Let us allow now the second heavy RH neutrino to be close to the lightest one,. How does the overall picture change? There are two crucial points to understand: The second RH neutrino will contribute to the wash-out of the asymmetry produced by the first one if the mass difference is sufficiently small. The CP asymmetry is enhanced as for The first effect is well illustrated in the following two plots: Leptogenesis beyond the limit of hierachical heavy neutrino masses* Introduction Steve Blanchet, in collaboration with Pasquale Di Bari Max-Planck-Institut für Physik, Munich Leptogenesis was first proposed by Fugukita and Yanagida in It stands for the generation of a lepton asymmetry (L) via the decay of very heavy right-handed (RH) neutrinos and its subsequent conversion into a baryon asymmetry (B) by the so-called non-perturbative sphaleron processes. Thus, it allows an explanation to one of the most outstanding problems in modern cosmology: where does the present baryon asymmetry of the Universe come from, or why is the Universe made of matter rather than anti-matter ? This scenario has become more and more appealing with the growing evidence for non-zero neutrino masses in the 90´s, which are “naturally” included in it. Leptogenesis is indeed based on the following minimal extension of the Standard Model: The N 1 -dominated scenario In the vast majority of the papers on leptogenesis, a hierarchical heavy neutrino spectrum is assumed, which allows for a number of simplifications. E.g. it usually implies that only the asymmetry generation of the lightest RH neutrino should be taken into account, the other two contribution being efficiently washed out: this is the N 1 -dominated scenario. For the present purposes, considering only decays and inverse decays is sufficient to capture all the essential features of leptogenesis. The Boltzmann equations in the N 1 -dominated scenario are then the following, with a : where the capital letters D and W denote decay and wash-out, respectively. Key quantities for leptogenesis The asymmetry produced by each decay is given by the CP asymmetry parameter : At high energies, this simple extension will allow, via the Yukawa coupling h, the heavy (Majorana) neutrinos to decay asymmetrically into leptons and Higgs bosons (L, and thus B-L are produced), and the lepton asymmetry will be partly converted into a baryon asymmetry by the sphaleron processes (B-L conserving). At low energies, below spontaneous symmetry breaking scale, very small masses compared to all the other fermions for the “usual” (active, mainly left- handed,…) neutrinos (also Majorana) will be naturally generated thanks to the well-known see-saw mechanism, because the right-handed neutrinos are taken to be very heavy,, and the active neutrinos masses are correspondingly suppressed: For the thermodynamical description of the decays of the right-handed neutrinos, a useful quantity is the decay parameter where is the Hubble parameter. The so-called strong wash-out, for jjölkjöll is favoured over the weak wash-out because it allows the whole picture to be self-contained, with no dependence on the initial conditions. Besides decays, there are other processes, especially inverse decays, which are relevant not only for producing the RH neutrinos, but also for washing out part of the asymmetry produced by the decays. The effect of production and wash-out are accounted for by the efficiency factors, so that the final B-L asymmetry can be written where and are the Higgs and left-handed lepton doublets, respectively, and. where we denote by any particle number or asymmetry X calculated in a portion of comoving volume containing one heavy neutrino in ultra-relativistic thermal equilibrium. Finally, the baryon-to-photon number ratio at the recombination time is calculated as where is the sphaleron conversion coefficient. Beyond the hierarchical limit These equations can be solved, i.e. the efficiency factor can be found as a function of the temperature (or ). We show here one case of weak wash-out (initial thermal abundance) and one of strong wash-out: The difference between the two cases is striking: In the weak wash-out regime (left panels), each decay contributes to the final asymmetry for any value of. In the strong wash-out regime (right panels), all the asymmetry produced at d is efficiently washed out by inverse decays, so that only decays occurring at give a contribution to the final asymmetry. On the left panel, one clearly sees how well the hierarchical limit (and thus the N 1 -dominated scenario) is recovered, because the asymmetry is generated by N 1 when the wash-out from N 2 inverse decays are already switched off. On the right panel, one notices that the final asymmetry (around ) is quite suppressed compared to the left panel, because the wash-out of the second RH neutrino is acting on the asymmetry generation by the first! One may wonder for which one recovers the hierarchical limit within a 10 % precision. This is shown in the next plot, both for the efficiency factor and for the final B-L asymmetry, which involves the CP asymmetry as well: * See hep-ph/ and references therein. ISAPP 2006, Neutrinos in Physics, Astrophysics and Cosmology, May 2006, Munich