baryogenesis through neutrino oscillations: a uni ed ...baryogenesis through neutrino oscillations:...
TRANSCRIPT
Baryogenesis through Neutrino Oscillations: A Unified Perspective
Brian Shuve1, 2, ∗ and Itay Yavin2, 1, †
1Perimeter Institute for Theoretical Physics 31 Caroline St. N, Waterloo, Ontario, Canada N2L 2Y5.2Department of Physics & Astronomy, McMaster University 1280 Main St. W. Hamilton, Ontario, Canada, L8S 4L8.
Baryogenesis through neutrino oscillations is an elegant mechanism which has found several real-izations in the literature corresponding to different parts of the model parameter space. Its appealstems from its minimality and dependence only on physics below the weak scale. In this work weshow that, by focusing on the physical time scales of leptogenesis instead of the model parameters, amore comprehensive picture emerges. The different regimes previously identified can be understoodas different relative orderings of these time scales. This approach also shows that all regimes requirea coincidence of time scales and this in turn translates to a certain tuning of the parameters, whetherin mass terms or Yukawa couplings. Indeed, we show that the amount of tuning involved in theminimal model is never less than one part in 105. Finally, we explore an extended model where thetuning can be removed in exchange for the introduction of a new degree of freedom in the form ofa leptophilic Higgs with a vacuum expectation value of the order of GeV.
I. INTRODUCTION
It is a theoretically attractive possibility to explain thebaryon asymmetry of the Universe through the mecha-nism of sterile neutrino oscillations [1, 2]. The model issimple, containing only right-handed sterile neutrinos inaddition to the Standard Model (SM). These neutrinosare light (∼ GeV) and do not require any new physicsat inaccessibly high energy scales. The model even holdsthe possibility that one of the sterile neutrinos can be thenon-baryonic dark matter of the Universe [2–5]. Severalpast works have found different choices of the parametersthat lead to the correct baryon asymmetry and identifiedseveral regimes [2, 5–10].
It is the purpose of this work to present a unifiedperspective on leptogenesis through neutrino oscillations,weaving the disjoint regimes previously identified in theliterature into a single continuous picture. Our analy-sis focuses on the three important time scales for baryo-genesis: the time of sterile neutrino oscillations, active-sterile neutrino equilibration time, and sphaleron decou-pling time. We identify the different regimes as differentrelative orderings of these time scales and demonstratethe continuity of separate parts of the parameter space.This allows us to point out the most important effectscontributing to the asymmetry in each regime. Along theway, we provide some improvement upon the calculationof the baryon asymmetry from neutrino oscillations byincluding the effects of scatterings between left-handed(LH) leptons and the thermal bath during asymmetrygeneration.
Aside from providing a unified framework, centeringthe discussion around the relevant time scales for baryo-genesis brings to the forefront the need for an accidentalcoincidence between the different, unrelated scales in theproblem. It is therefore no surprise that the framework
∗ [email protected]† [email protected]
is marred by the need for fine-tuning between its fun-damental parameters in order to successfully generatethe correct baryon asymmetry in the different regimes.We show that, while the fine-tuning present in differentregimes appears in different parameters, the total fine-tuning is always at least at the level of 1/105.
While possibly aesthetically unappealing, the persis-tent fine-tuning needed throughout the parameter spaceis not grounds to discount the framework. It is, how-ever, possibly an invitation to explore extensions of theminimal model which would either alleviate some of thenecessary tuning, or explain it as a small departure froma more symmetric phase. In the final part of this work,we therefore consider the possibility of an additionalHiggs doublet with a small electroweak vacuum expec-tation value (VEV), which is coupled to all the leptonsthrough larger-than-usual Yukawa couplings. We showthat, aside from ameliorating the fine-tuning needed forsuccessful leptogenesis, such a leptophilic Higgs doubletcan be searched for directly and indirectly in high-energyreactions accessible at the LHC.
The paper is organized as follows: in Section II, we re-view the qualitative features of baryogenesis through neu-trino oscillations, providing a unified perspective on thephysics responsible for leptogenesis in different regimes.In Section III, we discuss the different model parame-ters and their specific roles in leptogenesis. Section IVpresents the formalism for computing the baryon asym-metry and describes the modifications we make to theasymmetry evolution equations to account for SM ther-mal scatterings. In Section V, we confirm our unifiedperspective on leptogenesis with numerical studies of thedifferent parameter regimes. We explicitly show thetuning required to obtain the observed baryon asym-metry throughout the different regimes of the minimalmodel. We then discuss the baryon asymmetry in aleptophilic Two Higgs Doublet Model (2HDM) in Sec-tion VI, demonstrating how the asymmetry can be en-hanced relative to the minimal model, eliminating anyneed for tuning. We also discuss experimental implica-tions of this extended model. We close with a discussion
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and some remarks about the inclusion of dark matter inthe model.
II. REVIEW OF BARYOGENESIS THROUGHNEUTRINO OSCILLATIONS
It has been appreciated for some time that, in ex-tensions of the Standard Model with sterile neutrinos,oscillations of the sterile neutrinos can be responsiblefor baryogenesis [1, 2]. We discuss the basic frameworkin this section, emphasizing the essential features thatemerge from the model. We begin by presenting themodel, and follow with a qualitative account of the asym-metry generation and its dependence on the underlyingmodel parameter.
A. Neutrino Minimal Standard Model (νMSM)
We consider the Type I see-saw model for generatingLH neutrino masses, also known as the Neutrino Min-imal Standard Model (νMSM). In this scenario, theSM is supplemented by three sterile neutrinos, NI , withMajorana masses, MI . In the basis where the chargedlepton and sterile neutrino masses are diagonal, the La-grangian is (see ref. [11] for a recent review)
LνMSM = FαI LαΦNI + (MN )I NINI , (1)
where Φ is the electroweak doublet scalar responsible forgiving mass to the SM neutrinos, and Lα is the SM leptondoublet of flavour α. When the scalar doublet develops aVEV, 〈Φ〉 6= 0, these interactions generate a small massfor the LH neutrinos through the see-saw mechanism [12]of the order,
mν ∼F 2 〈Φ〉2MN
(2)
∼ 0.1 eV
( 〈Φ〉100 GeV
)2(F
10−7
)2(GeV
MN
).
Here, we used the parameters and masses most relevantfor the current work: a scalar VEV around the elec-troweak scale, 〈Φ〉 ∼ 100 GeV; sterile-neutrino massesaround or below the electroweak scale, MN ∼ GeV; andsmall Yukawa couplings, F ∼ 10−7 − 10−8. With weak-scale sterile neutrino masses, the Yukawa couplings ofthe neutrino are only somewhat smaller than the elec-tron Yukawa.
With such a small coupling between LH leptons andsterile neutrinos, active-sterile neutrino scattering is outof equilibrium in the early universe and does not becomerapid until T . TW, where TW ≈ 140 GeV is the tem-perature of the sphaleron decoupling at the electroweakphase transition [5]. If there is a negligible concentrationof sterile neutrinos immediately following inflation, thenthe sterile-neutrino abundance remains below its equilib-rium value until it re-thermalizes at T . TW. As is well
known, sphaleron processes active for T > TW violatebaryon number (B), but preserve the difference betweenbaryon and lepton numbers (B − L), processing the pri-mordial lepton asymmetry into a baryon asymmetry [13].Thus, the final baryon asymmetry of the universe ob-served today is determined by the lepton asymmetry atthe time of sphaleron decoupling, TW [14]. In everythingthat follows, we therefore assume the physics associatedwith sphalerons to be present and concentrate on a de-tailed understanding of leptogenesis alone.
B. Sakharov Conditions for Leptogenesis
The Sakharov conditions [15] necessary for generatinga total lepton asymmetry are satisfied in the νMSM:
1. Violation of Standard Model lepton number:The Yukawa coupling in Eq. (2) preserves a gen-eralized lepton number L − N under which bothSM and sterile neutrinos are charged. The L − Nsymmetry is broken by the sterile neutrino Majo-rana mass, but rates of (L−N)-violating processesare suppressed by a factor of M2
N/T2 relative to
(L−N)-preserving rates, and so total lepton num-ber violation is generally ineffective for T & TW.
However, scattering processes Lα → N†I throughthe Yukawa interactions in Eq. (2) violate Stan-dard Model lepton number, allowing the creationof equal asymmetries in L and N such that L−Nis still conserved. Sphalerons then convert the SML asymmetry into a baryon asymmetry.
2. CP violation: There are three CP phases in theYukawa couplings FαI . Together, these provide asufficient source for leptogenesis through neutrinooscillations.
3. Departure from equilibrium: As discussedabove, if MN . TW, sterile neutrino scatterings areout of equilibrium provided there is no abundanceof sterile neutrinos at the earliest times followinginflation. Unlike many models of baryogenesis, theout-of-equilibrium condition is satisfied for an ex-tended period in the early universe, with equili-bration only occurring after sphaleron decoupling,T . TW.
In the following subsection, we elaborate on the physi-cal processes responsible for the production (and destruc-tion) of the lepton asymmetry, and clarify which param-eters most strongly control the size of the baryon asym-metry.
