, we will take eV as a typical possibility and we will explain how the results vary when this value is changed.
Let us introduce now some notation. In the SUSY framework the effective mass term for the three light (lefthanded) neutrinos in the flavour basis is given by a term in the superpotential
(5) 
The mass matrix, , is diagonalized in the usual way, i.e. , where and is a unitary ‘CKM’ matrix, relating flavour to mass eigenstates
(6) 
Here and denote and , respectively. In the following we will label the mass eigenvectors as and , where ( is thus the most split eigenvalue). In this notation, constraint (4) reads
(7) 
As it has been put forward by Georgi and Glashow in ref. [2], a scenario of nearly degenerate neutrinos should be close to a bimaximal mixing, which constrains the texture of the mass matrix to be essentially [2, 3]
(8) 
where gives an overall mass scale. can be diagonalized by a matrix
(9) 
leading to exactly degenerate neutrinos, , and , .
It is quite conceivable that could be generated at some high scale by interactions obeying appropriate continuous or discrete symmetries [13]. However, in order to be realistic, at low energy should be slightly different from to account for the mass splittings given in eqs.(1–3). We will explore whether the appropriate splittings (and mixing angles) can be generated or not through radiative corrections; more precisely, through the running of the RGEs from the high scale down to low energy. As discussed in ref.[9], the output of this analysis can be of three types:
 i)

All the mass splittings and mixing angles obtained from the RG running are in agreement with all experimental limits and constraints.
 ii)

Some (or all) mass splittings are much larger than the acceptable ranges.
 iii)

Some (or all) mass splittings are smaller than the acceptable ranges, and the rest is within.
Case (i) is fine, while case (ii) is disastrous. Case (iii) is not fine, but it admits the possibility that other (not specified) effects could be responsible for the splittings. Concerning the mixing angles, it has been stressed in refs. [6, 9] that due to the two degenerate eigenvalues of , is not uniquely defined. Hence, once the ambiguity is removed thanks to the small splittings coming from the RG running, the mixing angles may be very different from the desired ones. If such cases correspond to the previous (iii) possibility, they could still be rescued since the modifications on (of nonspecified origin) needed to reproduce the correct mass splittings will also change dramatically the mixing angles.
In section 2, we examine the general case in which the neutrino masses arise from an effective operator, remnant from new physics entering at a scale . In this framework, we assume a bimaximalmixing mass structure at the scale as an initial condition and do not consider possible perturbations of that initial condition coming from the new physics entering at . If is small the LAMSW scenario in this case is of the undecidable type [possibility (iii) above], but for above a certain value which we compute, it is excluded (the VO solution is excluded for any ).
In section 3 we consider in detail a particularly well motivated example for the new physics beyond the scale introduced before: the seesaw scenario. We include here the high energy effects of the new degrees of freedom above the scale (identified now with the mass scale of the righthanded neutrinos). We find regions of parameter space where the neutrino spectrum and mixing angles fall naturally in the pattern required to explain solar (LAMSW solution) and atmospheric neutrino anomalies, which we find remarkable. We complement the numerical results, presented in section 4, with compact analytical formulas which give a good description of them, and allow to understand the pattern of mass splittings and mixing angles obtained. We also present plausible textures for the neutrino Yukawa couplings leading to a good fit of the oscillation data.
Section 5, still in the seesaw framework, discusses several possible implications of the effect of large neutrino couplings on: triviality limits on the Majorana mass; the infrared fixed point value of the top Yukawa coupling, with consequences for the lower limit on and the value of the Higgs mass; and gauge coupling and bottomtau unification. Finally we draw some conclusions.
