Quantum Zeno effect and the impact of flavor in leptogenesis

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

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

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

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

Quantum Zeno effect and the impact of flavour in leptogenesisFigure 2. Relevance of flavour effects in schematic regions of parameters K 1 andM 1 . The region above the horizontal dashed line corresponds to the condition (18)for z Bα = z max . The vertical dotted–dashed line is the border between theweak and the strong wash-out regime. The region below the inclined dottedline corresponds to the condition (22) forz Bα = z max .unity and the equations are summed over flavours, the kinetic equation for N B−L holdingin the unflavoured regime is recovered (cf equation (3)). Therefore, we require1012 GeVM 1 (22)2 W1 ID (z Bα )as an approximate condition for the fully flavoured behaviour.In figure 2 we summarize the different possible cases in the plane of parameters K 1and M 1 .ForM 1 5 × 10 11 GeV, above the dashed line, flavour effects are not importantindependently of K 1 . The condition equation (22), in the most restrictive case whenz Bα = z max and W1 ID ≃ 0.3 K 1 , is satisfied below the inclined dotted line. This casetypically occurs in a one-flavour-dominated scenario, as we explain below. The verticaldotted–dashed line is the border that separates the weak from the strong wash-out regimein the unflavoured case. In the flavoured case the condition K 1 3.3 still implies theweak wash-out regime because flavour effects can only reduce the wash-out. However,the condition for the strong wash-out regime can be more restrictive than K 1 3.3, asdiscussed in [15]. For K 1 3.3, flavour effects modify the final asymmetry only marginallyand more specifically only if the initial abundance vanishes, as indicated in figure 2. Onthe other hand, for K 1 3.3 and below the diagonal line, flavour modifications of thefinal asymmetry can be large, especially in the one-flavour-dominated scenario.There is a region in parameter space where neither condition equation (17) nor(22)holds. This intermediate regime can become very large in the case of a one-flavourdominatedscenario, where an order-of-magnitude enhancement of the final asymmetryis possible. In this case one of the two projectors is very small compared to the otherand so the wash-out is very asymmetric in the two flavours. On the other hand, if thetwo-flavoured CP asymmetries are comparable, then the final asymmetry is dominantlyproduced into one flavour and deviations from the unflavoured regime can become veryJCAP03(2007)012Journal of Cosmology and Astroparticle Physics 03 (2007) 012 (stacks.iop.org/JCAP/2007/i=03/a=012) 9

Quantum Zeno effect and the impact of flavour in leptogenesislarge. This scenario is realized, in particular, when the absolute neutrino mass scaleincreases, relaxing the traditional neutrino mass bound.However, the condition equation (22) strongly restricts the applicability of the oneflavour-dominatedscenario. Even though the reduction of the wash-out is driven byK 1α ≪ K 1 , implying z Bα ≪ z B , the possibility for flavour effects to be relevant relieson the dominance of the charged-lepton Yukawa interaction rate compared to the RHneutrino inverse decay rate that, however, is still driven by K 1 . Therefore, increasing K 1 ,one can enhance the asymmetry in the one-flavour-dominated scenario compared to theunflavoured case, if smaller and smaller values of the projector P1α 0 are possible. On theother hand, the inverse decay rate increases so that the fully flavoured behaviour may nolonger apply. In particular notice that the maximum enhancement of the asymmetry isobtained when K 1 ≫ K 1α ≃ 1, when z Bα ≃ z max and the condition (22) is maximallyrestrictive.4. Neutrino mass boundOne possible consequence of flavour effects is to relax the traditional upper bound on theneutrino mass that is implied by successful leptogenesis. In order to explore the impactof our modified criteria we first recall the origin of this bound in the unflavoured case.Maximizing the final value of the asymmetry over all see-saw parameters except M 1 andm 1 yields [26]ηB max(M( ) 1.2 ( ) [1,m 1 ) m⋆ M1 matm≃ 3.8exp − w M(1 ¯m2]ηBCMBm 1 10 10 GeV m 1 + m 3 z B 10 10 GeV eV)≥ 1, (23)wherewehaveapproximatedκ(K 1 ) ≃ 0.5 K1 −1.2 [30] and we have neglected the dependenceof z B on K 1 in the derivative. This constraint translates into m 1

Quantum Zeno effect and the impact of flavour in leptogenesisFigure 3. Relevance of flavour effects similar to figure 2, now mapped toschematic regions of parameters m 1 and M 1 . The region above the horizontaldashed line corresponds to the condition equation (18) for the applicability of theunflavoured regime. The region below the inclined thick dashed line correspondsto the condition equation (22) calculated for that value of z Bα that maximizesthe final asymmetry in the one-flavour-dominated scenario and for K 1 = m 1 /m ⋆ .In this same case, the lower inclined dotted–dashed lines includes also the effectof scatterings in the condition equation (22) while the upper inclined and thehorizontal dotted–dashed lines include the effect of oscillations. The area betweenthe two inclined dotted–dashed lines gives an estimation of the uncertainty on thecondition for the fully flavoured regime to hold. The area between the horizontalthin dotted–dashed line and the horizontal thick dashed line gives an estimationof the uncertainty on the condition for the unflavoured regime to hold. The twothick solid lines border the region where successful leptogenesis is possible: onthe left in the unflavoured regime and above in the fully flavoured regime. Thethin solid line is a more restrictive border obtained for a specific choice of the seesaworthogonal matrix and the thin dashed line is the corresponding condition,equation (22). In this case one has K 1 >m 1 /m ⋆ .JCAP03(2007)012whereε(M 1 ) ≡ 316π( )M 1 m atm≃ 10 −6 M1 ( matm). (27)v 2 10 10 GeV 0.05 eVIt is then possible to find the value of P1α 0 that maximizes the asymmetry as a functionof K 1 and the corresponding value of z Bα .ImposingηBmax ≥ ηBCMB implies a lower boundon M 1 as a function of K 1 .This limit can be translated into a lower bound on M 1 as a function of m 1 by replacingK 1 with its minimum value m 1 /m ⋆ . In this way the wash-out is always minimizedand the final efficiency factor and the final asymmetry are maximized. Notice that thesingle-flavoured CP asymmetries, like the total, vanish for K 1 = m 1 /m ⋆ . Therefore,this lower bound cannot be saturated. Notice, moreover, that for m 1 m ⋆ the oneflavour-dominatedscenario does not necessarily hold because it is possible that K 1 1.Actually for K 1 → 0, flavour effects disappear and one has to recover the usual asymptoticJournal of Cosmology and Astroparticle Physics 03 (2007) 012 (stacks.iop.org/JCAP/2007/i=03/a=012) 11

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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