C. Asymmetry Creation and Washout
The basic stages leading to the creation of a total SMlepton asymmetry are shown in Fig. 1. Immediately fol-
3
Lα NI
H∗
LβNJ
coherentoscillations
Y∆L1= 0
∑α
Y∆Lα= 0
Y∆L2, Y∆L3 = 0
Y∆L1> 0
Y∆L2, Y∆L3 < 0∑α
Y∆Lα= 0
Lα NI
H∗H
Y∆L1> 0
Y∆L2, Y∆L3 < 0∑α
Y∆Lα"= 0
time
FIG. 1. The basic stages leading to the creation of a total lepton asymmetry from left to right: out-of-equilibrium scatteringof LH leptons begin to populate the sterile neutrino abundance at order O(|F |2); after some time of coherent oscillation, asmall fraction of the sterile neutrinos scatter back into LH leptons to create an asymmetry in individual lepton flavours atorder O(|F |4); finally, at order O(|F |6), a total lepton asymmetry is generated due to a difference in scattering rate into sterileneutrinos among the different active flavours.
lowing inflation, there is no abundance of sterile neutri-nos, and out-of-equilibrium scatterings mediated by theYukawa couplings begin to populate the sterile sector, asshown on the left side of Fig. 1. The sterile neutrinosare produced in a coherent superposition of mass eigen-
states1 and remain coherent as long as the active-sterileYukawa coupling remains out of equilibrium, since in theminimal model there are no other interactions involvingthe sterile neutrinos.
Some time later, a subset of the sterile neutrinos scat-ter back into LH leptons, mediating Lα → Lβ transi-tions as shown in the centre of Fig. 1. Since the sterileneutrinos remain in a coherent superposition in the in-termediate time between scatterings, the transition rateLα → Lβ includes an interference between propagationmediated by the different sterile neutrino mass eigen-states. The different mass eigenstates have differentphases resulting from time evolution; for sterile neutri-nos NI and NJ , the relative phase accumulated during asmall time dt is e−i(ωI−ωJ ) dt, where
ωI − ωJ ≈(MN )2
I − (MN )2J
2T≡ (MN )2
IJ
2T. (3)
In the interaction basis, this CP -even phase results froman oscillation between different sterile neutrino flavours,and explains the moniker of leptogenesis through neu-trino oscillations.
When combined with the CP -odd phases from theYukawa matrix, neutrino oscillations lead to a differencebetween the Lα → Lβ rate and its complex conjugate,
Γ(Lα → Lβ)− Γ(L†α → L†β) ∝∑I 6=J
Im
[exp
(−i∫ t
0
M2IJ
2T (t′)dt′)]
× Im[FαIF
∗βIF
∗αJFβJ
]. (4)
1 This is true assuming generic parameters with no special align-ment of the sterile-neutrino interaction and mass eigenstates.
In the absence of efficient washout interactions, which isensured by the out-of-equilibrium condition, this differ-ence in rates creates asymmetries in the individual LHlepton flavours Lα.
Denoting the individual LH flavour abundances (nor-malized by the entropy density, s) by YLα ≡ nLα/s andthe asymmetries by Y∆Lα ≡ YLα −YL†
α, we note that the
processes at order O(|F |4) discussed thus far only convertLα into Lβ , conserving total SM lepton number,
Y∆Ltot =∑
α
Y∆Lα = 0 at O(|F |4). (5)
Since sphalerons couple to the total SM lepton number,it follows that no baryon asymmetry is generated at thisorder as well, Y∆Btot
= 0. Total lepton asymmetry is,however, generated at order O(|F |6): the excess in eachindividual LH lepton flavour due to the asymmetry fromEq. (4) leads to a slight increase of the rate of Lα → N†
vs. L†α → N . The result is that active-sterile lepton scat-terings can convert individual lepton flavour asymmetriesinto asymmetries in the sterile neutrinos. But, since therates of conversion, Γ(Lα → N†), are generically differentfor each lepton flavour α, this leads to a depletion of someof the individual lepton asymmetries at a faster rate thanothers, leading to an overall SM lepton asymmetry andan overall sterile neutrino asymmetry. Because Ltot −N
4
is conserved for T mN , this gives
dY∆Ntot
dt=dY∆Ltot
dt=∑
α,I
Y∆Lα Γ(Lα → N†I ). (6)
Therefore, while the sum of the LH lepton flavour asym-metries vanishes at O(|F |4) as in Eq. (5), the fact thatthe LH leptons scatter at different rates into the sterilesector results in a non-vanishing total lepton asymmetryat O(|F |6).
The asymmetry generated at early times can be de-stroyed later when the different lepton flavours establishchemical equilibrium with the sterile neutrinos. This oc-curs when the transition rate exceeds the Hubble rate,Γ(Lα → N†) ∼ (FF †)αα T & H. At this time, the par-ticular lepton flavour Lα reach chemical equilibrium withthe sterile neutrinos, but not yet with the other SM lep-tons since lepton flavour is still conserved to a good ap-proximation. Suppose, for concreteness, that Lτ comesinto equilibrium with the sterile neutrinos, but Lµ and Leremain out of equilibrium. Then, before Lτ comes intoequilibrium, we can use the approximate conservationof ∆Ltot −∆N = 0 to write
Y∆Le ≡ x, Y∆Lµ ≡ y, Y∆Lτ ≡ z,
Y∆N = Y∆Ltot = x+ y + z. (7)
Once Lτ comes into equilibrium with the sterile neutri-nos, the Yukawa coupling FτI LτHNI leads to the equi-librium chemical potential relation2 µLτ = −µN . Be-cause the fields Lτ and N both have two componentsand opposite-sign chemical potentials, their asymmetriesare therefore equal and opposite: Y∆Lτ = −Y∆N . Theflavour asymmetries ∆Lµ and ∆Le remain unchangedbecause they are not yet in equilibrium. Together withthe fact that the ∆Ltot −∆N ≈ 0 at high temperature,this implies that after Lτ equlibration:
Y∆N = Y∆Ltot=x+ y
2. (8)
Therefore, at the point of equilibration of the τ flavour,the total lepton asymmetry rapidly changes from x+y+zto (x + y)/2, and the ∆Lτ asymmetry rapidly changesfrom z to −(x + y)/2. We see that nothing remains ofthe original ∆Lτ flavour asymmetry z by the time thetau flavour comes fully into equilibrium. Both the to-tal lepton asymmetry and Lτ flavour asymmetry changedramatically when equilibrium is reached, with the asym-metry after equilibrium being fixed by the remainingasymmetries in the flavours Le and Lµ. It is also ap-parent that if all of the flavours come into equilibrium,∆Le = ∆Lµ = ∆Lτ = ∆N = 0. Clearly, sphaleron in-teractions must decouple before this time is reached orno baryon asymmetry results from sterile neutrino oscil-lation.
2 Strictly speaking, the relation is µLτ+µΦ = −µN ; for illustrativepurposes, and because µΦ is typically small compared to the LHSM lepton flavour asymmetries, we set µΦ = 0 here.
D. Qualitative Dependence of Asymmetry onModel Parameters
In the discussion above we have highlighted threeimportant elements for generating a non-zero baryonasymmetry: the coherent oscillation and interferenceof different sterile neutrino states in Lα → Lβ scat-tering; the magnitudes of the Yukawa couplings whichdetermine the rates of processes generating the leptonasymmetry; and the presence of differences in scatter-ing rates of individual LH lepton flavours into sterileneutrinos. We now elaborate on each, as they haveimplications for what parts of parameter space maximizethe baryon asymmetry in the minimal model, and hownew interactions can enhance the asymmetry.
Sterile neutrino mass splitting: In the absenceof a sterile-neutrino mass splitting, the sterile-neutrinomasses and couplings can be simultaneously diagonal-ized, and there is no coherent oscillation/interference asrequired in Fig. 1. The size of the mass splitting dic-tates the time scale at which the phases of the coher-ently evolving sterile neutrino eigenstates become sub-stantially different. From Eq. (4) and the Hubble scalein a radiation-dominated universe [16],
H =1.66√g∗ T
2
MPl(9)
(g∗ is the number of relativistic degrees of freedom), wehave an O(1) phase from oscillation at
tosc ≈(
3√MPl
1.66g1/4∗√
2 (M2N3 −M2
N2)
)2/3
. (10)
The oscillation time is later for smaller mass splittings.At later times, the rates of scattering between active-sterile neutrinos is larger relative to the Hubble scale,and the asymmetry is consequently larger. Therefore,small but non-zero sterile neutrino mass splittingsenhance the size of the baryon asymmetry [1, 2], aslong as active-sterile neutrino scattering is not so rapidas to decohere the sterile neutrinos prior to coherentoscillation. At even later times, t tosc, the oscillationrate is rapid compared to the Hubble scale, and sterileneutrinos produced at different times have differentphases; averaging over the entire ensemble results ina cancellation of the asymmetry production from eachsterile neutrino. Therefore, the lepton flavour asymme-tries are dominated by production at t ∼ tosc.
Magnitude of Yukawa couplings: The rate of pro-duction of individual lepton flavour asymmetries in (4)is O(|F |4); therefore, increasing the magnitude of theYukawa couplings gives a substantial enhancement to theindividual flavour asymmetries. Also, larger Yukawa cou-plings give a more rapid transfer rate from the individuallepton flavour asymmetries into a total lepton asymmetry
5
at order O(|F |6), further enhancing the baryon asymme-try. However, increase in the Yukawas also enhances thewashout processes. The characteristic time scale associ-ated with the washout of lepton flavour α is
tαwashout ∼1
Γ(Lα → N†)∼ 1
(FF †)ααT. (11)
If the Yukawa coupling is too large, then washoutoccurs before the electroweak phase transition, and alllepton flavour asymmetries are driven to zero in theequilibrium limit. The final asymmetry is maximal whenthe Yukawa couplings are large enough to equilibrateall lepton species immediately after the electroweakphase transition, but not larger. Since sphaleronsdecouple at TW, the baryon asymmetry is frozen in eventhough the lepton asymmetry is rapidly damped awayshortly after the phase transition. This is true, providedthat the Yukawa couplings are not so large that thesterile neutrinos decohere before the time of asymmetrygeneration, which can occur if the oscillation time isvery late (∆MN/MN 1).
