Scholarly article on topic 'The LMA solution from bimaximal lepton mixing at the GUT scale by renormalization group running'

The LMA solution from bimaximal lepton mixing at the GUT scale by renormalization group running Academic research paper on "Physical sciences"

CC BY
0
0
Share paper
Academic journal
Physics Letters B
OECD Field of science
Keywords
{"Renormalization group equation" / "Neutrino masses" / "LMA Solution"}

Abstract of research paper on Physical sciences, author of scientific article — Stefan Antusch, Jörn Kersten, Manfred Lindner, Michael Ratz

Abstract 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.

Academic research paper on topic "The LMA solution from bimaximal lepton mixing at the GUT scale by renormalization group running"

Physics Letters B 544 (2002) 1-10

www. elsevier. com/locate/npe

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

Stefan Antusch, Jörn Kersten, Manfred Lindner, Michael Ratz

Physik-Department T30, Technische Universität München, James-Franck-Straße, 85748 Garching, Germany Received 17 June 2002; received in revised form 31 July 2002; accepted 7 August 2002 Editor: P.V. Landshoff

Abstract

We show that in see-saw models with bimaximal lepton mixing at the GUT scale and with zero CP phases, the solar mixing angle 6\2 generically evolves towards sizably smaller values due to renormalization group effects, whereas the evolution of 613 and $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. © 2002 Elsevier Science B.V. All rights reserved.

PACS: 11.10.Hi; 14.60.Pq

Keywords: Renormalization group equation; Neutrino masses; LMA Solution

1. 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 012 [1-4]. 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 012 from maximal mixing, while keeping 023 maximal and 013 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 [7-10]. Others focused on the case of nearly degenerate neutrinos [11-16], on the existence of fixed points [17], or on the effect of non-zero Majorana phases on the stability of the RG evolution [18].

E-mail addresses: santusch@ph.tum.de (S. Antusch), jkersten@ph.tum.de (J. Kersten), lindner@ph.tum.de (M. Lindner), mratz@ph.tum.de (M. Ratz).

0370-2693/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S03 70-2693(02)02470-X

Table 1

Experimental data for the neutrino mixing angles and mass squared differences. For the solar angle $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 6 e [0°, 45°]) C.L.

012 [°] 32.9 26.1^3.3 99%(3a) e23 [ °] 45.0 33.2-45.0 99% (3a)

013 [ °] - 0.0-9.2 90% (2a) Am2ol [eV2] 5 x 10-5 2.3 x 10-5-3.7 x 10-4 99% (3a) |Am2tm I [eV2] 2.5 x 10-3 1.2 x 10-3-5 x 10-3 99% (3a)

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 [20-25].

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.

2. 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

„bimax

= v[ — , 0, — I -diag(mi,m2, m3) • V I —,0, — ) =

'a — b c —cs

c ab \ — c b a /

/ C12C13 Î12C13 S13 \

V (012,013,023) =1 — C23S12 — S23S13C12 C23C12 — S23S13C12 S23C13I

V S23S12 — C23S13C12 —S23C12 — C23S13S12 C23C13 /

with sij = sin 0ij and cij = cos 0ij is the (orthogonal) CKM matrix in standard parametrization, and 1

a = — (m 1 + rn.2 + 2 »13), 1

b = — (—mi — m2 + 2 »13), «2 — «1

(3a) (3b) (3c)

Fig. 1. Possible mass hierarchies for the light neutrinos. We use the convention that m\ and m2 are chosen in such a way that 0 < 6\2 ^ 45° The LMA solution then requires m2 > m 1.

Inverting Eqs. (3) yields the mass eigenvalues

m\=a — b — V2c, (4a)

m2=a — b + V2c, (4b)

m3 = a + b. (4c)

From Eq. (3a) we see that a > 0. Eqs. (4) imply that the solar mass squared difference Am2ol = m 2 — m 2 is related to c, while the atmospheric one, Am^tm = m2 — m^, 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, m1 < m2, otherwise m1 > m2. Hence, Am2ol 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 < 012 < 450. This can always be accomplished by the replacement c ^—c.

In our see-saw scenario, the effective mass matrix of the light neutrinos is V 2

mbimax=^yrM-lyv (5)

at the high-energy scale, with (0) = ^fiL ~ 174 GeV. Obviously, the singlet Yukawa and mass matrices Yv and M cannot be determined uniquely from this relation, i.e., there is a set of {YV,M} configurations that yield bimaximal mixing. After choosing an initial condition for YV, M (and thus the see-saw scales) is fixed by the see-saw formula (5).