2 as an effective operator
In this section we will simply assume that the effective mass matrix for the lefthanded neutrinos, , is generated at some high energy scale, , by some unspecified mechanism. Assuming that below the effective theory is the MSSM with unbroken parity, the lowest dimension operator in the superpotential producing a mass of this kind is [14]
(10) 
where is a matricial coupling and is the neutral component of the Higgs field (the one coupled to the quarks). Obviously, . Between and , the effective coupling runs with the scale with a RGE [14]
(11) 
where , and are the gauge couplings, the (MSSM) top Yukawa coupling and the (MSSM) matrix of Yukawa couplings for the charged leptons respectively. Between and runs with the SM RGE [14]
(12) 
where is the SM quartic Higgs coupling and , correspond to the SM Yukawa couplings [the matching at is , ].
In a scenario of almost degenerate neutrinos, the simplest assumption for the initial form of the matricial coupling, is just the bimaximal mixing texture of eq.(8), and this was also the assumption made in refs.[6, 9]. In consequence,
(13) 
where . The last terms in eqs.(11, 12), i.e. those depending on , are generationdependent and will modify the texture, thus generating mass splittings and changing the mixing angles. It is easy to see [6, 9] that, in first approximation, the splittings have the form
(14) 
where, neglecting for a while all the charged lepton Yukawa couplings but , and working in the approximation of constant RGE functions, is given by
(15) 
Thus, the SUSY and the SM corrections have opposite signs. If , as it is the usual case, has negative sign and the most split eigenvalue is the smallest one [thus the convention of labels used in eq.(14)]. Notice also that the pure SM case is recovered setting . As in the SM case, the previous spectrum is not realistic, i.e. the splittings are barely able to reproduce simultaneously the and splittings given in eqs.(1–3). Actually, the size of the splittings is larger than in the pure SM case (due to the coefficients of the RGEs and, especially, to the dependence on ). This makes the scenario potentially more difficult than the SM one (see the discussion in the Introduction).
Fig.1 shows the complete numerical evaluation of the RGEs for eV and , which corroborates the structure of eqs.(14, 15). The splittings are always much larger than those required for the VO solution to the solar neutrino problem, eV. Therefore, the effect of the RGEs for this scenario is disastrous in the sense discussed in the Introduction for the possibility (ii). In consequence, as for the pure SM case, the VO solution to the solar neutrino problem is excluded^{3}^{3}3 Notice from the figure and from eq.(15) that there is a value of (close to ) for which the splittings vanish, but of course the finetuning required for to be close enough to that point, so that the splittings are within the VO range, is enormous.. For the LAMSW solution to the solar neutrino problem, things are a bit different. The smallest mass splitting is always within (or not far larger than) the LAMSW range, which indicates that we are in the situation explained in the Introduction (not satisfactory, but it may be rescued by extra physics). When is varied the results change with the scaling law .
The dependence of the splittings on is quite strong. Fig.2 shows this dependence for GeV. Even for moderate values of the smallest splitting is much larger than the LAMSW range, thus spoiling that solution. Therefore, for a given value of and , the viability of the supersymmetric scenario of nearly degenerate neutrinos puts an upper bound on . A reasonable estimate for this upper bound is given in Fig.3, which shows the value of (vs. ) for which the smallest splitting generated is larger (and ten times larger), than the maximum acceptable LAMSW value. The upper curve is therefore a conservative bound. These bounds become weaker if the neutrino mass, , decreases, as it follows from the previously mentioned scaling law.
Concerning the mixing angles, in a first approximation the ‘CKM’ matrix, , is given by
(16) 
which leads to mixing angles
(17) 
These values are far away from the bimaximal mixing ones. In consequence, they are not acceptable, stressing the fact that the simplified scenario discussed in this section does not work. However, as explained in the Introduction, if extra effects are able to modify the form of in order to produce the correct splittings, this modification will also change drastically the mixing angles, hopefully in a positive direction. This leads us to the seesaw scenario, which is analyzed in the next section.