Lepton flavour dependence in scattering rates:As discussed in Section II C, the generation of individ-ual flavour asymmetries at order O(|F |4) due to sterileneutrino oscillations is insufficient. A total SM leptonasymmetry is generated only at O(|F |6) due to flavour-dependent scattering rates according to Eq. (6). In theabsence of lepton flavour-dependent effects, the individ-ual scattering rates are all equal, Γ(Le → N†) = Γ(Lµ →N†) = Γ(Lτ → N†), and the total lepton asymmetryremains zero even at higher orders in F . Therefore, dif-ferences in lepton flavour rates are crucial to generatea baryon asymmetry. Fortunately, there is already evi-dence for lepton flavour dependence in interactions withneutrinos. First, the structure of the LH neutrino massesand mixing angles tells us that their Yukawa couplingsare non-universal, and proportional to the mixing anglesθij . Second, the CP phases appear in very particularterms in the interaction: for instance, the Dirac phase δappears only in terms proportional to sin θ13. Changingthe phase can lead to constructive or destructive inter-ference of rates involving a specific lepton flavour. Thetotal lepton asymmetry is maximized in regions of param-eter space that accentuate the differences between leptonflavour interaction rates. The importance of flavour ef-fects was recently emphasized in [10].
Considering all of these effects, we identify regimes ofparameter space depending on the relative time scales ofoscillation, sterile neutrino equilibration, and sphalerondecoupling. Because these are the only time scales in theproblem, this results in three regimes that completelycharacterize the minimal model. Regime I is defined bytosc . tW teq and was considered in the original workson baryogenesis from neutrino oscillations [1, 2]. Equili-bration occurs long after the electroweak phase transitionand so washout effects are entirely irrelevant. However,the total baryon asymmetry is also generally too small
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Regime III
FIG. 2. Time evolution of the individual lepton flavour asym-metries and the total lepton asymmetry in different parameterregimes. The dashed vertical line indicates the electroweakphase transition. (Top) Regime I: The Yukawa couplingsare small enough that washout effects are always irrelevant;(Centre) Regime II: The Yukawa couplings are large enoughthat equilibration of the sterile neutrinos occurs at the elec-troweak phase transition; (Bottom) Regime III: Large lep-ton flavour dependence in the L → N† rates, such thatΓ(Le → N†) Γ(Lµ → N†), Γ(Lτ → N†). The Yukawacouplings are even larger than in Regime II, such that Lτand Lµ equilibrate completely with the sterile neutrinos atT TW, and Le comes into equilibrium at T ∼ TW.
because the small Yukawa couplings implied by the latewashout also suppress the rates in Eq. (6). To account forthe observed baryon asymmetry, the oscillation time scalemust be made as large as possible and thus close to theelectroweak scale, tosc . tW. This coincidence of scalesis achieved through a fine-tuning of the sterile-neutrino
6
mass splitting ∆MN ≡MN3−MN2
. (10−6−10−8)MN .In Regime II, the sterile neutrinos come into equilib-rium around the weak scale, and the sphaleron processesfreeze out “just in time” to avoid washing out the en-tire asymmetry tosc teq ∼ tW. This alleviates some ofthe tuning necessary in the mass splitting ∆MN , but thecoincidence between the equilibration scale and the elec-troweak scale requires some tuning in the Yukawa cou-plings. Finally, in Regime III, first identified in [10],the Yukawa couplings are made even larger. This gener-ically results in the equilibration time, teq, being evenearlier than the weak scale. This would have been a phe-nomenological disaster with the entire asymmetry beingwashed-out too early, except that destructive interferencein the scattering rate for one of the lepton flavours makesit much smaller than the others. The result is a time scaleordering tosc teq, α teq, β ≈ tW. In this regime, evenless tuning is necessary in the mass splitting, but the un-naturally early equilibration time for two of the flavoursrequires a substantial tuning in the Yukawa couplings.We quantify this combined tuning below in section V D.
In Fig. 2, we show the time evolution of the individuallepton flavour asymmetries and total lepton asymmetryfor each of the regimes discussed above. In Regime II,the downward spikes in the total lepton asymmetry andthe Lµ asymmetry are the result of a change in sign ofthe asymmetry, as shown in Eqs. (7)-(8) and the relateddiscussion. The spike indicates the time of the equili-bration of Lµ around T = 200 GeV ∼ TW. In RegimeIII, these spikes occur due to Lµ and Lτ equilibrationaround T & 3 TeV. Indeed, the hallmark of Regime IIIis that the equilibration temperatures of Lµ and Lτ aremore than an order of magnitude larger than that of Leat T . 100 GeV, whereas in Regime II, all flavours comeinto equilibrium near the same scale.
III. MODEL PARAMETERIZATION
The model we consider is the νMSM [2], with the La-grangian given in Eq. (1). At low temperatures T MI ,the SM neutrinos acquire a mass in the effective theory,
(mν)αβ = 〈Φ〉2(FM−1
N FT)αβ. (12)
This is the usual see-saw suppression of the SM neutrinomasses, and its parametric scaling was discussed in rela-tion with Eq. (2) . The observed masses and mixings ofthe SM neutrinos are [17],
• |∆m2atm| = 2.35+0.12
−0.09 × 10−3 eV2,
m2sol = 7.58+0.22
−0.26 × 10−5 eV2,
and∑imνi . eV.
• sin2 θ12 = 0.312+0.018−0.015,
sin2 θ23 = 0.42+0.08−0.03,
and sin2 2θ13 = 0.096± 0.013.
The data are consistent with one of the LH neutrinosbeing massless, and one of the sterile neutrinos beinglargely decoupled from the SM. This decoupled sterileneutrino, which we take for concreteness to be N1, isa possible dark matter candidate, but does not play arole in leptogenesis. Therefore, we consider an effectivetheory with only N2 and N3 as the two sterile neutrinos.
The assumption of one massless LH neutrino fixesthe other LH neutrino masses, up to a discrete choiceof mass hierarchy. We consider the normal hierarchy,where mν1 = 0, mν2 =
√m2
sol ≈ 9 meV, and mν3 =√|∆m2
atm| ≈ 49 meV. It is also possible for the set-up tobe realized in the inverted hierarchy (m3 = 0, m2 ∼ m1),but the qualitative dependence of the baryon asymme-try on model parameters is similar to the normal hier-archy, while the value of the baryon asymmetry can besomewhat larger with an inverted hierarchy [8]. We fo-cus exclusively on the normal hierarchy as a benchmark,since our results also qualitatively hold in the invertedhierarchy and easily generalize to that scenario.
The magnitudes of the Yukawa couplings Fα2, Fα3 arecrucial for successful baryogenesis. To better understandthe connection between the Yukawa couplings and physi-cal parameters, we decompose the Yukawa couplings withthe Casas-Ibarra parameterization [18],
F =i
〈Φ〉Uν√mν R
∗√MN . (13)
Here Uν is the MNS matrix containing mixing angles andCP phases from the LH neutrino mixing [19], mν is a di-agonal matrix of LH neutrino masses, and MN is a diago-nal matrix of sterile neutrino masses. The matrix R is anorthogonal matrix specifying the mixing between sterileneutrino mass and interaction eigenstates; in the case ofa normal neutrino hierarchy with two sterile neutrinos,R has the form
RαI =
0 0cosω − sinωsinω cosω
, (14)
where ω is a complex angle parameterizing the misalign-ment between sterile neutrino mass and interaction eigen-states. We refer the reader to Appendix A for the explicitform of the decomposition in Eq. (13). We note thatwhen ω = 0, the Yukawa interactions can be diagonal-ized in the sterile neutrino mass basis, and the interfer-ence/oscillation necessary for leptogenesis is absent.
The parameters that emerge out of the decompositionof the Yukawa couplings can be grouped as follows:
• Parameters currently fixed by experiment:
– Three LH neutrino masses, (mν)i;
– Three LH neutrino mixing angles from theMNS matrix: θ12, θ13, θ23.
7
• Parameters constrained by experiment:
– The VEV 〈Φ〉. If Φ is the SM Higgs, it hasthe value 〈Φ〉 = 174 GeV; otherwise, it is un-determined but constrained by bounds on newsources of electroweak symmetry breaking (seeSection VI C).
• Unconstrained parameters:
– Two phases from LH neutrino mixing: a Ma-jorana phase, η, and a Dirac phase, δ;
– Two sterile neutrino masses, (MN )I ;
– A complex mixing angle, ω.
These parameters, in turn, influence the dynamics ofbaryogenesis in three key ways: by setting the magni-tude of the Yukawa couplings, controlling lepton flavourdependence in scattering rates, and providing the CPviolation necessary for baryogenesis. We consider eachin turn.
Magnitude of Yukawa couplings: The scaling of theYukawa magnitude is generally fixed by the see-saw re-lation mν ∼ FFT〈Φ〉2/mN . In particular, increasingeither the sterile or LH neutrino masses enhances theYukawa couplings, as does decreasing 〈Φ〉. The pres-ence of the complex parameter ω, however, allows for anenhancement of the Yukawas well beyond the naıve see-saw value. The reason is that, while FFT is fixed bythe LH neutrino masses, physical rates depend on thequantities FF † and F †F . It is therefore possible thatsquaring the complex terms in ω leads to large cancella-tions among the terms in F such that FFT FF †. TheYukawa couplings are actually enhanced exponentially bythe imaginary part of ω as
|F |2 ∝ mνMN
〈Φ〉2 cosh(2Imω) (15)
when |Imω| is large. The parameter Imω does not other-wise have any impact on experimentally observed quan-tities, and it can be thought of as a dial to enhance therates of sterile neutrino production and scattering.
