Leptogenesis in models with a left-right symmetric seesaw mechanism
We study leptogenesis in supersymmetric models with a left-right symmetric seesaw mechanism, including flavour effects and the contribution of the next-to-lightest right-handed neutrino. Assuming and hierarchical light neutrino masses, we find that successful leptogenesis is possible for 4 out of the 8 right-handed neutrino mass spectra that are compatible with the observed neutrino data. An accurate description of charged fermion masses appears to be an important ingredient in the analysis.
pacs:12.10.DmUnified theories and models of strong and electroweak interactions and 14.60.StNon-standard-model neutrinos, right-handed neutrinos, etc.
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Testing the seesaw mechanism seesaw () is almost certainly an hopeless goal, except for specific low-energy realizations. The main reasons we have to believe in it are its elegance and the fact that it fits so nicely into unification. This motivates us to investigate its observable implications, such as leptogenesis FY86 () and, in supersymmetric theories, lepton flavour violation.
So far most studies of leptogenesis have been done in the framework of the type I (heavy right-handed neutrino exchange) seesaw mechanism, or assumed dominance of either the type I or the type II (heavy scalar triplet exchange) seesaw mechanism. It is interesting, though, to investigate whether the generic situation where both contributions are comparable in size can lead to qualitatively different results. A further motivation to do so comes from the well-known fact that successful leptogenesis is difficult to achieve in models with a type I seesaw mechanism, which generally111This might not be the case in models where the relation receives large corrections from Yukawa couplings involving a or Higgs representation, or from non-renormalizable interactions. present a very hierarchical right-handed neutrino mass spectrum, with lying below the Davidson-Ibarra bound DI02 ().
2 Right-handed neutrino spectra in the left-right symmetric seesaw mechanism
2.1 The left-right symmetric seesaw mechanism
In left-right symmetric extensions of the Standard Model, the light neutrino mass matrix is often given by the following formula seesawII ():
In Eq. (1), is the scale of breaking, is the electroweak scale, and is the vev of the heavy triplet. A discrete left-right symmetry ensures that a single symmetric matrix determines both the couplings of the triplet to lepton doublets, to which the type II contribution (first term) is proportional, and the right-handed neutrino mass matrix , which enters the type I contribution (second term). The discrete symmetry also constrains the Dirac coupling matrix to be symmetric.
In order to study leptogenesis, the knowledge of the masses and couplings of the right-handed neutrinos and of the triplet is needed. Therefore, in a theory which predicts the Dirac matrix , one must solve Eq. (1) for the couplings, assuming a given pattern for the light neutrino masses and mixings. In Ref. AF05 (), it was shown that this “reconstruction” problem has exactly solutions for families, and explicit expressions for the ’s were provided up to . Here we use the alternative reconstruction procedure proposed in Ref. HLS ().
2.2 Reconstruction procedure
In order to solve Eq. (1), we first rewrite it as
with , and
where is a matrix such that , and is assumed to be invertible. Being complex and symmetric, can be diagonalized by a complex orthogonal matrix if its eigenvalues (i.e. the roots of the characteristic polynomial ) are all distinct:
The right-handed neutrino masses are obtained by diagonalizing with a unitary matrix , and the couplings of the right-handed neutrino mass eigenstates are given by .
Since each equation has two solutions and , there are 8 different solutions for the matrix , which we label in the following way: refers to the solution , to the solution , and so on. It is convenient to define and such that, in the limit:
With this definition, the large limit () of solutions and corresponds to the “pure” type I and type II cases, respectively:
The remaining 6 solutions correspond to mixed cases where the light neutrino mass matrix receives significant contributions from both types of seesaw mechanisms. In the opposite, small limit (), one has , which indicates a partial cancellation between the type I and type II contributions to light neutrino masses.
2.3 Application to models
Let us now apply the reconstruction procedure to supersymmetric models with two s, a and a representations in the Higgs sector. The two s generate the charged fermion masses, leading to the well-known relations:
The and the contain the representations needed for the left-right symmetric seesaw mechanism. In particular, the triplet as well as the triplet whose vev breaks are components of the . The equality and the symmetry of are ensured by gauge symmetry.
Then, for a given choice of the light neutrino mass parameters and of the high energy phases contained in , the matrix is known222The implicit additional inputs are (we choose ) and the values of the up quark masses and of the CKM matrix at the seesaw scale. and can be reconstructed as a function of the breaking scale and of . Perturbativity of the couplings constrains and restricts the range of from above. In Fig. 1, we show the right-handed neutrino mass spectrum of three representative solutions as a function of for a hierarchical light neutrino mass spectrum. The 4 solutions with are characterized by a constant value of the lightest right-handed neutrino mass, GeV; the 2 solutions with and by GeV; and the 2 solutions with and by a rising .