3 from the seesaw mechanism
The simplest example of the kind of new physics appearing at a scale which can generate an effective mass term for the lowenergy neutrinos we observe is the socalled seesaw mechanism [15]. Its supersymmetric version has superpotential
(18) 
where is the superpotential of the MSSM. The extra terms involve three additional neutrino chiral fields (one per generation; indices are suppressed) not charged under the SM group: (). is the matrix of neutrino Yukawa couplings and is the hypercharge Higgs doublet. Now, the Dirac mass matrix is . Finally, is a Majorana mass matrix which does not break the SM gauge symmetry. It is natural to assume that the overall scale of , which we will denote by , is much larger than the electroweak scale or any soft mass. Below the theory is governed by an effective superpotential
(19) 
obtained by integrating out the heavy neutrino fields in (18). From this effective superpotential, the Lagrangian contains a mass term for the lefthanded neutrinos:
(20) 
with
(21) 
suppressed with respect to the typical fermion masses by the inverse power of the large scale .
The approximate degeneracy of neutrino masses in this framework follows from the initial condition for the neutrino mass matrix at the Planck scale: , which would lead to exactly degenerate neutrino masses. We do not address in this paper the possible origin of this very symmetric form. As explained in [9], to explore the simplest (and most natural) textures of and leading to this initial condition it is enough to consider the case in which the Majorana mass matrix is
(22) 
and all the structure is in the Yukawa matrix, which reads
(23) 
Here is the overall magnitude of and is a combination of two ‘boosts’
(24) 
with two free parameters . Having all the structure in the Majorana matrix and is equivalent to the case . We will present our results, for fixed values of and , in the plane demanding . If or are larger than 1.5 the matrix elements of are finetuned at least in a [9].
The neutrino masses, exactly degenerate at treelevel by assumption, will receive generation dependent radiative corrections which will lift that degeneracy, eventually reproducing the pattern of mass splittings necessary to interpret the experimental indications. The bulk of these radiative corrections is logarithmic and easy to compute by standard renormalization group techniques: one starts at with (22) and (23) as boundary conditions and integrates down in energy the relevant RGEs in a supersymmetric theory which is the MSSM with three righthanded neutrino chiral fields and the superpotential (18). At the scale the are decoupled and below this scale the running is performed exactly as in the previous section, except that now we have precise boundary conditions for the couplings and masses at .
From to the evolution of the relevant matrices is governed by the following renormalization group equations [16]:
(25) 
(26) 
where
(27) 
and
(28) 
(not yet given in the literature). Here and are the and gauge coupling constants, and are the Yukawa matrices for up quarks, down quarks and charged leptons.
At , decouple, and must be diagonalized to redefine the flavour basis of leptons [note that the last term in (26) produces nondiagonal contributions to ] affecting the form of the matrix. Then the effective mass matrix for the light neutrinos is .
From to , the effective mass matrix is run down in energy exactly as described in section 2.
The renormalization group equations are integrated with the following boundary conditions: and are chosen at so as to satisfy
(29) 
with the overall magnitude of fixed, for a given value of the Majorana mass , by the requirement . We will take eV as a guiding example. The boundary conditions for the other Yukawa couplings are also fixed at the low energy side to give the observed fermion masses. The free parameters are therefore and .
3.1 Analytical integration of the RGEs
It is simple and very illuminating to integrate analytically the renormalization group equations (11,25,26,28) in the approximation of constant right hand side. In this approximation (which works very well for our analysis), the effective neutrino mass matrix at lowenergy is simply plus some small perturbation. The overall mass scale is fixed to be of order 2 eV, so we need only to pay attention to the nonuniversal terms in the RGEs, which will be responsible for the mass splittings. Neglecting the , Yukawa couplings, we get the following analytical expressions (the labelling of mass eigenvalues below may not always correspond to the conventional order , ):
(30) 
where , , etc. These expressions are identical to the ones derived in [9] for the SM case, except for the numerical values of and , which are now given by:
(31) 
(32) 
The Yukawa couplings in these expressions should be chosen at an appropriate intermediate scale but the simple formulas (31,32) are good enough for our purpose. Note also that and have an implicit dependence on and, furthermore depends strongly on (due to the requirement eV).