When |Imω| 1, the see-saw relation only holds dueto a precise cancellation of parameters, and there is avery specific alignment of the Yukawa couplings associ-ated with this enhancement. While the Yukawa matrixis stable under radiative corrections, and therefore tech-nically natural in the sense of ’t Hooft [20], the phys-ically observed parameters (such as the LH neutrinomasses) change significantly under small perturbationsof the Yukawa matrix entries. This results in tuning inthe sense of Barbieri and Giudice [21], which is quanti-fied by observing how the physical masses mν changeunder perturbations of the Yukawa coupling, F . Forconcreteness, consider a Yukawa matrix decomposed ac-cording to eq. (13), which is then perturbed according
to F22 → (1 + ε)F22 but otherwise left unchanged. Inthe simplest case with θαI = Reω = 0, the LH neutrinomasses have a simple analytic form and the eigenvaluem2 changes according to
d logm2
dε= 1 + cosh(2Imω). (16)
With non-zero θ and Reω, the change in LH neutrinomasses from ε is apportioned among the differentmass eigenstates, but the overall shift is of the orderof eq. (16). We have also verified this numerically.Therefore, while it is possible to exponentially enhancethe rates relevant for baryogenesis in the minimal sterileneutrino model, it necessarily implies an exponentialtuning of the Yukawa couplings to obtain the observedLH neutrino masses. Since much of the viable parameterspace in the baryogenesis studies of [5, 8, 10] requires|Imω| 1, baryogenesis in these set-ups is unnatural inthe sense of ref. [21].
Lepton flavour-dependent scattering rates: As dis-cussed in Section II, lepton flavours must have differentscattering rates into sterile neutrinos in order to convertthe individual lepton flavour asymmetries into a totallepton asymmetry. There are several parameters in Fthat contribute differently to the asymmetry generationand scattering rates for each flavour. The LH neutrinomass hierarchy (m1 m2 m3) and hierarchy amongmixing angles (θ13 θ12 ∼ θ23) provides some differen-tiation among lepton flavours; this typically suppressesrates of electron scattering versus the corresponding ratesfor muons and taus.
The CP phases δ and η also play an important rolein distinguishing lepton flavours. Since they appear indifferent entries of the MNS matrix, these phases canlead to constructive or destructive interference amongprocesses involving specific lepton flavours. For instance,when δ + η = −π/2, there is destructive interference
between the Le → N†2 and Le → N†3 amplitudes,resulting in Γ(Le → N†) Γ(Lµ → N†) ∼ Γ(Lτ → N†)[10, 22]. Such interference effects can substantially alterthe rates of asymmetry creation or washout, modifyingthe total baryon asymmetry. We emphasize that thiseffect is independent of the role of the phases in CPviolation. In fact, over many regions of parameter space,this is the dominant contribution of the Majorana andDirac CP phases to the baryon asymmetry.
CP violation: There are three sources of CP violationin the theory: the Majorana phase η, the Dirac phaseδ, and the complex sterile neutrino mixing angle ω. Itsimaginary part, Imω, gives rise to two phases that arerelated to one another,
tanφ1 ≡ tan(arg cosω) = − tan(Reω) tanh(Imω),(17)
tanφ2 ≡ tan(arg sinω) = cot(Reω) tanh(Imω). (18)
According to the Sakharov conditions, a source of CP vi-olation is required to generate a baryon asymmetry. Ex-cept where δ, η, and Imω all vanish, there is generically
8
an O(1) phase contributing to baryogenesis coming froma combination of these individual phases. Indeed, evenin the regions where the phases are aligned to give con-structive or destructive interference in scattering rates(as discussed above), the combination of the LH neu-trino phases δ + η is generally non-zero. For instance,with Imω 1, constructive (destructive) interference inΓ(Le → N†) occurs when δ + η is π/2 (−π/2), which isalso the phase alignment that gives maximal CP viola-tion in the LH lepton sector. Therefore, CP violationis generally O(1) over the entire parameter space of theminimal model. The vanishing of one phase (such as theexperimentally accessible Dirac phase δ) does not neces-sarily constrain the other phases relevant for leptogene-sis [6] or determine the relative lepton flavour scatteringrates.
Finally, we note that there have been some statementsin the literature implying that the limit of large |Imω|somehow leads to more CP violation. As shown inEqs. (17)-(18), the CP violating phases φ1 and φ2 de-pend on tanh(Imω) and quickly saturate with increasedImω. Instead, as discussed above, the main effect of|Imω| 1 is an exponential enhancement of the Yukawacouplings, not the presence of an additional source of CPviolation.
IV. ASYMMETRY EVOLUTION EQUATIONS
In this section, we review the formalism for comput-ing the baryon asymmetry from sterile neutrino oscilla-tions. Along the way we discuss corrections we madeto the existing formalism to account for the contribu-tion of equilibrium processes to the evolution of the lep-ton asymmetries. Because of the central role of coherent
oscillations among the sterile neutrino, the density ma-trix formalism is well-suited for tracking the evolution ofabundances and coherences between states. Such an ap-proach was used in [1], with [2] being the first analysisto include all of the relevant terms in the evolution equa-tions, with various factors corrected in subsequent work[7, 9]. More sophisticated approaches have also beentaken, such as separately evolving different momentummodes in the density matrix [9] or using non-equilibriumquantum field theory [10]. These results are very similarto those using thermally averaged density matrix evolu-tion; for example, ref. [9] find enhancements of the totalbaryon asymmetry of factors of 10-40% when computingthe asymmetries separately in each momentum mode rel-ative to thermal averaging. Because of the computationalsimplicity of thermal averaging, and the small changesto the total asymmetry when using more sophisticatedmethods, we employ thermal averaging in the currentwork.
As in earlier works, we follow the evolution of the ster-ile neutrino and anti-neutrino density matrices, ρN , andρN , as well as the LH lepton asymmetry density matrix,ρL−L. Here, L refers to a single component of the SU(2)lepton doublet, and is a matrix in lepton flavour space.The diagonal elements of the density matrices are equalto the abundances of the corresponding fields normalizedby the equilibrium abundance,
ρii(t) =Yi(t)
Y eqi (t)
, (19)
and the off-diagonal terms correspond to the coherencesbetween the fields.
The evolution equations have the general form [2],
dρNdt
= −i[H(t), ρN ]− 1
2Γ(L† → N)2×2, ρN − ρeq
LI2×2 −
1
2γavT F †ρL−LF, (20)
dρNdt
= −i[H(t), ρN ]− 1
2Γ(L→ N†)2×2, ρN − ρeq
L I2×2+1
2γavT FTρL−LF
∗, (21)
dρL−Ldt
= −1
4Γ(L→ N†)3×3, ρL−L+
1
2γavT
(FρNF
† − F ∗ρNFT). (22)
Here (, ) [, ] denotes a matrix (anti-) commutator.The evolution equations satisfy the relation Tr(ρN−N −2ρL−L) = 0, which reflects the conservation of the globalL − N charge. The interested reader can find a de-tailed account of these equations in Appendix B. Herewe instead concentrate on a qualitative understandingof the significance of the different terms that appear inEqs. (20)-(22).
1. The Hamiltonian terms induce coherent oscil-lations between diagonal and off-digonal compo-nents of the density matrix. Such a term is notincluded for the LH leptons because these rapidlydecohere in the thermal bath of the early universe,and therefore only on-diagonal density matrix com-
ponents are non-zero.
2. The terms proportional to Γ(L → N†) lead toboth production of the sterile neutrino abundance(through the terms ∝ ρeq
L ) and destruction (orwashout) of the asymmetries in both N and L.These interactions drive the fields towards theirequilibrium distributions and erase any asymme-tries. These scattering rates are proportional tothe temperature T and can often be factorized intothe form
Γ(L→ N†) = γav(T )T |F |2. (23)
The factor γav(T ) parameterizes the rate after fac-toring out the Yukawa couplings and has been com-
9
Lα
N †I
Φ∗
Lα N †I
H∗
t†L
Lα
tcR
N †I
V
Lα
N †I
Φ∗
Lα N †I
Φ∗
t†L
Lα
tcR
N †I
V
Lα
N †I
Φ∗
Lα N †I
H∗
t†L
Lα
tcR
N †I
V
FIG. 3. Representative Feynman diagrams for sterile neutrinocreation through (Top) 1 → 2 processes; (Centre) gaugeboson 2→ 2 scattering; (Bottom) quark 2→ 2 scattering.
puted completely at leading order [23]. Representa-tive diagrams contributing to the process are shownin Fig. 3. The rates from ref. [23] sum over bothlepton doublet components.
3. The final terms in each line produce the asym-metries in each sector. For ρL−L, the coherentoscillations of the sterile neutrinos (encoded in theoff-diagonal terms of ρN ) create an asymmetry inindividual LH lepton flavours at O(|F |4). For thesterile neutrinos, an excess of Lα over Lα translatesinto an excess of scattering into N† vs. N , sourc-ing an asymmetry in the sterile sector as well atO(|F |6). The rate of transfer of SM lepton flavourasymmetries into sterile neutrino asymmetries de-pends on the asymmetry in the flavour Lα and the
rate Lα → N†I ; the matrix ρN appears between theF † and F matrices because of this sensitivity tolepton flavour effects.
The initial conditions are such that there are initiallyno sterile neutrinos ρN (0) = ρN (0) = 0 and no primor-dial lepton asymmetry ρL−L(0) = 0. In existing works inthe literature, these initial conditions are used to evolveEqs. (20)-(22) down to the weak scale to determine the
asymmetry ρL−L(tW). At the weak scale, chemical po-tential relations relate the size of the baryon asymmetryto the total LH lepton asymmetry [24]:
Y∆B(tW) = −28
79
∑
α
Y∆Lα(tW). (24)
Since sphalerons decouple at the electroweak phase tran-sition, the final baryon asymmetry is frozen at this timeand is simply Y∆B(tW) as given in Eq. (24).