3. 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

d (n) (n) /r

(n) (n) (n) (n) (n)

for all the parameters Xi, X j £ { k ,Yv, 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.

4. 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 012 changes drastically, while the changes in 013 and 023 are comparatively small. This agrees remarkably well with the experimentally favored scenario.

Fig. 2. RG evolution of the mixing angles from the GUT scale to the SUSY-breaking scale (taken to be ^ 1 TeV) in the MSSM extended by heavy singlets for a normal mass hierarchy and Yv = X diag(1,s, s2) with tan fi = 5, s = 0.525, a = 0.0675 eV and X = 1. In this example, the lightest neutrino has a mass of 0.025 eV. The kinks in the plots correspond to the mass thresholds at the see-saw scales. The grey-shaded regions mark the various effective theories.

Fig. 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 Yv = X diag(1,s, s2) with s = 0.65, a = 0.0655 eV and X = 1. In this example, the lightest neutrino has a mass of 0.024 eV.

5. 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

16tc fi-—mv ■ dx

= Ce\Y}Ye\mv + Cemv[Y}Ye] + C^Y^Y^1 mv + Cvm^Y]Yv]

+ terms with trivial flavour structure (7)

with Ce = -3/2, Cv = 1/2 in the SM and Ce = Cv = 1 in the MSSM. Analogously to Eq. (1), mv is parametrized by

mv(t) = V(0i2(t),0i3(t),023(t^mdiag(t)yr(0i2(t),0i3(t),023(t^,

where /x is the renormalization scale, t := In jj-. and wdiag := diag(/«i. mi- m3). In general, the real Yv can be written as 0

Yv = V(^12,^13,^32)diag(yi,y2,y3)VT (012,013,032).

However, the effective mixing matrix mv is invariant under the transformations Yv ^ VTYv and M ^ VTMV, which correspond to a change of basis for the heavy sterile neutrinos. Thus, V(f12,f13,f32) in Eq. (9) can be absorbed into M, leading to the simpler parametrization

Yv(yi,y2,y3,012,013,032) = diag(yi,y2,y3)V (012,013,032).

Furthermore, we use the approximation that the effect of the charged lepton Yukawa matrices Ye can be neglected compared to that of the neutrino Yukawa matrix. Note that in the MSSM a large tan j can yield a relatively large Ye, which can also have sizable effects that are neglected in this approximation.

We now differentiate Eq. (8) w.r.t. t and insert the RGE (7). For the evolution of the mixing angles at the GUT scale with bimaximal mixing as initial condition, we thus obtain both in the SM and in the MSSM the ratios

012 2\fl(m\m2)(m3 — m\)(m3 — m2)F\

013 Imgut (m2 — TMI)[8(»Z3 — m\m2)F2 + 4-^/2 (m2 — m^m^F^]

1 m2 + mi Fi 2V2 m2 - mi F2 1 Arn2atm Fx

2V2 At

for hierarchical neutrino masses1, for degenerate neutrino masses,

012 023

2-1/2 (mi + 7M2)(/M3 — m\)(m3 — m2)F\ MGUT (m2 ~ tmi)[8(tm2 - m\)m3F2 + 4^2 (mj - m\m2)F3\

1 m 2 + m 1 F1

2 m 2 - m 1 F3

1 Am2tm F1

for hierarchical neutrino masses1 , for degenerate neutrino masses

Fi = (y2 - y|){cos(20i2^(cos(20i3) - 3) sin(2023) - 6cos2 (0O)] - 4cos(2023) sin(20i2) sin(0i3)}

1 Note that this approximation is also valid for a relatively weak hierarchy, where «3 is a few times larger or smaller than mi, «2.

+ (y? + - 2y3) [cos(20i3)(sin(2<fe) - 3) + (1 + sin(2<fe))],

F2 = 2(y2 - yf) cos(<fe) sin(2012^sin(023) - cos(<fe)) - (y2 + y^ - 2y32 + (y2 - yf) cos(2012)) X sin(2013^cos(023) + sinOfo)),

F3 = (y2 - y!)[cos(2012^cos(2013) - 3)cos(2<fe) + 4sin(2012) sin(<fe) sin(2<fe)] + 2(y! + y2 - 2y!) cos2(013)cos(2<fe).