There are important differences with respect to the SM case presented in [9]: 1) is insensitive to the Majorana threshold, its sign is negative (it was positive in the SM) and it grows in magnitude for increasing ; 2) is twice larger than in the SM and decreases slightly when increases.
In the case , the mass splittings are independent of and similar to those found in section 2, that is, not satisfactory (remember that this case is equivalent to having all the structure in , while is proportional to the identity and thus universal).
For a given , there is a critical value of below (above) which is smaller (larger) than . This critical value of increases with . When , we can further expand the neutrino masses in powers of finding
(33) 
Here we clearly see that the small mass splitting (solar) is controlled by the small parameter , proportional to the squared Yukawa couplings of the charged leptons, while (proportional to the square of the larger Yukawa coupling ) is responsible for the larger mass difference (atmospheric). Moreover, the two neutrino eigenstates closest in mass () have masses of opposite sign, which is exactly what is required to fulfill the neutrinoless double decay condition, eq.(7). In the SM case, for values of between and GeV, and have the right orders of magnitude to account for , [9]. In the supersymmetric scenario it is still the case that gives the correct atmospheric mass splitting, but for eV gives typically a one order of magnitude too high (recall here that by lowering this splitting becomes smaller following the approximate law ). This means that in this case only in particular regions of the plane, in which some mild cancellation is taking place, we will obtain the right mass splitting to solve the solar neutrino problem. This cancellation does occur in regions around the lines and for which, in the approximation (33), . In those regions then, we have a natural explanation for the mass splitting while requires only a mild finetuning of parameters.
If , then a different expansion shows that there is no natural hierarchy of mass splittings, which tend to be of the same order. In these cases, a stronger finetuning is required to get small enough and the regions in parameter space where this occurs shrink, again around . However, is still naturally of the right order of magnitude.
Turning to the mixing angles, it is easy to see that the eigenvectors of the perturbed matrix are of the form
(34) 
where are the eigenstates corresponding to the bimaximal mixing matrix, [see eq.(9)]
(35) 
In the approximation , the parameters are given by
(36) 
The vectors define the new ‘CKM’ matrix from which the mixing angles are extracted. In the case of interest, the spectrum of neutrinos consists of two lightest states () very close in mass and a heavier one . The relative ordering in mass of is not fixed and this affects the relative order of and in the matrix. This has an effect on the sign of , which is important for the MSW condition (written using the conventional order ). This will be satisfied as long as corresponds to the lightest mass eigenvalue. In other words, this condition requires that the negative mass eigenvalue, see eqs.(30, 33), corresponds to the lightest neutrino. On the other hand, the relative ordering of the two lightest neutrinos does not change the values of .
In this approximation, if just one of the two parameters is vanishing, then , i.e. exactly the bimaximal mixing case. Also, whenever , are sizeable (i.e. away from ), , and thus we are close to the bimaximal case. Therefore, it is not surprising that in most of the parameter space this will be in fact the case. This is remarkable, because it gives a natural origin for the bimaximal mixing, which was not guaranteed a priori due to the ambiguity in the diagonalization of the initial matrix, as was explained in the Introduction.
The only free parameter in the matrix is the ratio and one obtains the relations:
(37) 
where . It is instructive to invert the last relation, find in terms of , and then substitute back in the two previous relations.
Figures 4a,b present as functions of obtained in this way. For clarity, figure 4b focuses on the region of close to 1, which is the interesting one. We can impose the experimental limits on these angles directly in this figure. We see that the limit (dashed line) translates into a lower limit (the region to the left of the dashed line is forbidden). In a similar way, (dotted lines) requires either (but this solution gives and is not acceptable) or (the region to the right of the dotted line in figure 4b is allowed). Note that if we impose the stringent condition , derived from some fits to solar data as discussed in the Introduction, then no viable solution would exist. Without such condition we find that this framework can accomodate the angles required by the data with
(38) 
Since, in first approximation, the perturbed will have eigenvectors of the form (34), this result and the previous discussion are also valid for the SM case [9] and any model starting with . Actually, the results shown in eq.(37) and Fig. 4 may be considered as predictions of such a kind of scenarios.