The above approach to solving for the late-time baryonasymmetry is incorrect, as it neglects the effects of inter-actions between SM fields, which are rapid compared tothe active-sterile neutrino scatterings. Because of thesescatterings, the asymmetries in individual lepton flavourscreated by sterile neutrino oscillations are rapidly dis-tributed among all SM fields. Since the asymmetries aredestroyed only through interactions of the LH leptonswith sterile neutrinos, this modifies the relative rates ofasymmetry creation and destruction.
The effects of equilibrium scatterings on the evolutionof an asymmetry have been understood and corrected ina different context, namely of baryogenesis through weak-scale dark matter scatterings [25]. Our approach here issimilar: we include in our density matrices only quan-tities that are preserved by the equilibrium SM interac-tions, justifying the absence of such rapid interactionsin the evolution equations. Specifically, we exchange theanomalous asymmetries in individual lepton flavours Lαin (22) for asymmetries of B − 3Lα, which are exactlyconserved by SM scatterings. This modifies the densitymatrix equations: because Eq. (22) now represents theevolution of ∆(B − 3Lα), instead of ∆Lα, all terms area factor of −2× 3 larger than for the individual LH lep-ton species, due to the re-definition of the charge andan SU(2) factor from summing over charged and neutralcomponents of Lα. The last term in Eq. (22), which cre-ates the individual ∆Lα asymmetries, is otherwise un-modified, since the coherent scattering N† → Lα gen-erating the asymmetry is not sensitive to the details ofSM thermal scatterings. By contrast, the first term inEq. (22), which destroys the individual lepton flavourasymmetries, is proportional only to the individual asym-metry ∆Lα, not the total ∆(B−3Lα) asymmetry3. Thisis because the asymmetry in fields carrying B − 3Lα isdivided by the equilibrium scatterings among many SMfields, such as the quarks and RH charged leptons, whichhave no direct coupling to sterile neutrinos. To write thecorrect evolution equation, we relate the LH lepton abun-dances from Eq. (22) to the B−3Lα charges through thechemical potential constraints of the SM interactions:
(ρL−L)α = − 1
2133
∑
β
[221 δαβ − 16(1− δαβ)] ρB−3Lα .
(25)
3 The same is true for the terms in the kinetic equations generatingthe sterile neutrino asymmetries.
10
We see immediately that the rates are suppressed by atleast ≈ 6× 221/2133 ≈ 0.6 due to the fact that much ofthe asymmetry leaks out of the LH leptons, and possi-bly more due to the other negative terms in the rotationmatrix. Having made this substitution, we express thedensity matrix evolution equations purely in terms of ρN ,ρN , and ρB−3Lα . These quantities are then evolved downto the weak scale, and instead of Eq. (24) the final baryonasymmetry is
Y∆B(tW) =28
237
∑
α
Y∆(B−3Lα)(tW). (26)
The equations are actually easiest to solve in a differentbasis: we provide the complete evolution equations inthis basis and a few details on numerical integration inAppendix B. In that same appendix, we also include theeffects arising from a phase-space suppression of certaindiagrams contributing to the ∆Lα destruction rates; suchmodifications were first discussed in [9].
V. BARYON ASYMMETRY IN THE νMSM
This section is devoted to a quantitative confirmationof the qualitative features discussed above. We demon-strate the continuous transition between the different pa-rameter regimes discussed in section II and depicted inFig. 2. We also show that, over the entire parameterspace of the minimal model, the observed baryon asym-metry requires a very specific alignment of the model pa-rameters to within at leastO(10−5). We do not perform acomprehensive scan of possible parameters in the νMSM,as such a scan was carried out in [5, 8] for Regimes I andII with kinetic equations similar to those we use here,and our results are in qualitative agreement with theirs.However, our qualitative picture provides a simple un-derstanding for the parameters maximizing the baryonasymmetry found in ref. [5, 8]. Finally, with regime IIIwe explore a new region of parameter space whose exis-tence was first demonstrated in ref. [10] with a couple ofrepresentative points in a model with three sterile neu-trinos (instead of the two we consider here). We expandupon ref. [10] with a comprehensive scan of the parame-ter space and demonstrate the emergence of Regime IIIas a continuous part of the other regimes.
The observed baryon asymmetry of the Universe is [26]
Y∆B ≈ 8.6× 10−11. (27)
Throughout this section, we take TW = 140 GeV asthe temperature of the electroweak phase transition, cor-responding to a SM Higgs mass of 126 GeV [5]. Forthe numerical solutions of the evolution equations inall regimes, we take values consistent with Section III:m1 = 0, m2 = 9 meV, m3 = 49 meV, sin θ12 = 0.55,sin θ23 = 0.63, and sin θ13 = 0.16. For concreteness, weuse Reω = π/4 throughout our analyses, as this gives anappreciable asymmetry for all values of Imω. The asym-metry does not change substantially with this angle: for
example, Reω = π/2 gives a comparable baryon asym-metry, with somewhat larger values than Reω = π/4 forImω ≈ 0.5−3, and smaller values elsewhere (see Fig. 7).
A. Regime I
The generation of a baryon asymmetry in this regimeis maximized when the rates of LH lepton scatteringinto sterile neutrinos are very different for different lep-ton flavours. However, the Yukawa couplings here areset to their small, natural values expected from the see-saw relation, Eq. (2). Therefore, even with large differ-ences in lepton flavour scattering rates, the baryon asym-metry in Regime I is not large enough to account forthe observed asymmetry unless the coherent oscillationtime is maximal, which requires a strong degeneracy be-tween the masses of N2 and N3. A mass degeneracy of∆MN/MN ∼ O(10−6 − 10−8) is needed, depending onImω.
To illustrate this, we show in Fig. 4 the baryon asym-metry for a set of parameters where the magnitude ofthe Yukawa coupling F is held fixed, while the relativerates of Le → N†, Lµ → N†, and Lτ → N† are al-lowed to vary. We see that the asymmetry is largestwhen flavour dependence on the scattering rates is sig-nificant, i.e. the rates of Γ(Lα → N†) are very different,especially the rates of Lµ and Lτ . In this regime, the con-tribution from the Le flavour asymmetry and scatteringrate is subdominant, due to the smallness of θ13 and thefact that m3 has the largest Yukawa coupling to the ster-ile sector. Over the parameters scanned in Fig. 4, therate Γ(Le → N†) is typically . 3% of the correspond-ing Lµ and Lτ , and the asymmetry in Le is . 10% ofthe Lµ and Lτ asymmetries4. The asymmetry vanisheswhen Γ(Lµ → N†)/Γ(Lτ → N†) ≈ 0.8, which is whenthe difference in Lµ → N† and Lτ → N† rates exactlycompensates for the difference in flavour asymmetries.
This behaviour is easily understood with the qualita-tive lessons learned in previous sections. In this regime,the Yukawa couplings are sufficiently small that equili-bration of the active and sterile neutrinos is irrelevantas it occurs long after the electroweak time scale. Asa result, the asymmetries in individual lepton flavoursremain unchanged from their generation at the time ofcoherent sterile neutrino oscillation to the sphaleron de-coupling at TW. The baryon asymmetry is determinedby the slow transfer of asymmetry from individual leptonflavours into the sterile sector, Eq. (6), which generates atotal lepton asymmetry. It is therefore maximized withlarge differences in rates associated with different leptonflavours.
4 This is true except in a small window around Γ(Lµ →N†)/Γ(Lτ → N†) ≈ 0.7, where the Lτ asymmetry becomesvery small and the Le asymmetry is important.
11
0.5 1.0 1.5
0
5. ´ 10-11
1. ´ 10-10
1.5 ´ 10-10
GHΜ ® N+L GHΤ ® N+L
YD
B
FIG. 4. Illustration of the importance of flavour-dependent ef-fects in generating a non-zero baryon asymmetry in Regime I.The baryon asymmetry is plotted as a function of the ratio be-tween the muon-sterile neutrino scattering rate and the corre-sponding tau-sterile rate. The asymmetry is maximized whenthe washout rates for µ and τ are substantially different, aspredicted by Eq. (6). The mass splittings are ∆MN = 3×10−8
GeV (blue, short dash), 10−7 GeV (purple, solid), 3 × 10−7
GeV (black, long dash). The ratio of muon to tau rates is var-ied by changing the relative values of δ − η, while the overallMNS CP phase and other parameters are held fixed (MN = 1GeV, ω = π/4 − i/2, δ + η = π/2). The horizontal dashedline indicates the observed baryon asymmetry.
B. Regime II
As discussed in Section II, the maximization of thebaryon asymmetry in this region is done by increasingthe scattering rates as much as possible while avoidingequilibration (and subsequent washout) before the elec-troweak time scale, Γ(Lα → N†) ∼ H when T ∼ TW.This optimization is achieved by setting the Yukawa cou-plings to be much larger than their natural see-saw rela-tions through a careful alignment of the different Yukawacouplings so that |F |2/F 2 ∼ cosh(2Imω) ∼ O(102).
