Table Of ContentBaryogenesis 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.
Baryogenesisthroughneutrinooscillationsisanelegantmechanismwhichhasfoundseveralreal-
izations in the literature corresponding to different parts of the model parameter space. Its appeal
stems from its minimality and dependence only on physics below the weak scale. In this work we
showthat,byfocusingonthephysicaltimescalesofleptogenesisinsteadofthemodelparameters,a
morecomprehensivepictureemerges. Thedifferentregimespreviouslyidentifiedcanbeunderstood
asdifferentrelativeorderingsofthesetimescales. Thisapproachalsoshowsthatallregimesrequire
acoincidenceoftimescalesandthisinturntranslatestoacertaintuningoftheparameters,whether
in mass terms or Yukawa couplings. Indeed, we show that the amount of tuning involved in the
minimalmodelisneverlessthanonepartin105. Finally,weexploreanextendedmodelwherethe
tuning can be removed in exchange for the introduction of a new degree of freedom in the form of
4
a leptophilic Higgs with a vacuum expectation value of the order of GeV.
1
0
2
I. INTRODUCTION is marred by the need for fine-tuning between its fun-
n
damental parameters in order to successfully generate
a
J Itisatheoreticallyattractivepossibilitytoexplainthe the correct baryon asymmetry in the different regimes.
3 baryon asymmetry of the Universe through the mecha- We show that, while the fine-tuning present in different
2 nism of sterile neutrino oscillations [1, 2]. The model is regimes appears in different parameters, the total fine-
simple, containing only right-handed sterile neutrinos in tuning is always at least at the level of 1/105.
] addition to the Standard Model (SM). These neutrinos While possibly aesthetically unappealing, the persis-
h
are light ( GeV) and do not require any new physics tent fine-tuning needed throughout the parameter space
p
- atinaccess∼iblyhighenergyscales. The modelevenholds is not grounds to discount the framework. It is, how-
p thepossibilitythatoneofthesterileneutrinoscanbethe ever, possibly an invitation to explore extensions of the
e non-baryonic dark matter of the Universe [2–5]. Several minimal model which would either alleviate some of the
h
[ pastworkshavefounddifferentchoicesoftheparameters necessary tuning, or explain it as a small departure from
thatleadtothecorrectbaryonasymmetryandidentified a more symmetric phase. In the final part of this work,
2 several regimes [2, 5–10]. we therefore consider the possibility of an additional
v It is the purpose of this work to present a unified Higgs doublet with a small electroweak vacuum expec-
9
perspectiveonleptogenesisthroughneutrinooscillations, tation value (VEV), which is coupled to all the leptons
5
4 weaving the disjoint regimes previously identified in the through larger-than-usual Yukawa couplings. We show
2 literature into a single continuous picture. Our analy- that, aside from ameliorating the fine-tuning needed for
. sis focuses on the three important time scales for baryo- successful leptogenesis, such a leptophilic Higgs doublet
1
0 genesis: the time of sterile neutrino oscillations, active- canbesearchedfordirectlyandindirectlyinhigh-energy
4 sterile neutrino equilibration time, and sphaleron decou- reactions accessible at the LHC.
1 pling time. We identify the different regimes as different Thepaperisorganizedasfollows: inSectionII,were-
: relative orderings of these time scales and demonstrate viewthequalitativefeaturesofbaryogenesisthroughneu-
v
i the continuity of separate parts of the parameter space. trino oscillations, providing a unified perspective on the
X This allows us to point out the most important effects physics responsible for leptogenesis in different regimes.
r contributingtotheasymmetryineachregime. Alongthe In Section III, we discuss the different model parame-
a
way, we provide some improvement upon the calculation ters and their specific roles in leptogenesis. Section IV
of the baryon asymmetry from neutrino oscillations by presents the formalism for computing the baryon asym-
including the effects of scatterings between left-handed metry and describes the modifications we make to the
(LH) leptons and the thermal bath during asymmetry asymmetry evolution equations to account for SM ther-
generation. mal scatterings. In Section V, we confirm our unified
Aside from providing a unified framework, centering perspectiveonleptogenesiswithnumericalstudiesofthe
the discussion around the relevant time scales for baryo- different parameter regimes. We explicitly show the
genesis brings to the forefront the need for an accidental tuning required to obtain the observed baryon asym-
coincidencebetweenthedifferent,unrelatedscalesinthe metry throughout the different regimes of the minimal
problem. It is therefore no surprise that the framework model. We then discuss the baryon asymmetry in a
leptophilic Two Higgs Doublet Model (2HDM) in Sec-
tion VI, demonstrating how the asymmetry can be en-
hanced relative to the minimal model, eliminating any
∗ [email protected] need for tuning. We also discuss experimental implica-
† [email protected] tions of this extended model. We close with a discussion
2
and some remarks about the inclusion of dark matter in known, sphaleron processes active for T > T violate
W
the model. baryon number (B), but preserve the difference between
baryon and lepton numbers (B L), processing the pri-
−
mordialleptonasymmetryintoabaryonasymmetry[13].
