Quantum Zeno effect and the impact of flavor in leptogenesis

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Quantum Zeno effect and the impact of flavour in leptogenesisvalue of the lower bound obtained in the unflavoured case for thermal initial abundance,M 1 4 × 10 8 GeV. For intermediate values of K 1 one can use a simple interpolation. Thefinal result is shown in the bottom part of figure 3 as a thick solid line.In figure 3 we also show the condition equation (22), calculated for the same valueof z Bα that maximizes the final asymmetry, but replacing K 1 with its minimum valuem 1 /m ⋆ (thick dashed line). Since W1 ID increases with K 1 , this produces a necessary,but not sufficient, condition in the m 1 –M 1 plane for the fully flavoured behaviour. Thiscondition matches the validity of the unflavoured regime at m 1 ≃ 3 × 10 −3 eV and thelower bound on M 1 at m 1 ≃ 2 eV. This means that for m 1 2 eV the fully flavouredbehaviour does not obtain. Notice also that this upper limit is quite conservative becausethe lower bound on M 1 has been obtained neglecting that, for K 1 = m 1 /m ⋆ , the flavouredCP asymmetry vanishes and thus the bound cannot be saturated. Moreover we haveassumed that P1α 0 can always assume the value that maximizes the asymmetry.In figure 3 we also show (thin solid line) a lower bound M 1 (m 1 ) in a specificscenario [15], corresponding to a particular choice of the orthogonal see-saw matrix [32]Ω=R 13 , that represents a complex rotation in the 13-plane, where Ω 2 13 is taken purelyimaginary. In this case the value of z Bα is not necessarily the same that maximizes theasymmetry in the one-flavour-dominated scenario and K 1 >m 1 /m ⋆ . Therefore, a specificcalculation is necessary in order to work out correctly the condition equation (22). Theresult is shown in figure 3 with a thin dashed line.In this case the upper limit on m 1 for the applicability of the fully flavoured regime ismuch smaller, m 1 ≃ 0.1 eV. Allowing for a non-vanishing real part of Ω 2 13 , slightly largervalues are possible. It should, however, be kept in mind that these values are indicativesince they rely on a condition for the fully flavoured regime that comes from a simple ratecomparison.5. Limitations of a simple rate comparisonJCAP03(2007)012We have exploited a somewhat qualitative rate comparison for the determination of theregion where the fully flavoured regime obtains. While we believe that our approachnicely illustrates the modifications that derive from our more restrictive criterion for thesignificance of flavour effects, there are also important shortcomings. First, we havesimply compared the inverse decay rate with the charged-lepton Yukawa interaction rate,ignoring flavour oscillations caused by the flavour-dependent lepton dispersion relation inthe medium. If the oscillations are much faster than the inverse decay rate, they alsocontribute effectively, together with inelastic scatterings, to project the lepton state onthe flavour basis. Therefore, including oscillations will tend to enlarge the region wherethe fully flavoured behaviour obtains (the inclined upper dotted–dashed line in figure 3)and to reduce that one where the unflavoured behaviour obtains (the horizontal upperdotted–dashed line in figure 3). In our case, the oscillation frequency is comparable to Γ αand so the two estimations are not too far off.Moreover, we have also neglected ΔL = 1 scatterings. They also contribute, likeinverse decays, both to generate the asymmetry and to the wash out and therefore,together with inverse decays, contribute to preserving the flavour direction of the leptons.At the relevant z ∼ z Bα ∼ 2, the ΔL = 1 scattering rate is actually larger than theinverse decay rate and thus tends to reduce the region where the fully flavoured behaviourJournal of Cosmology and Astroparticle Physics 03 (2007) 012 (stacks.iop.org/JCAP/2007/i=03/a=012) 12
Quantum Zeno effect and the impact of flavour in leptogenesisobtains (the lower inclined dotted–dashed line in figure 3). Therefore, the effects ofoscillations and of ΔL = 1 scatterings may partially cancel each other. In figure 3 theregion between the two inclined dotted–dashed lines gives therefore an indication of thetheoretical uncertainty on the determination of the region where the fully flavoured regimeholds. It can be seen that current calculations cannot establish whether the upper boundholding in the unflavoured regime is nullified, just simply relaxed or still holding, whenflavour effects are included.Only a full quantum-kinetic treatment can give a final verdict on the effectiveness offlavour effects in leptogenesis and its impact on the neutrino mass limit. While we haveverified that our rate criteria are borne out by the quantum kinetic equations stated in [13],these equations are not necessarily complete in that the term describing the generationof the asymmetry has been added by hand. Moreover, the ‘damping rate’ caused bythe flavour-sensitive Yukawa interactions ultimately derives from a collision term in thekinetic equation [20]. Extending the pioneering treatment of [13] toallowforacompleteunderstanding of flavour effects remains a challenging task.6. ConclusionsFlavour effects can play a very important role in leptogenesis. However, we have shownthat the condition for the fully flavoured behaviour is more restrictive because one needs tocompare the speed of the charged-lepton Yukawa interactions with that of N ↔ l+Φ, notwith the cosmic expansion rate. This distinction makes a significant difference particularlyin the strong wash-out regime where, for some of the leptogenesis epoch, the rate ofN ↔ l + Φ is faster than the cosmic expansion.We are especially interested in the question if the traditional neutrino mass boundof the unflavoured treatment [8] can be circumvented by flavour effects. We have foundthat the see-saw parameters that correspond to m 1 0.1 eV do not fall into the strictlyunflavoured or the fully flavoured regimes. The intermediate regime requires a detailedquantum kinetic treatment, so at present it is not possible to decide if the upper boundneutrino mass bound can be circumvented (relaxed or nullified) by flavour effects.JCAP03(2007)012AcknowledgmentsIt is a pleasure to thank W Buchmüller, S Davidson, M Losada, M Plümacher and E Rouletfor comments and discussions. We also wish to thank A Riotto for sending us the draft of aforthcoming manuscript with A De Simone on the same subject. This work was supported,in part, by the Deutsche Forschungsgemeinschaft (DFG) under grant no. SFB-375 andby the European Union under the ILIAS project, contract no. RII3-CT-2004-506222, andunder the Marie Curie project ‘Leptogenesis, Seesaw and GUTs,’ contract no. MEIF-CT-2006-022950.References[1] Minkowski P, μ → eγ at a rate of one out of 10 9 muon decays?, 1977 Phys. Lett. B 67 421 [SPIRES][2] Yanagida T, Horizontal gauge symmetry and masses of neutrinos, 1979 Proc. Workshop on the BaryonNumber of the Universe and Unified Theories (Tsukuba, Japan, Feb. 1979) p95[3] Gell-Mann M, Ramond P and Slansky R, Complex spinors and unified theories, 1979 Supergravity edP van Nieuwenhuizen and D Freedman (Amsterdam: North-Holland) p 315Journal of Cosmology and Astroparticle Physics 03 (2007) 012 (stacks.iop.org/JCAP/2007/i=03/a=012) 13
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