To illustrate this, we show in Fig. 5 the magnitude ofthe baryon asymmetry as a function of Yukawa couplingmagnitude. We set the Yukawa coupling magnitude bychanging Imω. Unlike in Regime I, the Le flavour hereis important; although the flavour asymmetry in Le istypically smaller than in Lµ or Lτ , it also equilibratesmore slowly, and so can be comparable to the asymme-tries in Lµ or Lτ if the latter are partially washed out.As expected, the magnitude of the baryon asymmetry in-creases monotonically with the Yukawa coupling, exceptfor a small region where the asymmetry changes sign dueto the onset of Lτ → N† equilibration, which modifiesthe total lepton asymmetry as discussed in Section II.The total baryon asymmetry is maximized in the regionwith Γ(Lα → N†) ∼ H(TW). Any further increase be-yond this point results in an equilibration time earlierthan the electroweak scale and a precipitous decrease inthe baryon asymmetry.
For the parameter points in Fig. 5, we see that theobserved baryon asymmetry is obtained in Regime II
1 5 10 50 100 500 100010-15
10-14
10-13
10-12
10-11
10-10
10-9
coshH 2 Im w L»Y D
B»
1 5 10 50 100 500 100010-17
10-16
10-15
10-14
10-13
10-12
GHGe
VL
electronmuontau
Regime I Regime II
FIG. 5. Baryon asymmetry (Bottom) and lepton washoutrates at T = 140 GeV (Top) as a function of the Yukawa cou-pling magnitude, which is parameterized by Imω. The plotshows the continuous evolution from Regime I to Regime II.The washout rates are shown for taus (top), muons, and elec-trons (bottom); the dashed line in the upper plot shows theHubble scale at TW = 140 GeV, while the dashed line in thelower plot shows the observed Y∆B = 8.6×10−11. The asym-metry is maximized around Γ(Le → N†) ∼ Γ(Lµ → N†) ≈H. Other parameters held fixed: MN = 1 GeV, ∆MN = 10−5
GeV, η = −π/4, δ = 3π/4, Imω > 0.
with ∆MN/MN ∼ 10−5, which is less degenerate thanin Regime I. However, this reduction in parameter tun-ing from mass degeneracy is compensated by a tun-ing of the Yukawa couplings that goes like one part incosh 2 Imω ∼ O(102) in Regime II. Therefore, the pa-rameter space giving the observed baryon asymmetry isno less tuned than in Regime I, but the tuning arisesfrom alignments in both the sterile neutrino masses andthe Yukawa couplings.
C. Regime III
In Regime III, the effects of large Yukawa couplingsand maximal differences in active-sterile neutrino scat-tering rates combine to give the largest possible asym-metry. The Yukawa couplings are set even larger thanin Regime II, enhancing the generated baryon asymme-
12
try. As expected, such large Yukawas also cause theequilibration time scale to occur earlier than the elec-troweak scale. If this were true for all lepton flavours, itwould have resulted in complete washout of the asymme-try, which is phenomenologically unacceptable. However,in this part of parameter space, destructive interferencein the Le → N† rate allows the Le asymmetry to avoidearly washout by remaining out of equilibrium until theelectroweak scale when Γ(Le → N†) ≈ H(TW). This ef-fect was first discussed in the context of leptogenesis withthree sterile neutrinos [10].
A strong destructive interference in the rate Γ(Le →N†) is possible when Imω 1 and [10, 22]
tan θ13 =m2
m3sin θ12, (28)
cos(δ + η) = −1. (29)
The parameters in Eq. (28) are fixed by oscillation data.Interestingly, the current best-fit value of θ13 happensto be very close to satisfying Eq. (28), leading to verystrong suppression of Γ(Le → N†) when the Majoranaand Dirac phases satisfy Eq. (29). In practice, the de-structive interference is still very effective if δ + η arewithin about 10% of this critical value, but cannot devi-ate much more than this. The Le asymmetry generationrate is proportional to sin(2Reω) in the destructive in-terference limit cos(δ+ η)→ 0, and so the baryon asym-metry is maximized for Reω ≈ π/4.
In Fig. 6, we show the baryon asymmetry for a choiceof parameters satisfying (29). The Yukawa couplings arechanged by varying Imω, and we demonstrate how theasymmetry varies continuously from Regimes I to III. Asin Fig. 5, the magnitude of the baryon asymmetry mono-tonically increases (except when it changes sign at thepoint of Lτ → N equilibration). Once again, the baryonasymmetry is maximal when the Yukawa couplings havea value such that Le is just coming into equilibrium atTW, while Lµ and Lτ equilibrate at earlier times.
As a result of the enhanced asymmetry in Regime III,the required mass degeneracy for the baryon asymmetryof the universe is the smallest of any region of parameterspace. However, even here a degeneracy of ∆MN/MN ∼10−3 is necessary, as is a tuning of the Yukawa couplingof one part in 104 and an alignment of the CP phasesδ + η ≈ −π/2 to within 10%. As in all of the otherregimes, the combined tuning in mass degeneracy andYukawa couplings is larger than one part in 105.
D. Tuning
To close this section, we show in Fig. 7 the tuning nec-essary in the different regimes in one continuous plot.We take the tuning to be the product of the tuningof the mass splitting, MN/∆MN , and the alignment ofthe Yukawa couplings, cosh(2Imω). While such tun-ings/alignments are technically natural, there is no ex-planation for their structure in the minimal model. This
1 10 100 1000 10410-14
10-13
10-12
10-11
10-10
coshH 2 Im w L»Y D
B»
1 10 100 1000 10410-17
10-16
10-15
10-14
10-13
10-12
GHGe
VL
electronmuontau
Regime I Regime II Regime III
FIG. 6. Baryon asymmetry (Bottom) and lepton washoutrates at T = 140 GeV (Top) as a function of the Yukawacoupling magnitude, which is parameterized by Imω. Theplot shows the continuous evolution from Regime I to RegimeIII for parameters exhibiting strong destructive interferencein Γ(Le → N†). The curves are the same as Fig. 5. Theasymmetry is again maximized around Γ(Le → N†) ≈ H.Other parameters held fixed: MN = 1 GeV, ∆MN = 10−3
GeV, η = δ = −π/4, Imω > 0.
might simply be a feature of nature, but it could also bea hint for additional structure beyond the νMSM. Thecase for considering an extended model becomes partic-ularly cogent if it brings with it new observable effects.We present one such extension in the following section,show that it entirely alleviates the needed tuning, anddiscuss its observability in on-going collider searches.
VI. BARYON ASYMMETRY WITH ALEPTOPHILIC HIGGS
In the previous section, we saw that increasing theYukawa couplings can generally lead to larger asymmetrygeneration. Schematically, we have that [2, 6, 7],
asymmetrygeneration rate
∝ Im(FF †)2αα ∼
m2νM
2N
〈Φ〉4 cosh(2Imω).
(30)
13
1 10 100 1000 104
105
106
107
108
109
1010
coshH 2 Im Ω L
Tunin
gM
easu
re
FIG. 7. Fine-tuning required to obtain the observed baryonasymmetry, Y∆B = 8.6 × 10−11. The tuning measureis cosh(2Imω)MN/∆MN . The other parameters used are(black, solid) η = δ = Reω = π/4; (purple, dotted) η = δ =−Reω = −π/4; (blue, dashed) η = 2.42, δ = 0.5, Reω = π/2.For all points, ∆MN is fixed by imposing the observed baryonasymmetry.
For very small Yukawas (regime I), the asymmetry gen-eration rate is correspondingly small and one must tunethe mass splitting of the sterile neutrino so that the os-cillation time scale is as long as possible and approachesthe electroweak time scale. We saw that it is possibleto increase the asymmetry generation rate, and therebyalleviate some of this tuning, by increasing the Yukawacouplings (while keeping the seesaw relations intact) withan alignment controlled by the parameter Imω (regimeII). However, this increase cannot proceed indefinitelysince, at some point, the equilibration time scale becomesas early as the electroweak time scale. At this point,the washout processes that kick-in and act to reduce theasymmetry scale as
asymmetrywashout rate
∝ (FF †)αα ∼mνMN
〈Φ〉2 cosh(2Imω).
(31)
The scaling relations Eqs. (30) and (31) show that, whileone can initially enhance the asymmetry by increasingthe alignment in the Yukawas, cosh(2Imω) 1, thisgain is saturated once the time scale of equilibration co-incides with the electroweak scale and washout processesbecome relevant. Regime III circumvents this saturationand supports even stronger alignment in the Yukawas byhaving a different equilibration time scale for the differentlepton flavours.
However, the above scaling relations suggest an al-ternative approach. The asymmetry generation rate,Eq. (30), depends strongly on the value of the electroweakVEV, 〈Φ〉. It can be greatly enhanced if the LH neutrinomasses arise from a new source of electroweak symme-try breaking. Such a scenario can arise in a leptophilicTwo Higgs Doublet Model (2HDM), with one scalar cou-pling exclusively to leptons and acquiring a much smallerVEV. This idea was also mentioned in ref. [10] as a way
to alleviate the needed alignment in the Yukawa cou-plings. Here, however, we see that a smaller leptophilicHiggs VEV has a much more pronounced effect than justtuning the Yukawa couplings to be large in the mini-mal model (|Imω| 1): the asymmetry generation andwashout rates scale very differently with the VEV, whilethey have the same scaling with Imω.
A. The Model
A Z2 symmetry is typically required to prevent bothHiggs fields from coupling to the same fermions, induc-ing tree-level flavour-changing neutral currents [27]. Onepossible choice is the leptophilic (or Type IV) 2HDM:
Lleptophilic = µ21|Φ1|2 − µ2
2|Φ2|2 −λ1
4|Φ1|4 −
λ2
4|Φ2|4
+ λuQΦ1uc + λdQΦ∗1d
c
+ λ` LΦ∗2Ec + F LΦ2N + h.c. (32)
In this model, Φ1 is a SM-like Higgs giving mass to thequarks, and Φ2 is a leptophilic Higgs giving mass to thecharged leptons and LH neutrinos. It is motivated bythe observation that the heaviest lepton masses are muchsmaller than the heaviest quark masses, and might obtaintheir masses through a field with a smaller VEV. Indeed,λτ ∼ O(1) for 〈Φ2〉 ≈ 2 GeV.