II. REVIEW OF BARYOGENESIS THROUGH Thus, the final baryon asymmetry of the universe ob-
NEUTRINO OSCILLATIONS served today is determined by the lepton asymmetry at
the time of sphaleron decoupling, T [14]. In everything
W
It has been appreciated for some time that, in ex- that follows, we therefore assume the physics associated
tensions of the Standard Model with sterile neutrinos, with sphalerons to be present and concentrate on a de-
oscillations of the sterile neutrinos can be responsible tailed understanding of leptogenesis alone.
for baryogenesis [1, 2]. We discuss the basic framework
in this section, emphasizing the essential features that
emerge from the model. We begin by presenting the B. Sakharov Conditions for Leptogenesis
model,andfollowwithaqualitativeaccountoftheasym-
metry generation and its dependence on the underlying The Sakharov conditions [15] necessary for generating
model parameter. a total lepton asymmetry are satisfied in the νMSM:
1. ViolationofStandardModelleptonnumber:
A. Neutrino Minimal Standard Model (νMSM) The Yukawa coupling in Eq. (2) preserves a gen-
eralized lepton number L N under which both
−
We consider the Type I see-saw model for generating SM and sterile neutrinos are charged. The L N
−
LH neutrino masses, also known as the Neutrino Min- symmetry is broken by the sterile neutrino Majo-
imal Standard Model (νMSM). In this scenario, the ranamass,butratesof(L N)-violatingprocesses
−
SM is supplemented by three sterile neutrinos, NI, with are suppressed by a factor of MN2/T2 relative to
Majorana masses, M . In the basis where the charged (L N)-preserving rates, and so total lepton num-
lepton and sterile neuItrino masses are diagonal, the La- ber−violation is generally ineffective for T (cid:38) TW.
grangian is (see ref. [11] for a recent review) However, scattering processes Lα → NI† through
the Yukawa interactions in Eq. (2) violate Stan-
νMSM =FαILαΦNI +(MN)ININI, (1) dard Model lepton number, allowing the creation
L
of equal asymmetries in L and N such that L N
where Φ is the electroweak doublet scalar responsible for −
is still conserved. Sphalerons then convert the SM
givingmasstotheSMneutrinos,andL istheSMlepton
α L asymmetry into a baryon asymmetry.
doubletofflavourα. Whenthescalardoubletdevelopsa
VEV, Φ = 0, these interactions generate a small mass 2. CP violation: There are three CP phases in the
(cid:104) (cid:105) (cid:54)
fortheLHneutrinosthroughthesee-sawmechanism[12]
Yukawa couplings F . Together, these provide a
αI
of the order,
sufficient source for leptogenesis through neutrino
oscillations.
F2 Φ 2
m (cid:104) (cid:105) (2)
ν
∼ MN 3. Departure from equilibrium: As discussed
Φ 2 F 2 GeV above,ifMN (cid:46)TW,sterileneutrinoscatteringsare
∼0.1 eV 100(cid:104)G(cid:105)eV 10−7 M . out of equilibrium provided there is no abundance
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) N (cid:19) of sterile neutrinos at the earliest times following
Here, we used the parameters and masses most relevant inflation. Unlike many models of baryogenesis, the
for the current work: a scalar VEV around the elec- out-of-equilibrium condition is satisfied for an ex-
troweak scale, Φ 100 GeV; sterile-neutrino masses tended period in the early universe, with equili-
(cid:104) (cid:105) ∼
aroundorbelowtheelectroweakscale, M GeV; and bration only occurring after sphaleron decoupling,
N
small Yukawa couplings, F 10−7 10−8.∼With weak- T (cid:46)T .
∼ − W
scale sterile neutrino masses, the Yukawa couplings of
the neutrino are only somewhat smaller than the elec- Inthefollowingsubsection, weelaborateonthephysi-
tron Yukawa. calprocessesresponsiblefortheproduction(anddestruc-
With such a small coupling between LH leptons and tion) of the lepton asymmetry, and clarify which param-
sterile neutrinos, active-sterile neutrino scattering is out eters most strongly control the size of the baryon asym-
ofequilibriumintheearlyuniverseanddoesnotbecome metry.
rapid until T (cid:46) T , where T 140 GeV is the tem-
W W
≈
perature of the sphaleron decoupling at the electroweak
phasetransition[5]. Ifthereisanegligibleconcentration C. Asymmetry Creation and Washout
of sterile neutrinos immediately following inflation, then
thesterile-neutrinoabundanceremainsbelowitsequilib- The basic stages leading to the creation of a total SM
rium value until it re-thermalizes at T (cid:46)T . As is well lepton asymmetry are shown in Fig. 1. Immediately fol-
W
3
time
coherent
oscillations
Lα NI NJ Lβ Lα NI
H H H
∗ ∗
Y =0 Y >0 Y >0
∆L1 ∆L1 ∆L1
Y ,Y =0 Y ,Y <0 Y ,Y <0
∆L2 ∆L3 ∆L2 ∆L3 ∆L2 ∆L3
Y =0 Y =0
α Y∆Lα =0 α ∆Lα α ∆Lα "
! !
