Quantum Zeno effect and the impact of flavor in leptogenesis

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Quantum Zeno effect and the impact of flavour in leptogenesisContents1. Introduction 22. Unflavoured case 32.1. Unflavoured leptogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2. When are flavour effects important? . . . . . . . . . . . . . . . . . . . . . . 63. Maximum flavour effects 84. Neutrino mass bound 105. Limitations of a simple rate comparison 126. Conclusions 13Acknowledgments 13References 131. IntroductionThe see-saw mechanism [1]–[6] is an elegant way to understand neutrino masses andmixing and, at the same time, with leptogenesis [7], provides a natural mechanism forgenerating the matter–antimatter asymmetry of the Universe by virtue of CP-violatingdecays N → l + Φ of heavy right-handed Majorana neutrinos N into ordinary leptons land Higgs particles Φ. In comparison with other baryogenesis scenarios, leptogenesis hasthe unique advantage of relying on an ingredient of physics beyond the standard model,neutrino masses, that is experimentally established. Moreover, the neutrino mass valuessuggested by flavour oscillation experiments are optimal in the sense that leptogenesiswould operate in a mildly ‘strong wash-out regime’ [8]. This means that the inverseprocesses l +Φ → N are efficient enough to wash out all contributions to the finalasymmetry that depend on initial conditions, but not too efficient to prevent successfulleptogenesis. This is true for hierarchical light neutrino schemes, while in the case ofquasi-degenerate neutrinos, leptogenesis provides a stringent upper bound on neutrinomasses, m 1 ≤ 0.1 eV, holding when a hierarchical right-handed neutrino spectrum isassumed [8].Traditionally, flavour effects were neglected in leptogenesis, i.e. the lepton asymmetryproduced by N → l + Φ and the wash-out effects caused by the inverse processeswere treated as if l had no flavour properties. Recently it was recognized, however,that important modifications arise if some of the charged-lepton Yukawa interactionsare in equilibrium [9]–[13]. In particular, there is an additional source of CP violationthat can make the single-flavoured CP asymmetries larger than the total. Thisadditional source includes a dependence on low-energy phases, in contrast to unflavouredleptogenesis [11, 12]. In fact, successful leptogenesis stemming only from Majorana andDirac phases is possible [14]–[18]. A second consequence is a reduction of the wash-outefficiency, generally by different amounts in each flavour. These modifications togethercontribute to enhance the final asymmetry, an effect that increases for a larger neutrinoJCAP03(2007)012Journal of Cosmology and Astroparticle Physics 03 (2007) 012 (stacks.iop.org/JCAP/2007/i=03/a=012) 2
Quantum Zeno effect and the impact of flavour in leptogenesismass scale. Therefore, flavour effects can relax the neutrino mass bound of the unflavouredscenario [13].The purpose of our present study is to explore the conditions for flavour effects to berelevant in leptogenesis and if the neutrino mass bound can indeed be circumvented. Thepast literature is unclear on this point in the following sense. It was stressed that the linteractions with the background medium are flavour-sensitive and the coherence of thel flavour content defined by N → l + Φ can be destroyed if some of the charged-leptonYukawa couplings are in thermal equilibrium during the leptogenesis epoch. On the otherhand, the reactions N ↔ l + Φ constantly regenerate the l flavour composition and ifthese reactions are fast, l is frozen in its coherent flavour state by the quantum Zenoeffect [19, 20]. In other words, the charged-lepton Yukawa couplings and the N ↔ l +Φreactions generally single out different directions in flavour space. The l density matrixin flavour space will dominantly stay in the direction favoured by the fastest term ofthe kinetic equation. Therefore, we expect flavour effects to be important if the ‘flavourmeasurements’ by the background medium are fast relative to the N ↔ l + Φ reactions, acondition that was also stated in [12]. On the other hand, the actually used requirementin the previous literature was the weaker condition that the flavour-sensitive reactions arefast on the cosmic expansion scale. Of course, at the end of the leptogenesis epoch whenthe N ↔ l+Φ reactions no longer track equilibrium, this weaker condition is correct. Ourmain concern is that previous treatments may not properly cover the important strongwash-out case, where N ↔ l + Φ is fast relative to the expansion rate for most of theleptogenesis epoch.In order to identify the conditions for flavour effects to be relevant for the finalbaryon asymmetry we split the problem into two parts. First, in section 2, wederivea condition for the unflavoured treatment to be adequate. Next, in section 3, weobtaina condition for flavour effects to be maximally effective (‘fully flavoured regime’). Inthese cases two different sets of classical Boltzmann equations apply to calculate the finalasymmetry. In section 4 we consider the neutrino mass bound to hold in the unflavouredcase and show that it cannot be circumvented in the fully flavoured case. However, thereis an intermediate regime between the fully flavoured and unflavoured cases where a fullquantum kinetic equation is required to decide on the neutrino mass bound. In section 5we briefly discuss the limitations of our simple rate comparison. We summarize ourfindings in section 6.JCAP03(2007)0122. Unflavoured case2.1. Unflavoured leptogenesisIn order to derive a condition for the unflavoured leptogenesis treatment to be adequate, wefirst retrace the steps leading to the final asymmetry prediction. Adding to the standardmodel three right-handed (RH) neutrinos with a Majorana mass term M and Yukawacouplings h, after spontaneous symmetry breaking a Dirac mass term, m D = vh, isgenerated by the vev v of the Higgs boson. In the see-saw limit, M ≫ m D , the spectrumof neutrino masses splits into two sets, a very heavy one, M 3 ≥ M 2 ≥ M 1 , almostcoinciding with the eigenvalues of M, and a light one, m 3 ≥ m 2 ≥ m 1 , correspondingto the eigenvalues of the light neutrino mass matrix. It is given by the see-sawJournal of Cosmology and Astroparticle Physics 03 (2007) 012 (stacks.iop.org/JCAP/2007/i=03/a=012) 3
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