Corrigendum to"Flavour Covariant Transport Equations: an Application to Resonant Leptogenesis"

We amend the incorrect discussion in Nucl. Phys. B 886 (2014) 569 [1] concerning the numerical examples considered there. In particular, we discuss the viability of minimal radiative models of Resonant Leptogenesis and prove that no asymmetry can be generated at $\mathcal{O}(h^4)$ in these scenarios. We present a minimal modification of the model considered in [1], where electroweak-scale right-handed Majorana neutrinos can easily accommodate both successful leptogenesis and observable signatures at Lepton Number and Flavour Violation experiments. The importance of the fully flavour-covariant rate equations, as developed in [1], for describing accurately the generation of the lepton asymmetry is reconfirmed.

there is related to the usage, in our numerical analysis, of the incorrect formulae (2.9) and (2.13) of [2], reported in (5.10) of [1].

No-go theorem for minimal radiative RL at O(h 4 )
The relevant heavy-neutrino Lagrangian is given by where Φ = iσ 2 Φ * is the isospin conjugate of the Higgs doublet Φ and the superscript C denotes charge conjugation. In minimal radiative scenarios, the masses of these heavy neutrinos N α (α = 1, 2, 3) are assumed to be degenerate at a high scale µ X ∼ 10 16 GeV, thanks to an approximate O(3) symmetry: M N (µ X ) = m N 1 3 . At the scale m N , relevant for leptogenesis, the mass matrix M N is obtained by the RG evolution from µ X to m N : where, in the minimal radiative RL scenario, ∆M RG N is taken to be the only O(3)-breaking correction to the mass matrix and is given by However, as we are going to show below, this minimal scenario is not viable at O(h 4 ), because of the following no-go theorem for minimal radiative RL at O(h 4 ). The right-handed (RH) neutrino mass matrix given by (2) and (3) where the caret ( ) denotes the mass eigenbasis. At leading order, i.e. O(h 2 ), the Yukawa couplings in (3) can be taken at the scale m N . Since O is real and orthogonal, both are also separately diagonal. On the other hand, the leptonic asymmetry ε lα in the decay where m N,α is the physical mass of N α and α = β. Therefore, the leptonic asymmetry ε lα ∝ Re ( h † h) αβ , being proportional to the off-diagonal entries of a diagonal matrix, vanishes identically at O(h 4 ) in minimal radiative models, where no other source of O(3) flavour breaking is present.

A next-to-minimal radiative RL model
To avoid the no-go theorem of suppressed leptonic asymmetries as derived in Section 1, we proceed differently from [1,2]. We include a new source of flavour breaking ∆M N , which is not aligned with Re(h † h) at the input scale µ X . More explicitly, the heavy-neutrino mass matrix takes on the following form: For the purposes of this note, we consider a minimal breaking matrix ∆M N of the form where ∆M 2 is needed to make the light-neutrino mass matrix rank-2, thus allowing us to fit successfully the low-energy neutrino data. Instead, ∆M 1 governs the mass difference between N 1 and N 2,3 , and its inclusion is sufficient to obtain successful leptogenesis.
In order to protect the lightness of the left-handed neutrinos in a technically natural manner, we consider a RL τ model that possesses a leptonic symmetry U (1) l . In this scenario, the Yukawa couplings h α l have the following structure [3,4]: where, in order to protect the τ asymmetry from an excessive washout and at the same time guarantee observable effects in low-energy neutrino experiments, we take |c| |a|, |b| ≈ 10 −3 − 10 −2 . The leptonic flavour-symmetry-breaking matrix is taken to be To leading order in the symmetry-breaking parameters of ∆M N and δh, the tree-level light-neutrino mass matrix is given by the seesaw formula where . Assuming a particular mass hierarchy between the light-neutrino masses m ν i and for given values of the CP phases δ, ϕ 1,2 , we determine the following model parameters appearing in the Yukawa coupling matrix (9): Therefore, the Yukawa coupling matrix (9) in the RL τ model can be completely fixed in terms of the heavy neutrino mass scale m N and the input parameters c and ∆M 2 . Notice that, whereas (11) and (13) coincide formally with the corresponding formulae in [1,2], the latter are incorrect for the model considered therein.
We amend the three benchmark points considered in [1] as detailed in Table 1. The input parameters ∆M 1 and c are easily chosen such that leptogenesis is successful. Instead, ∆M 2 has been tuned in order to reproduce exactly the predictions for the Lepton Number and Flavour Violation (LNV and LFV) observables discussed in [1]. In particular, Table 4 of [1] is unaltered, thus confirming the observable effects in LNV and LFV experiments predicted by this class of models, while simultaneously allowing for successful leptogenesis. The CP phases of the light neutrinos have been chosen as φ = −π and φ 2 = δ = 0.
The discussion in Section 5.3.2 of [1], concerning the approximate analytic solution for the charged-lepton decoherence effect, requires modifications. In particular, some of the approximations adopted there are no longer valid. In light of this, (5.22) of [1] becomes where {, } denotes anti-commutators in flavour space and we need to introduce also the K-factor [ K − ] nklm = κ γ LΦ LcΦc − γ LΦ LΦ nklm , which is no longer subdominant. Correspondingly, (5.26) of [1] is modified to It is not easy to perform further approximations in this equation and so it is convenient to solve the linear system for the variables [δ η L ] lm numerically. We then obtain the semianalytic contribution of mixing and charged-lepton (de)coherence to the asymmetry in the strong-washout regime where the diagonal asymmetries [δ η L ] ll are obtained by solving the linear system (15).
Finally, Figures 8-11 of [1] are modified too. The amended numerical results are shown in Figures 1-4 of this note. The main qualitative difference with respect to those given in [1] is that the contribution of the charged-lepton off-diagonal number densities now suppresses the total asymmetry for the three benchmark points considered here, rather than enhancing it, as in Figures 10-11 of [1]. Nevertheless, successful leptogenesis is still comfortably realized. Thus, we may conclude that the salient features discussed in Sections 5 and 6 of [1] remain valid, namely the joint possibility of successful leptogenesis and observable signatures in LNV and LFV experiments. Moreover, as is evidenced by the disparity between the asymmetries predicted by the partially flavour off-diagonal treatments in Figures 3 and 4, the use of fully flavour-covariant rate equations, as developed in [1], remains of paramount importance for obtaining accurate quantitative predictions in this class of models.    Figure 1: The deviation of the heavy-neutrino number densities η N αβ = η N αβ /η N eq − δ αβ from their equilibrium values for the three benchmark points given in Table 1 Table 1. The top panel shows the comparison between the total asymmetry obtained using the fully flavour-covariant formalism (thick solid lines, with different initial conditions) with those obtained using the flavour-diagonal formalism (dashed lines). Also shown (thin solid line) is the semi-analytic result (16). The bottom panel shows the diagonal (solid lines) and offdiagonal (dashed lines) elements of the total lepton number asymmetry matrix in the fully flavour-covariant formalism. δ η L ee and δ η L µµ are coincident. For details, see the text and [1].   Table 1. The labels are the same as in Figure 2. δ η L ee and δ η L µµ are coincident.  Figure 4: Lepton flavour asymmetries as predicted by the BP3 RL τ model parameters given in Table 1. The labels are the same as in Figure 2. δ η L ee and δ η L µµ are coincident.