If Φ2 acquires a VEV through a negative mass-squaredterm in the potential, a prediction of the associated scalarmasses would be m2 ∼ λ〈Φ`〉2. Collider searches ruleout the existence of any such charged states below 100GeV. Therefore, Φ2 must instead acquire a VEV througha linear tadpole term in its potential. We assume thiscomes from a mixing with the SM Higgs, which arisesfrom additional terms in the 2HDM potential:
V2HDM ⊃ µ2mixΦ1Φ∗2 + h.c. (33)
µ2mix can be naturally smaller than µ2
1 and µ22 since it
breaks the Z2 symmetry5. The ratio of VEVs is
tanβ ≡ 〈Φ1〉〈Φ2〉
≈ µ22
µ2mix
, (34)
which can account for the observed pattern 〈Φ1〉 〈Φ2〉for µ2 µmix.
B. Asymmetry Generation
To demonstrate the larger asymmetries possible in theleptophilic 2HDM, we compute the baryon asymmetriesobtained in both the leptophilic 2HDM and the mini-mal model, and we show their ratio in the left panel of
5 For our purposes, other Z2-breaking terms such as |Φ1|2Φ1Φ∗2can be absorbed into a redefinition of µmix.
14
Fig. 8. To make a direct comparison, we choose a com-mon set of parameters for both models in Regime III,where the effects are most pronounced: MN = 1 GeV,∆MN = 1 GeV, η = δ = −Reω = −π/4. For theleptophilic 2HDM, we fix Imω = 1 and take as a freeparameter the VEV 〈Φ2〉. For the minimal model, wechoose the value of Imω such that the electron washoutrate Γ(Le → N†) is equal to the rate in the leptophilic2HDM for each 〈Φ2〉. The ratio of the baryon asymmetryin the leptophilic 2HDM compared to the minimal modelgrows quadratically as Φ2 decreases from the SM value,as predicted by Eq. (30). We see in the right panel ofFig. 8 that enhancements to the baryon asymmetry ofO(103 − 104) are possible in the leptophilic Higgs modelover the minimal model. The resulting baryon asymme-try in the leptophilic 2HDM is sufficiently large that nomass degeneracy is required to obtain the observed value.Therefore, the leptophilic 2HDM removes the need forany tuning in both the masses of the sterile neutrinosand the Yukawa couplings FαI if tanβ ≈ 20 − 80. Thesame conclusion holds true for many other values of theCP phases and mixing angles.
One might wonder why such a complete elimination ofthe tuning in the mass splitting is possible in the lep-tophilic Higgs model but not in the νMSM. In otherwords, why is it now possible to obtain the correct asym-metry with a much shorter coherent oscillation time ascompared to the minimal model? The answer is foundin the scaling relations, Eqs. (30) and (31). In the min-imal model, one can only increase the Yukawa couplingsthrough alignment, cosh(2Imω) 1. But since theasymmetry generation and washout rates scale in thesame way with cosh(2Imω), this increase is saturatedwhen the equilibration rate is as early as the electroweakscale. Unfortunately, the baryon asymmetry at the pointof saturation is still too small, unless one tunes the sterile-neutrino mass splitting to achieve a longer coherent os-cillation time. In our extended model with smaller lep-tophilic Higgs VEV, the asymmetry generation scales like∼ 〈Φ2〉−4, which rises much faster with a smaller VEVthan the washout rate, ∼ 〈Φ2〉−2. The point of satu-ration is still reached in the extended model, but be-cause the asymmetry generation rate increases more thanthe washout rate, the point of saturation gives a baryonasymmetry which is orders of magnitude larger.
C. Experimental Implications of the LeptophilicHiggs Model
In the leptophilic Higgs νMSM, which leads to the ob-served baryon asymmetry without any tuning, there ex-ist new, weakly charged scalars. In a natural theory, thenew scalar masses are expected to lie below ∼TeV. Thus,experimental studies of the Higgs sector can also act asprobes of physics related to leptogenesis.
2 5 10 20 50 100
1
10
100
1000
104
leptophilic VEV HGeVL
YD
BHle
pto
phil
icL
YD
BHm
inim
alL
2 5 10 20 50 100
10-14
10-13
10-12
10-11
10-10
10-9
leptophilic VEV HGeVLÈY
DB
È
Leptophilic model
Minimal model
FIG. 8. (Top) Ratio of the baryon asymmetry in the lep-tophilic Higgs model with Imω = 1 to the asymmetry inthe minimal, single Higgs model with Imω set such that theLe washout rate is the same in both scenarios for each datapoint. The enhancement to the asymmetry from changing〈Φ2〉 is quadratically larger than the tuned, minimal model.(Bottom) Baryon asymmetry in the leptophilic Higgs model(top, purple) and minimal model (bottom, black) with the Lewashout rate the same in both scenarios for each data point.The dashed line indicates the observed baryon asymmetry ofthe universe. (Both) Other parameters are fixed at MN = 1GeV, ∆MN = 1 GeV, η = δ = −Reω = −π/4.
The baryon asymmetry is largest in models where
tanβ ≡ 〈Φ1〉〈Φ2〉
1, (35)
in which the Yukawa couplings of Φ2 to leptons are en-hanced. In the 2HDM with scalar mixing induced by thepotential (33), the ratio of VEVs is given by (34), andthe CP -even mixing angle is
sinα ≈ µ2mix
m2h − µ2
2
, (36)
where m2h = λ〈Φ1〉2/4 and we assume µ2
mix m2h − µ2
2.When µ2 ≈ mh, the above approximations break down,and sinα ≈ 1/
√2.
Constraints from SM Higgs searches: Currently,the strongest constraints on the leptophilic Higgs modelare from measurements of the observed SM-like Higgs
15
1000500200 300 7001.0
2.0
1.5
leptophilic Higgs mass HGeVL
GHH
®ΤΤ
LG
HH®
ΤΤL SM LHC 7+8 TeV HCMSL
LHC 14 TeV, 300fb
ILC 250 GeV, 250fb
FIG. 9. Signal strength of SM Higgs decay to τ+τ− as afunction of the leptophilic Higgs mass h` with tanβ = 20.The enhancement of the τ+τ− signal strength comes fromthe modification of the SM Higgs coupling to taus (37). Thehorizontal solid line is the current CMS 7+8 TeV 2σ bound[28], and the horizontal dashed (dotted) lines show the 2σreach for LHC14 at 300 fb−1 (ILC at 250 GeV, 250 fb−1).The reach estimates are from [29].
decays. In the leptophilic model, the SM-like Higgs hasa modified τ Yukawa coupling
λτ → λτ tanβ sinα ≈ λτ(
µ22
m2h − µ2
2
). (37)
Even though the SM Higgs doublet does not directlycouple to leptons, we see that its coupling to taus isactually enhanced due to a combination of mixing withΦ2 and the tanβ enhancement of the lepton Yukawacouplings. With µ2 ∼ mh ≈ 126 GeV, the SM-like Higgscoupling to τ+τ− is so large that it is excluded by dataof SM Higgs decays into taus from the Large HadronCollider (LHC); the current bound is µ2 & 220 GeV. InFig. 9, we show the current exclusion [28], along with the2σ reach of the 14 TeV LHC with 300 fb−1 of data, andthe 2σ reach of a 250 GeV International Linear Collider(ILC) with 250 fb−1 of data [29]. For this analysis, wecalculated the ΦSM → τ+τ− signal strength with 2HDMC[30].
Direct searches for leptophilic Higgs: Theleptophilic-Higgs-like scalars couple to the electroweakgauge bosons and can be directly produced at colliders.Such searches are currently weaker than the above con-straints, but are relevant in extended models where theΦ` VEV and mixing with the SM are not determinedcompletely by (33), and therefore the modification of theSM Higgs coupling is not as large as (37). Direct searchesmay also be more relevant for higher luminosities at theLHC. There is one new CP -even scalar H0
` , a CP -oddscalar A0
` , and charged scalars H±` . The dominant pro-
duction modes are pp → H0` /A
0` + H±` → 3τ + ντ (see
Fig. 10). There is also a 4τ final state, but the produc-tion cross section is smaller. The best channel to use insearches for such final states has the same-sign taus decayleptonically and the other tau(s) decay hadronically [31];
q
q
W +
H+ℓ
H0ℓ
τ+
ντ
τ+
τ−
FIG. 10. Feynman diagram for production of the leptophilicHiggs states at the LHC and their decays.
the current constraints from CMS with 8 TeV, 19 fb−1
are mH` = mA` . 150 GeV [32]. The search in same-sign dileptons + hadronic taus has a discovery potentialof m` ≈ 300 GeV for LHC14 with ∼ 200 fb−1. Combin-ing this channel with other proposed search modes (suchas the all-hadronic channel [33]) could have even higherreach.
