Quantum Zeno effect and the impact of flavor in leptogenesis

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Quantum Zeno effect and the impact of flavour in leptogenesisformula [1]–[6],1m ν = −m DM mT D. (1)Neutrino oscillation experiments measure two light neutrino mass squared differences. Ina normal scheme one has m3 2 − m 2 2 =Δm2 atm and m 3 2 − m 1 2 =Δm2 sol , whereas in an√inverted scheme one has m3 2 − m2 2 =Δm 2 sol and m2 2 − m1 2 =Δm 2 atm. For m 1 ≫ m atm ≡Δm2atm +Δm 2 sol the spectrum is quasi-degenerate, while for m 1 ≪ m sol ≡ √ Δm 2 solit isfully hierarchical.In the early Universe, the decays of the RH neutrinos into leptons and Higgs bosonsgenerally produce a net lepton number that is partly converted into a net baryon numberby sphaleron processes if the temperature exceeds about 100 GeV. We will follow commonpractice and assume that the final asymmetry is determined only by the decays of thelightest RH neutrino, although a scenario where the asymmetry is produced by the nextto-lighteststates is also possible [21, 22].For now we also assume that the flavour composition of the leptons produced inthe decays does not affect the final asymmetry (unflavoured limit). Neglecting thermaleffects in ΔL = 1 scatterings [23] and spectator processes [24, 25], the final asymmetry isdetermined by a simple set of two kinetic equations, one for the RH neutrino abundanceand one for the B–L asymmetry [26],dN N1dzdN B−Ldz= −D ( N N1 − N eqN 1)= ε 1 D ( N N1 − N eqN 1)−(WID1 +ΔW ) N B−L , (3)where z ≡ M 1 /T . With N B−L and N N1 we denote the abundances per RH neutrino N 1 ,taken to be in ultra-relativistic thermal equilibrium. The actual N 1 equilibrium abundanceis N eqN 1= z 2 K 2 (z)/2, where K i (z) are the modified Bessel functions.Introducing the decay parameter K 1 ≡ ˜Γ 1 /H T =M1 , defined as the ratio of the totaldecay width to the expansion rate at T = M 1 , the decay term can be written as(2)JCAP03(2007)012D ≡ Γ DHz = K 1 z 〈〉γ1−1 , (4)where 〈γ1 −1 〉 = K 1(z)/K 2 (z) is the thermally averaged Lorentz dilation factor. Theexpansion rate is√8 π3 g ⋆H ≃90M 2 1M Pl1z 2 ≃ 1.66 √ g ⋆M 2 1M Pl1z 2 (5)and g ⋆ = g SM = 106.75 is the total number of thermally excited degrees of freedom.The wash-out term caused by inverse decays is, after subtracting the resonant ΔL =2processes,W ID1 (z) ≡ 1 2Γ ID1 (z)H(z) z = 1 4 K 1 K 1 (z) z 3 , (6)where Γ ID1 =Γ D N eqN 1/N eqlis the inverse decay rate. The non-resonant ΔL = 2 contributionJournal of Cosmology and Astroparticle Physics 03 (2007) 012 (stacks.iop.org/JCAP/2007/i=03/a=012) 4
Quantum Zeno effect and the impact of flavour in leptogenesisdominates in the non-relativistic regime and is approximatelyΔW (z) ≃ w () 2 M1 m, (7)z 2 10 GeV)( 10 eVwherew = 9√ 5 M Pl × 10 −8 GeV 34π 9/2 √ ≃ 0.186 (8)g l g⋆ v 4and m ≡ √ ∑i m2 i .The evolution of N B−L is explicitly, denoting initial quantities with the subscript ‘in’,with the efficiency factor∫ zN B−L (z) =NB−L {−in exp z inκ 1 (z) ≡−∫ zdz ′ dN N 1z indz ′dz ′ [ W ID1 (z′ )+ΔW (z ′ ) ]} + ε 1 κ 1 (z) (9)[ ∫ z]exp − dz ′′ [W1 ID (z ′′ )+ΔW (z ′′ )] . (10)z ′An approximate analytic expression for the final efficiency factor at the end of theleptogenesis epoch and valid in the strong wash-out regime (K 1 ≫ 1) is[κ f 1 (m 1,M 1 ,K 1 ) ≃ κ(K 1 )exp −w (M1( ¯m) 2](11)z B (K 1 ) 10 GeV) 10 eVwhere2 (κ(K 1 ) ≃) 1 − e−K 1 z B (K 1 )/2. (12)K 1 z B (K 1 )Further,z B (K 1 ) ≃ 2+4K1 0.13 e −2.5/K 1(13)is an approximate expression for that value of z where W1 ID (z B ) ≃ 1, i.e. where thewash-out term from inverse decays becomes ineffective [27]. In the weak wash-out regimethe expression for the final efficiency factor depends on the value of NN in1. For an initialthermal equilibrium abundance the given expression still holds. For an initial vanishingabundance the final efficiency factor is the sum of two contributions with opposite signand approximate analytic expressions can be found in [26]. The weak wash-out, actingless efficiently on the positive contribution, prevents a full cancellation.Assuming either that the initial asymmetry is negligible or that it is efficiently washedout, the final B–L asymmetry is NB−L f ≃ ε 1 κ f 1. Assuming further a standard thermalhistory of the Universe and accounting for the sphaleron conversion coefficient a sph ∼ 1/3,the final baryon-to-photon ratio at recombination (rec) isJCAP03(2007)012NB−Lf η B = a sph ≃ 0.96 × 10 −2 εNγrec1 κ f 1 . (14)This is the number to be compared with the measured value [28]ηB CMB =(6.1 ± 0.2) × 10 −10 . (15)Journal of Cosmology and Astroparticle Physics 03 (2007) 012 (stacks.iop.org/JCAP/2007/i=03/a=012) 5
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