In the next section we show that there are regions in parameter space where both the mass splittings and the mixing angles have the right values to explain the solar and atmospheric neutrino data.
3.2 Numerical Results
Figures 5 to 10 present our results for the mass splittings and mixing angles at low energy, after numerical integration of the RGEs from to as described in the previous sections. We stress that our convention in all the figures is to assign the indices 1 and 2 to the neutrino states closest in mass, with and leave the index 3 for the other neutrino state. In the interesting regions, where the parameters would fit well the solar and atmospheric data, the third neutrino is the heaviest. Hence, the neutrino spectrum has the same structure as in the SM case [9].
We start with the results for GeV and chosen as a typical example; the dependence of the results with and is discussed afterwards.
Figure 5, left plot, shows contour lines of constant (the squared mass difference between the lightest neutrinos) in the plane . The black (grey) region is excluded because there (), which is too small (large) to account for the oscillations of solar neutrinos (LAMSW solution). The white area is thus the allowed region. The line in it corresponds to . Note how the allowed area is not too large and extends around the parametric line , as explained in the previous section.
Figure 5, right plot, gives contour lines of constant . The black (grey) region is excluded because there (), which is too small (large) to account for the oscillations of atmospheric neutrinos. Again, the white area is allowed. The small black areas correspond in fact to the “undecidable” case discussed in the Introduction: they might be rescued by unspecified extra effects. We do not present a plot for because it can always be inferred from and . Moreover, in the interesting case, , one has .
The intersection of the white areas in both plots is nonzero and would give the allowed area concerning mass splittings. It is always the case that the area surrounding the origin is excluded. There, the mass differences are always of the same order, and follow the same pattern discussed in section 2 (). In any case we conclude that, away from the origin, there is a nonzero region of parameter space where , in accordance with the values required to explain the solar and atmospheric neutrino anomalies simultaneously.
For mixing angles, figure 6, left plot, gives contours of constant (one of the mixing angles relevant for atmospheric neutrino oscillations). The grey (white) area has larger (smaller) than 0.64 and is disfavored (favored) by the data (SK CHOOZ) at 99% C.L. according to the most recent analysis (last paper of ref. [10]). The line singledout corresponds to (maximum allowed value at 90% C.L. according to the same reference). Figure 6, right plot, shows contours of constant (the other mixing angle relevant for atmospheric neutrinos). The grey (white) area corresponds to smaller (larger) than 0.82, and is thus disallowed (allowed). The additional line included has .
Finally, figure 7, left plot, presents contours of constant which is relevant for oscillations of solar neutrinos. The grey (white) region has larger (smaller) than 0.99. If one is willing to interpret the existing data as implying an upper bound of 0.99 on , then the grey region would be excluded. The plotted curves give . Figure 7, right plot, shows the region of the parameter space accomplishing the resonance condition (), which is required for an efficient MSW solution of the solar anomaly (see however the first paper of ref. [10] for caveats on this issue).
The region of parameter space where all constraints on mixing angles and mass splittings are satisfied is given by the intersection of all white areas in figures 5, 6 and 7 (right plot). If is imposed, then that intersection region, including now figure 7 (left plot), is empty and no allowed region remains. It should be noticed that this fact does not come from an incompatibility between the previous constraint and the obtained from neutrinoless double decay limits, eq. (7), in the approximation. If this were the case, it could be easily solved by decreasing the overall size of the neutrino masses, , in eq.(7), and this is not the case. Indeed, eq.(7) is satisfied in nearly all the parameter space. Even where , this is still true thanks to the contribution of . What actually forbids the whole parameter space if is imposed is the incompatibility between acceptable , and angles to fit simultaneously all the neutrino oscillation data, as can be seen from the figures and in agreement with the discussion of the previous subsection, see eq.(38). This fact remains when is decreased. In fact, the effect of decreasing is essentially an amplification of the figures shown here, which comes from the fact that for a given Majorana mass, the neutrino Yukawa couplings become smaller (the effect is similar to decreasing , which is discussed below).
