Quantum Zeno effect and the impact of flavor in leptogenesis

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Quantum Zeno effect and the impact of flavour in leptogenesisFigure 1. Comparison between the wash-out term W1 ID (z) (thick solid lines),defined in equation (6) and plotted for the three indicated values of K 1 ,andthecharged-lepton Yukawa interaction term F τ ,definedinequation(20) and plottedfor the indicated values of M 1 .The wash-out rate by inverse decays reaches a maximum W1 ID (z max ) ≃ 0.3 K 1 atz max ≃ 2.4. In the weak wash-out regime, when K 1 3.3, one has W1 ID (z) < 1forany value of z. In this case the wash-out is negligible and the final asymmetry dependson the initial conditions. In the strong wash-out regime, when K 1 3.3, there is aninterval [z on ,z off ]whereW1 ID ≥ 1. The asymmetry produced at z z off is very efficientlywashed out and thus the final asymmetry is essentially what is produced around z B ≃ z offby out-of-equilibrium decays of the residual RH neutrinos whose number correspondsapproximately to the final value of the efficiency factor.In figure 1 we show the wash-out term from inverse decays for three different valuesof K 1 .ForK 1 = 100 we show the interval [z on ,z off ]. The maximum value W1 ID (z max ) ≃ 33is reached at z max ≃ 2.4. For K 1 ≃ 3.3 one has z on ≃ z max ≃ z off . Thiscanbetakenasthethreshold value distinguishing between the strong and the weak wash-out regimes. ForK 1 =10 −1 one has W1 ID ≪ 1 for any value of z. Notice, however, that even in this casethe weak wash-out can be important for successful leptogenesis if the initial abundancevanishes, since it prevents a full cancellation between two different sign contributions tothe final asymmetry [26].JCAP03(2007)0122.2. When are flavour effects important?We now turn to the question when flavour effects will modify these results. The crucialeffect is caused by charged-lepton Yukawa interactions [9] that occur with a rate [29]Γ α ≃ 5 × 10 −3 Th 2 α. The largest one is for α = τ where(Γ τ 10 12H ≃ GeVT). (16)Therefore, if T 10 12 GeV, charged-lepton Yukawa interactions are not effective and allprocesses in the early Universe are flavour blind, justifying the unflavoured treatment.Journal of Cosmology and Astroparticle Physics 03 (2007) 012 (stacks.iop.org/JCAP/2007/i=03/a=012) 6
Quantum Zeno effect and the impact of flavour in leptogenesisFor T 10 12 GeV, the τ Yukawa couplings are strong enough that the scatteringsτ L ¯τ R → Φ † are in equilibrium. However, as stressed in the introduction, this conditionis not necessarily sufficient for important flavour effects to occur because we need tocompare the speed of the Yukawa interactions with that of the RH neutrino decays andinverse decays. To this end we study the weak and strong wash-out regimes separatelyand consider only a two-flavour case because the τ lepton Yukawa coupling causes themain modification.In the weak wash-out regime, assuming a vanishing initial abundance, the productionof RH neutrinos through inverse decays occurs around T ∼ M 1 . At this epoch, inversedecays are, by definition, slower than the expansion rate. Therefore, the conditionT 10 12 GeV is sufficient to conclude that the charged-lepton Yukawa interactions arefaster than the inverse decay rate. This translates into the condition M 1 10 12 GeVbecause the RH neutrino production occurs at T ∼ M 1 , in agreement with the previousliterature [12, 13].However, this condition does not guarantee that flavour effects indeed have an impacton the final asymmetry, because this impact depends on wash-out playing some role. Fora vanishing initial abundance this is the case in that wash-out effects prevent a full signcancellation between the asymmetry produced when N N1 < N eqN 1and the asymmetryproduced later on. On the other hand, for a thermal initial abundance, no such effectarises from the weak wash-out and flavour effects do not modify the final asymmetry. Inany case, one can say that in the limit K 1 → 0 flavour cannot have effects on the finalasymmetry for any initial abundance. We will come back to this point.In the strong wash-out regime the situation is very different. The rate of RH neutrinoinverse decays at T ∼ M 1 is larger than the expansion rate. Therefore, we need to comparethe charged-lepton Yukawa rate Γ τ with the RH neutrino inverse decay rate Γ ID1 .Fortheunflavoured treatment to be valid for z z fl ≤ z B then requiresM 1 1012 GeV2 W1 ID(zfl) , (17)where z fl is that value of z where the two rates are equal. This condition guarantees thatat temperatures T > T fl = M 1 /z fl flavour effects will not be able to break the coherentpropagation of lepton states. The final asymmetry is dominantly produced around z ∼ z B .Therefore, the condition for flavour effects to be negligible isM 1 5 × 10 11 GeV, (18)similar to the weak wash-out regime. However, the corresponding condition on thetemperatureT 1012 GeV(19)2 z B (K 1 )is now less restrictive.If one starts with a non-vanishing initial abundance, then the final asymmetry isalso determined by how efficiently the initial value is washed out; this is described bythe integral in equation (9). In this case even a value of z fl
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