Finally, we comment on the possibility that Φ2 onlygives mass to the neutrinos, while the charged leptonsacquire a mass through 〈Φ1〉. In this scenario, the phe-nomenology changes dramatically; instead of decayingthrough the large τ Yukawa interaction, Φ2 can only de-cay through either the coupling to sterile neutrinos orthe mixing with the SM-like Higgs. The latter is themore likely possibility due to the smallness of the ster-ile neutrino Yukawas FαI , in which case Φ2 looks ex-actly like a heavy SM Higgs but with a much smallersingle-production cross section and with enhanced pairproduction. There are no constraints on the leptophilicscalars H` in this scenario, although future searches inthe “golden channel” Φ2 → 4` may eventually provideconstraints. A linear collider may prove to be a betterprobe of such final states. If H` instead decays throughthe Yukawa coupling, then H±` → `±+N , and H±` lookslike a slepton decaying to a massless neutralino. The lep-ton is most likely to be a µ or τ because, in the normal hi-erarchy, these couple strongest to N . For the non-tunedmodels of leptogenesis, the Yukawa couplings are largeenough that H±` decays promptly; the slepton boundsconstrain mH` . 300 GeV with decays to muons [28, 34],and there are no constraints with decays to taus abovethe LEP bound of 90 GeV [35].
VII. CONCLUSIONS
In this paper, we attempted to provide a comprehen-sive and coherent overview of the mechanism of baryogen-esis through neutrino oscillations. Focusing on the phys-ical time scales involved in the problem rather than theunderlying model parameters, we identified three broadregimes depending on the relative ordering of this timescales (the neutrino oscillation time scale, the equilibra-tion time scale, and the sphaleron decoupling time scale).
16
While these regimes are not new - they have been identi-fied in past works [2, 5–10] either through scans or withindividual points - our work endeavours to clarify thephysical basis for these regimes and for their intercon-nectedness. On a more quantitative level, our calculationalso includes an improvement upon previous calculationsof the baryon asymmetry by including the effects of scat-terings between left-handed leptons and the thermal bathduring asymmetry generation.
One of the less appealing features of this mechanismis the need to fine-tune some of the model parameters.We showed that this is fundamentally related to a cer-tain coincidence required of the physical time scales andthat a tuning of no less than one part in 105 is necessarythroughout the parameter space. In one regime the tun-ing is entirely in the mass terms of the sterile neutrinos,while in other regimes it is also manifested strongly in acertain alignment of the Yukawa couplings, which allowsthem to be much larger than what the see-saw relation,Eq. (2), would naıvely imply.
The fine-tuning of model parameters is an unavoidablefeature of the minimal model. In the last part of thiswork we considered an extended model with an additionalelectroweak Higgs boson that predominantly couples tothe sterile neutrinos and SM leptons. We showed thatthe correct baryon asymmetry can be obtained with notuning of the sterile neutrino parameters if the leptophilicHiggs boson has a VEV of order ∼ GeV. This extra Higgsboson can be searched for, and discovered, at the LHC.
ACKNOWLEDGMENTS
We would like to thank Eder Izaguirre, Gordan Krn-jaic, Maxim Pospelov, and Carlos Tamarit for helpfuldiscussions. BS is supported in part by the CanadianInstitute of Particle Physics. IY is supported in partby funds from the Natural Sciences and Engineering Re-search Council (NSERC) of Canada. This research wassupported in part by Perimeter Institute for TheoreticalPhysics. Research at Perimeter Institute is supportedby the Government of Canada through Industry Canadaand by the Province of Ontario through the Ministry ofResearch and Innovation.
Appendix A: Casas-Ibarra parameterization
The Yukawa coupling FαI in Eq. (1) can be written as
FαI =i
〈Φ〉Uν√mνR
√MN , (A-1)
where mν is the diagonal matrix of LH neutrino massesas determined from oscillation data, Uν is the MNS LHneutrino mixing matrix, and MN is the diagonal matrixof sterile neutrino masses. For the normal hierarchy, and
with m1 = 0, the decomposition of Uν and R are
Uν =
1 0 00 cos θ23 sin θ23
0 − sin θ23 cos θ23
diag
(eiδ/2, 1, e−iδ/2
)
×
cos θ13 0 sin θ13
0 1 0− sin θ13 0 cos θ13
diag
(e−iδ/2, 1, eiδ/2
)
×
cos θ12 sin θ12 0− sin θ12 cos θ12 0
0 0 1
diag
(1, e−iη, 1
), (A-2)
R =
0 0cosω sinω− sinω cosω
. (A-3)
The θij are the usual mixing angles. In general, thereis another Majorana phase appearing in Uν , but it onlyappears in terms proportional to m1, which we assumeto be zero as explained in Section III.
Appendix B: Full density matrix evolution equations
In Section IV, we outlined the basic method for calcu-lating the baryon asymmetry, providing schematic evo-lution equations for ρN , ρN , and ρL−L in Eqs. (20)-(22).It turns out to be simpler to move to a different basis forthe sterile neutrino density matrices [6]:
δρ+ =ρN + ρN
2− ρeq
L I2×2, (B-1)
δρ− = ρN − ρN . (B-2)
In this basis, the sterile neutrinos are expressed in termsof deviations from equilibrium: δρ+ is the sterile CP -even deviation from equilibrium, while δρ− is the sterileCP -odd deviation from equilibrium. This basis has thenice property that, in thermal equilibrium, δρ+, δρ−, andρB−3Lα all vanish. Furthermore, when looking at individ-ual terms that generate an asymmetry, it can be conve-nient to separate out the action of CP violation on sterileneutrino oscillations themselves, which source δρ−, andthe presence of CP violation in L−N scattering, whichinvolves Imω, as well as the Dirac and Majorana CPphases from the MNS matrix.
Before providing the full kinetic equations used in thepaper, we highlight an additional modification of the evo-lution equations from [9]. In section IV we parametrizedthe rate Γ(L→ N†) in Eq. (23) with the aid of the func-tion γav(T ). This function is shown in Fig. (11). Asakaand Ishida observed that the assumption that the N pro-duction and L destruction rates are parameterized by thesame γav(T ) is erroneous: while at leading order, N pro-duction always proceeds from SM initial states, some Ldestruction processes have sterile neutrinos in the ini-tial state and are consequently suppressed by the out-of-equilibrium N phase space density. For both 2 → 2 and1↔ 2 scattering processes, 1/3 of the lepton destructionprocesses are suppressed by a factor of ρN .
17
100 104 106 1080.000
0.002
0.004
0.006
0.008
0.010
0.012
T HGeVL
Γav
1 ® 2 scattering
2 ® 2 scattering
total Γav
FIG. 11. Temperature dependence of the coefficient γav(T )in the sterile neutrino production rate as defined in Eq. (23).We show the contributions from 1 ↔ 2 scattering, 2 → 2scattering, and the total leading order rate. All values aretaken from [23].
The various contributions to γav(T ) can be extractedfrom the analysis in [23]. We then modify the leptondestruction rate such that the coefficient changes as
γav(T )→ 2
3γav(T ) +
ρN + ρN6
γav(T ). (B-3)
Putting together all of the modifications due to the ro-tation to the δρ+, δρ− basis, the transformation to theB − 3Lα lepton charges, and the above modifications tothe scattering rates, we have
idδρ+
dt= [ReHN , δρ+]− i
2ReΓN , δρ+ −
iT
4γav(T )(F †ρL−LF − F
TρL−LF∗)
− iT
24γav(T )
[F †ρL−LF, 2δρ+ + δρ− − FTρL−LF
∗, 2δρ+ − δρ−]
+i
2[ImHN , δρ−] +
1
4ImΓN , δρ−, (B-4)
idδρ−dt
= [ReHN , δρ−]− i
2ReΓN , δρ− −
iT
2γav(T )(F †ρL−LF + FTρL−LF
∗)
− iT
12γav(T )
[F †ρL−LF, 2δρ+ + δρ−+ FTρL−LF
∗, 2δρ+ − δρ−]
+ 2i[ImHN , δρ+] + ImΓN , δρ+, (B-5)
idρB−3L
dt=
3i
2ΓL, ρL−L+
iT
2γav(T )
[F (2δρ+ + δρ−)F † + F ∗(2δρ+ − δρ−)FT
]ρL−L
− 3iT
2γav(T )
[F (2δρ+ + δρ−)F † − F ∗(2δρ+ − δρ−)FT
]. (B-6)
Here, we have defined
(ΓN )IJ = γav T (F †F )IJ , (B-7)
(HN )IJ =T
8(F †F )IJ +
M2I −M2
J
2T, (B-8)
(ΓL)αβ = γav T Re(FF †)αβ , (B-9)
and ρL−L can be expressed in terms of ρB−3L accordingto Eq. (25).
As noted in [8], this system of differential equationsis “stiff” due to the fact that the sterile neutrino oscil-lation frequency increases monotonically with time, andintegrating from an early time to tW can be very compu-tationally intensive. Fortunately, there is a workaround:washout cannot equilibrate much before the weak scaleor else the asymmetry would be entirely wiped out, whileoscillation frequencies are very fast at tW only when theoscillations began at a much earlier time. This separationof tosc tW allow us to divide the equation evolutioninto two regimes. At early times, when the oscillationsare responsible for generating the lepton asymmetries,we solve the full kinetic equations above. At late times,
the oscillations are rapid and the contributions to theasymmetry average to zero, and so we solve the samesystem of equations except that we set the off-diagonalcomponents to zero. The solutions are matched at a timetmatch = 500tosc, where
tosc ≈(MPl/1.66
√g∗)
1/3
2(M23 −M2
2 )2/3(B-10)
is the time when oscillations become rapid. We havechecked numerically that at least 90% of the asymmetryis generated by tmatch, and we have also done extensivenumerical checks of the validity of this two-stage solu-tion. When tosc ∼ tW, then it is fast to integrate the fullequations right up to tW.
Integration can also be slow when we include an in-terpolating function representation of the numerical co-efficients γav(T ) for the solution of the full equations.Because these functions vary slowly with time, and therates are most relevant at the times of asymmetry gen-eration and washout, we set γav = γav(tosc) for the firststage of the solution. In the second stage of the solution,we incorporate the full time dependence of γav(T ).
18
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