This result can also be obtained from the formulae derived in [26]. The constants F1, F2 and F3 clearly depend on the choice of Yv(MGUT). However, unless the parameters {y1,y2,y3,$12,$13,$32} are fine-tuned, we expect the ratios F1 /F2 and F1 /F3 to be of the order one. Consequently, the RG change of 012 is larger than that of the other angles if the mass-dependent factors in Eqs. (11a) and (11b) are large. This is always the case for degenerate neutrino masses, since Am^lm > Am2ol. As (m1 - m2) is related to the small solar mass squared difference, it is also true for non-degenerate mass schemes, unless m1 is very small, in which case the ratio approaches 1. This corresponds to a normal mass hierarchy and a strongly hierarchical mass scheme. Finally, it can be shown that the running of 012 is always enhanced compared to that of 013 and 023 for inverted schemes. Hence, we conclude that this is a generic effect.

6. Parameter space regions compatible with the LMA solution

6.1. 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 Yv at the GUT scale. We assume that it is diagonal and of the form

Yv and M are now determined by the parameters {e, 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, e and a. The parameter e controls the hierarchy of the entries in Yv and thus the degeneracy of the see-saw scales, while a determines the mass of the lightest neutrino. The dependence of physical quantities on e 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.

6.2. 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 j = 5) with a normal mass hierarchy. We find that for the form of Yv 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 Yv at the GUT scale. One also has to ensure that the sign of Am2ol 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 Am2ol is not defined by the initial conditions. Using the analytic approximation

Yv = X diag(1,e,e2).

0.04 0.15 0.25 0.10 0.55 0.99

a [eV] £

(a) Parameter a 0) Parameter e

Fig. 4. Plot (a) shows the mass of the lightest neutrino (at low energy) as a function of a for the SM and the MSSM with normal mass hierarchy, X = 1 and e e [0.1, 0.99] (grey region). Plot (b) shows the degeneracy of the see-saw scales, parametrized by ln(M3/M\) (at the GUT scale), as a function of e for the same cases with a e [0.04 eV, 0.25 eV] (grey region).

0.04 0.15 0.25 0.04 0.15 0.25

a [eV] a [eV]

(a) SM (b) MSSM

Fig. 5. Parameter space regions compatible with the LMA solution of the solar neutrino problem for the example Yv = diag(1,e, e2). The initial condition at the GUT scale Mgut = 1016 GeV is bimaximal mixing, and the comparison with the experimental data is performed at the EW scale or at 1 TeV for the SM and the MSSM, respectively. The white regions of the plots are excluded by the data (LMA) at 3a. For this example, we consider the case of a normal neutrino mass hierarchy and X = 1 for the scale factor of the neutrino Yukawa couplings.

of Section 5, the sign just below the GUT scale can be calculated. We find Am2ol > 0 for F\ < 0 and vice versa. However, in order to predict the sign of Am2ol at low energy, the numerical RG evolution has to be used. This excludes some of the possible choices for the neutrino Yukawa coupling Yv at the GUT scale. For example, among the possibilities with diagonal Yv it excludes Yv = diag(e2,e, 1).

6.3. Dependence on the scale X of the neutrino Yukawa coupling

For small values of X, the contribution from Yv to the evolution of the mixing angles above the largest see-saw scale is suppressed by a factor of X2. 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 012 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.

Fig. 6. RG evolution in the SM for X = 0.01, e = 0.3, a = 0.0535 eV and a normal mass hierarchy. The running from bimaximal mixing to the LMA solution now takes place between the see-saw scales. In this example, the lightest neutrino has a mass of 0.017 eV.

8 10 log10WGeV)

Fig. 7. RG evolution in the SM with a negative CP parity for m2, X = 0.5, e = 3.5 x 10-3 and a normal mass hierarchy. The running from bimaximal mixing to the LMA solution takes place between the see-saw scales. In this example, we consider a strongly hierarchical mass spectrum. The lightest neutrino has a mass of 0.004 eV. Note that the cases $23 > 45°, Am^ > 0 and $23 := 90° - $23 < 45°, Am2,3 := -Ami^ are indistinguishable in neutrino oscillations.

6.4. 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 m2 is shown in Fig. 7. For this we have chosen a different diagonal structure for Yv,

Yv = X diag( e2,e, 1),

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 m1 and m2 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.

