We give analytic approximations to the baryon asymmetry produced by thermal leptogenesis with hierarchical right-handed neutrinos. Our calculation includes flavour-dependent washout processes and CP violation in scattering, and neglects gauge interactions and finite temperature corrections. Our approximate formulae depend upon the three CP asymmetries in the individual lepton flavours as well as on three flavour-dependent efficiency factors. We show that the commonly used expressions for the lepton asymmetry, which depend on the total CP asymmetry and one single efficiency factor, may fail to reproduce the correct lepton asymmetry in a number of cases. We illustrate the importance of using the flavour-dependent formulae in the context of a two right-handed neutrino model.Note Added: In the published version of this paper, we applied incorrectly the A matrix [8], which relates the asymmetries in B/3 − L α to the asymmetries Y αα carried by the lepton doublets. This over-estimated the baryon asymmetry by a factor of order one (or a few). In this revised version, we modify sections 4.1.4 and 4.2 to give a more correct relation between our analytic estimates for the flavour asymmetries, and the final baryon asymmetry. We thank E.J. Chun and M. Plumacher for discussions on this issue, and in particular A Strumia for details of his calculations and results.Second Note Added: There are various other errors (we thank S Antusch, P Di Bari, EJ Chun, and C Gonzalez-Garcia for comments) which we correct in this version:1. In the previous net version (modified to use more correctly the A matrix), the overall sign of the baryon asymmetry was wrong in eqns (55) and (57).2. The coefficient of the washout term in the Boltzmann Equations was a factor of 2 too large: in the last term of eqns (11), (18), and (36), and in the corresponding coefficients of the analytic solutions, the numerical factor should be 1/4 (not the 1/2 of the published version).3. The vev of the Higgs should be taken 175 GeV, not 246 GeV(given after eqn (1)). This changes m * from the 2 − 3 × 10 −3 eV used in the published version (at eqns (5),(20)) to m * ≃ 10 −3 eV. NB: in the analytic approximations this erroneous factor of 2 cancels against the one of point 2 above. So the analytic approximations remain acceptable , using m * ≃ 10 −3 eV. (22), (23) and (28) have been corrected.
Factors of two in eqns
5.In the published version, the captions of Figures 1,2, and 3 indicate that e was taken distinguishable from µ. However, for M 1 > 10 9 GeV, only the τ is distinguishable 1 . We therefore interchanged the e and τ indices on the inputs (ǫ ee → ǫ τ τ , K τ τ → K ee ,...). We also changed a washout factor in the caption of Figure 2 to account for points 2 and 3 above. We have remade the plots, with these modified inputs, and the plots remain unchanged.