If the condition is relaxed (as discussed in the Introduction), then the allowed region is given by the two islands in figure 8, which is nonnegligible. It is remarkable that the small regions where all the mass splittings are of the right size and the small regions where the mixing angles are the appropriate ones have a nonzero overlapping.
The dependence of the results with the Majorana mass is illustrated by figure 9, where the allowed regions in the plane are shown for two different Majorana masses, and GeV, keeping . For comparison with the allowed region for GeV (given in figure 8) we see that for small the allowed region grows and flies away from the origin, until it leaves the naturalness region . For GeV no allowed region inside the natural range for remains. Conversely, increasing reduces the allowed region, which gets closer to the origin (at GeV the allowed region becomes extinct).
Concerning the remaining parameter, , figure 10 gives the allowed regions for GeV and four different values of : 1.75, 3, 4 and 6.5, as indicated. The minimum in is dictated by the requirement of perturbativity of all couplings up to the Planck scale, banning the presence of a Landau pole below it, as discussed in section 5. The allowed region gets smaller and smaller when increases. As explained in the previous section, grows with making harder and harder a cancellation that gives the correct and mixing angles. Eventually, for the allowed region disappears (for GeV that value is ).
Finally, let us stress that, if the condition is imposed, the whole parameter space becomes disallowed for any value of and . We also find that, whenever there is a hierarchy in the mass splittings, the two lightest eigenvalues have opposite signs. This is just what is needed to have a cancellation occurring in the neutrinoless double decay constraint (7). This constraint is satisfied in almost the whole parameter space for any .
4 Examples of acceptable ansätze
It is possible to find examples of matrices which both fall inside the allowed areas presented in the previous section and are natural (perhaps pointing to a possible underlying symmetry).
An example, already presented in [9], which still survives in the supersymmetric case (for and ) is
(39) 
It corresponds to and . The mass splittings are
(40) 
and the mixing angles
(41) 
with at the border of the resonance condition for the MSW mechanism.
Another examples of working ansätze can be obtained. For instance, the following ansatz (corresponding to , )
(42) 
works correctly for GeV and , giving
(43) 
and the mixing angles
(44) 
with . It could be interesting to explore possible symmetries that may be responsible for the form of these ansätze and to analyze their implications for future longbaseline experiments [18].
5 Other relevant implications of neutrinoinduced radiative corrections
The previous sections focussed on the possibility of reproducing the “observed” neutrino mass splittings from supersymmetric radiative corrections. In particular we analyzed the case of nearly degenerate neutrinos, which is closely related to the bimaximal mixing scenario.
In this section, still working in a seesaw framework, we take a different point of view and analyze the physical impact of neutrinoinduced radiative corrections. The results of this section are quite generic. They are not associated to the bimaximal mixing scenario, not even to a scenario of degenerate neutrinos, but we will take this case as a representative example to illustrate the phenomena. Some of the following effects have already been mentioned along the paper, but here they are studied in greater detail.
The first topic concerns the appearance of Landau poles associated to the neutrinos and the corresponding implications. It is easy to check from eq.(25) that the neutrino Yukawa couplings, , have a supersymmetric RGE quite similar to the top Yukawa coupling. It is therefore not surprising that they can develop Landau poles at high energy in a similar way. Obviously, the larger the righthanded neutrino Majorana mass, , and the larger the low energy neutrino masses, , the larger the neutrino Yukawa couplings, thus lowering the scale at which the Landau pole appears. Consequently, if, in order to preserve perturbation theory (and the nice supersymmetric gauge coupling unification), we demand that the Landau pole does not appear below (or the preferred highenergy scale), this puts upper bounds on and (see ref.[17] for the analogue in the SM framework).
To analyze this effect, we will use the simplest textures of and , leading to a bimaximal mass matrix, namely and [ has exactly the form given in eq.(