6.5. Low scale values of 613 and 023

The mixing angles 013 and 023 are affected by the RG evolution as well, i.e., they do not stay at their initial values 013 = 0°and 023 = 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 A0n = 0.02°, which corresponds to sin2(2013) = 5 x 10-7, and A623 = 0.28°. On the other hand, other choices of Yv at the GUT scale produce A013 and A023 that come close to the experimental bounds. This can make it possible to discriminate between models with different initial values Yv (MGUT).

7. 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 012 evolves towards sizably smaller values, whereas the change of 013 and 023 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.

Acknowledgements

We would like to thank P. Huber for useful discussions. This work was supported in part by the "Sonderforschungsbereich 375 für Astro-Teilchenphysik der Deutschen Forschungsgemeinschaft".

References

[1] V. Barger, D. Marfatia, K. Whisnant, B.P. Wood, Phys. Lett. B 537 (2002) 179, hep-ph/0204253.

[2] A. Bandyopadhyay, S. Choubey, S. Goswami, D.P. Roy, hep-ph/0204286.

[3] J.N. Bahcall, M.C. Gonzalez-Garcia, P. Peña-Garay, hep-ph/0204314.

[4] P.C. de Holanda, A.Yu. Smirnov, hep-ph/0205241.

[5] T. Toshito, etal., Super-Kamiokande Collaboration, hep-ex/0105023.

[6] M. Apollonio, et al., CHOOZ Collaboration, Phys. Lett. B 466 (1999) 415, hep-ex/9907037.

[7] M. Tanimoto, Phys. Lett. B 360 (1995) 41, hep-ph/9508247.

[8] K.R.S. Balaji, A.S. Dighe, R.N. Mohapatra, M.K. Parida, Phys. Lett. B 481 (2000) 33, hep-ph/0002177.

[9] T. Miura, E. Takasugi, M. Yoshimura, Prog. Theor. Phys. 104 (2000) 1173, hep-ph/0007066.

[10] G. Dutta, hep-ph/0203222.

[11] J.R. Ellis, S. Lola, Phys. Lett. B 458 (1999) 310, hep-ph/9904279.

[12] J.A. Casas, J.R. Espinosa, A. Ibarra, I. Navarro, Nucl. Phys. B 556 (1999) 3, hep-ph/9904395.

[13] J.A. Casas, J.R. Espinosa, A. Ibarra, I. Navarro, Nucl. Phys. B 569 (2000) 82, hep-ph/9905381.

[14] P.H. Chankowski, A. Ioannisian, S. Pokorski, J.W.F. Valle, Phys. Rev. Lett. 86 (2001) 3488, hep-ph/0011150.

[15] M.-C. Chen, K.T. Mahanthappa, Int. J. Mod. Phys. A 16 (2001) 3923, hep-ph/0102215.

[16] M.K. Parida, C.R. Das, G. Rajasekaran, hep-ph/0203097.

[17] P.H. Chankowski, W. Krolikowski, S. Pokorski, Phys. Lett. B 473 (2000) 109, hep-ph/9910231.

[18] N. Haba, Y. Matsui, N. Okamura, Eur. Phys. J. C 17 (2000) 513, hep-ph/0005075.

[19] S.F. King, N.N. Singh, Nucl. Phys. B 591 (2000) 3, hep-ph/0006229.

[20] S. Antusch, J. Kersten, M. Lindner, M. Ratz, Phys. Lett. B 538 (2002) 87, hep-ph/0203233.

[21] P.H. Chankowski, Z. Pluciennik, Phys. Lett. B 316 (1993) 312, hep-ph/9306333.

[22] K.S. Babu, C.N. Leung, J. Pantaleone, Phys. Lett. B 319 (1993) 191, hep-ph/9309223.

[23] S. Antusch, M. Drees, J. Kersten, M. Lindner, M. Ratz, Phys. Lett. B 519 (2001) 238, hep-ph/0108005.

[24] S. Antusch, M. Drees, J. Kersten, M. Lindner, M. Ratz, Phys. Lett. B 525 (2002) 130, hep-ph/0110366.

[25] S. Antusch, M. Ratz, hep-ph/0203027.

[26] J.A. Casas, J.R. Espinosa, A. Ibarra, I. Navarro, Nucl. Phys. B 573 (2000) 652, hep-ph/9910420.

[27] S. Antusch, J. Kersten, M. Lindner, M. Ratz, in preparation.