Quantum Zeno effect and the impact of flavor in leptogenesis

Quantum Zeno effect and the impact of flavour in leptogenesisWe conclude that the condition equation (17) obeys the intuitive expectation thatthere is always a threshold value for K 1 above which the unflavoured case is recovered. Inthis case the temperature below which flavour effects play a role indeed becomes smallerand smaller. The situation is illustrated in figure 1 where we compare W ID withF τ ≡ 1 2Γ τHz ≃ 5 × 1011 GeVM 1, (20)the analogous quantity for the charged-lepton Yukawa interactions. For any value of M 1and K 1 , there is a value z fl such that F τ W ID for z>z fl . If M 1 2 × 10 12 GeV/K 1and K 1 3.3, corresponding to F τ W ID (z max ) in the strong wash-out regime, thenz fl = 0, meaning that flavour effects are important during the entire thermal history. Onthe other hand, for a fixed value of M 1 , one has z fl →∞for K 1 →∞, implying thatflavour effects tend to disappear for sufficiently large values of K 1 . Notice however thatif M 1 5 × 10 11 GeV, then z fl z off ≃ z B for any value of K 1 . This confirms that onlyfor M 1 5 × 10 11 GeV can flavour effects be neglected and the unflavoured regime isrecovered.3. Maximum flavour effectsWe now turn to the opposite extreme case when flavour effects are maximal, the ‘fullyflavoured regime’. In other words, the charged-lepton Yukawa interactions are now takento be so fast that the lepton flavour content produced in N → l + Φ on average fullycollapses before the inverse reaction can take place, i.e. the l density matrix in flavourspace is to be taken diagonal in the charged-lepton Yukawa basis. In this case each singleflavour asymmetry has to be calculated separately because generally the wash-out byinverse decays is different for each flavour. Moreover, the single-flavoured CP asymmetriesnow have an additional contribution compared to the total [12, 13]. Finally, the inversedecay involving a lepton in the flavour α does not wash out as much asymmetry as theone produced by one RH neutrino decay. The reduction is quantified by the probabilityPiα, 0 averaged over leptons and anti-leptons, that the lepton l i produced in the decay ofN i collapses into the flavour eigenstate l α . The relevant Boltzmann equations becomedN N1(= −D 1 NN1 − N eq )Ndz1dN Δαdz= ε 1α D 1(NN1 − N eqN 1)− P01α W ID1 N Δα .Since we are dealing with the two-flavour case, here α = τ or μ where the latter stands fora suitable superposition of the μ and e flavour. Notice also that we defined Δ α ≡ B/3−L αand therefore the total asymmetry is given by N B−L = N Δμ + N Δτ .As in the unflavoured case, we next identify the condition for the fully flavouredapproximation to hold. The final asymmetry in the flavour α is dominantly produced atz ≃ z Bα ≡ z B (K 1α ), where K 1α ≡ P1α 0 K 1. Therefore, one must require that Γ α Γ ID1holds already at z ∼ z Bα lest the wash-out reduction take place too late. We stress thatflavour effects modify the final asymmetry only if the flavour projection takes place beforethe wash-out by inverse decays freezes out. Otherwise the wash-out epoch is over and theunflavoured behaviour is recovered. It is easy to verify that. If the projectors are set to(21)JCAP03(2007)012Journal of Cosmology and Astroparticle Physics 03 (2007) 012 (stacks.iop.org/JCAP/2007/i=03/a=012) 8