3 Implications for leptogenesis
Since and in all solutions, one can safely assume that the triplet is heavier than the lightest right-handed neutrino. Then the dominant contribution to leptogenesis comes from out-of-equilibrium decays of (in some cases to be discussed below, the next-to-lightest neutrino will also be relevant). The CP asymmetry in decays, , receives two contributions: the standard type I contribution FY86 (); epsilonI (), and an additional contribution from a vertex diagram containing a virtual triplet epsilonII (); HS03 ():
where , , , , and . The final baryon asymmetry is given by:
The behaviour of the different solutions can be anticipated from the observation of the mass spectra in Fig. 1 HLS (). Indeed, successful leptogenesis requires , while for Eqs. (10) and (11) yield the upper bound HS03 ():
Thus, the 4 solutions with will fail to generate the observed baryon asymmetry from decays, a conclusion that generalizes a well-known fact in the type I case. However, decays can do the job if they generate a large asymmetry in a lepton flavour that is only mildly washed out by decays and inverse decays Vives05 (). The 2 solutions with and have a rising and should be able to reproduce the observed asymmetry, as in the pure type II case. Finally, the situation is less conclusive for the 2 solutions with and , for which flavour effects and the contribution of could be decisive.
It is clear from the above discussion that a careful study of leptogenesis requires the inclusion of the next-to-lightest right-handed neutrino and of flavour effects BCST00 (). As is well known in the type I case, flavour effects can significantly affect the final baryon asymmetry if there is a hierarchy between the washout parameters for different lepton flavours flavour (). We performed such an analysis in Ref. AHJL (), and present our results here. Fig. 2 shows the final baryon asymmetry as a function of for solutions , and . Not surprisingly, the solution leads to successful leptogenesis; however there is a tension with the upper bound on the reheating temperature from gravitino overproduction gravitino () above GeV, where GeV. By contrast, the solutions and fail to reproduce the observed baryon asymmetry333In Ref. ABHKO06 (), a different conclusion has been obtained for the solution in the case of an inverted light neutrino mass hierarchy. . In the case, flavour effects prevent an exponential washout of the asymmetry generated in decays ( decays alone would give ), but this is not sufficient for “ leptogenesis” to work.
However, this is not the whole story, since the above results were obtained assuming the mass relation , which is in gross conflict with experimental data. Corrections to this formula, e.g. from non-renormalizable operators of the form , will modify the reconstructed ’s by introducting a mismatch between the bases of charged lepton and down quark mass eigenstates. Fig. 3 shows how the final baryon asymmetry is modified when the effect of is taken into account. We can see that several choices for (the measured charged lepton and down quark masses do not fix all parameters in ) lead to successful leptogenesis in the case, but not in the case. There is some tension between successful leptogenesis and gravitino overproduction in the solution but, exactly as in the solution, the observed asymmetry is generated over a significant portion of the parameter space with GeV.
We have studied leptogenesis in supersymmetric models with a left-right symmetric seesaw mechanism, including flavour effects and the contribution of the next-to-lightest right-handed neutrino. Assuming the relation and a hierarchical light neutrino mass spectrum, we found that the “type II-like” solutions and , as well as the solutions and , can lead to successful leptogenesis. An accurate description of charged fermion masses was a crucial ingredient in the analysis. By contrast, the solution fails to generate the observed baryon asymmetry from decays, and a similar conclusion holds for the 3 other solutions with if one requires GeV.
Some comments about the generality of our results are in order: (i) Although the above results were obtained for , the same qualitative behaviour of the 8 solutions is expected for a more generic hierarchical Dirac matrix. Of course, whether leptogenesis is successful or not in a given solution can only be decided on a model-by-model basis; (ii) At the quantitative level, different input parameters (other than the various phases and ) can significantly affect the results presented in Figs. 1 to 3. This is most notably the case of the light neutrino mass parameters: , and the type of the mass hierarchy (see Ref. AHJL () for details). Also, corrections to the relation could have a significant impact, since e.g. both in the solution and in the solution are proportional to .
This work has been supported in part by the RTN European Program MRTN-CT-2004-503369, the Marie Curie Excellence Grant MEXT-CT-2004-014297, and the French Program “Jeunes Chercheurs” of the Agence Nationale de la Recherche (ANR-05-JCJC-0023). PH and SL would like to thank Carlos Savoy for a pleasant and fruitful collaboration on Ref. HLS ().
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