!
FIG. 1. The basic stages leading to the creation of a total lepton asymmetry from left to right: out-of-equilibrium scattering
of LH leptons begin to populate the sterile neutrino abundance at order O(|F|2); after some time of coherent oscillation, a
small fraction of the sterile neutrinos scatter back into LH leptons to create an asymmetry in individual lepton flavours at
orderO(|F|4);finally,atorderO(|F|6),atotalleptonasymmetryisgeneratedduetoadifferenceinscatteringrateintosterile
neutrinos among the different active flavours.
lowing inflation, there is no abundance of sterile neutri- states1 and remain coherent as long as the active-sterile
nos, and out-of-equilibrium scatterings mediated by the Yukawacouplingremainsoutofequilibrium,sinceinthe
Yukawacouplingsbegintopopulatethesterilesector, as minimal model there are no other interactions involving
shown on the left side of Fig. 1. The sterile neutrinos the sterile neutrinos.
are produced in a coherent superposition of mass eigen-
Some time later, a subset of the sterile neutrinos scat- In the absence of efficient washout interactions, which is
ter back into LH leptons, mediating L L transi- ensured by the out-of-equilibrium condition, this differ-
α β
→
tions as shown in the centre of Fig. 1. Since the sterile ence in rates creates asymmetries in the individual LH
neutrinos remain in a coherent superposition in the in- lepton flavours L .
α
termediate time between scatterings, the transition rate
Denoting the individual LH flavour abundances (nor-
L L includes an interference between propagation
α β malized by the entropy density, s) by Y n /s and
med→iated by the different sterile neutrino mass eigen- Lα ≡ Lα
theasymmetriesbyY Y Y , wenotethatthe
states. The different mass eigenstates have different ∆Lα ≡ Lα− L†α
processesatorder (F 4)discussedthusfaronlyconvert
phases resulting from time evolution; for sterile neutri- O | |
L into L , conserving total SM lepton number,
nos N and N , the relative phase accumulated during a α β
I J
small time dt is e−i(ωI−ωJ)dt, where
(M )2 (M )2 (M )2 Y = Y =0 at (F 4). (5)
ω ω N I − N J N IJ. (3) ∆Ltot ∆Lα O | |
I J
− ≈ 2T ≡ 2T α
(cid:88)
Intheinteractionbasis, thisCP-evenphaseresultsfrom
an oscillation between different sterile neutrino flavours, Since sphalerons couple to the total SM lepton number,
and explains the moniker of leptogenesis through neu- it follows that no baryon asymmetry is generated at this
trino oscillations. order as well, Y = 0. Total lepton asymmetry is,
∆Btot
When combined with the CP-odd phases from the however, generated at order (F 6): the excess in each
Yukawa matrix, neutrino oscillations lead to a difference individual LH lepton flavourOdue| t|o the asymmetry from
between the Lα Lβ rate and its complex conjugate, Eq. (4) leads to a slight increase of the rate of L N†
→ α
Γ(L →L )−Γ(L† →L†)∝(cid:88)Im(cid:20)exp(cid:18)−i(cid:90) t MI2J dt(cid:48)(cid:19)(cid:21) vs.L†α →N. Theresultisthatactive-sterilelepton→scat-
α β α β 2T(t(cid:48)) teringscanconvertindividualleptonflavourasymmetries
I(cid:54)=J 0 into asymmetries in the sterile neutrinos. But, since the
×Im(cid:2)FαIFβ∗IFα∗JFβJ(cid:3). (4) ratesofconversion,Γ(Lα N†),aregenericallydifferent
→
foreachleptonflavourα,thisleadstoadepletionofsome
oftheindividualleptonasymmetriesatafasterratethan
1 This is true assuming generic parameters with no special align- others, leading to an overall SM lepton asymmetry and
mentofthesterile-neutrinointeractionandmasseigenstates. an overall sterile neutrino asymmetry. Because L N
tot
−
4
is conserved for T m , this gives D. Qualitative Dependence of Asymmetry on
N
(cid:29)
Model Parameters
dY dY
∆Ntot = ∆Ltot = Y Γ(L N†). (6)
dt dt ∆Lα α → I
(cid:88)α,I In the discussion above we have highlighted three
Therefore, while the sum of the LH lepton flavour asym- important elements for generating a non-zero baryon
metries vanishes at (F 4) as in Eq. (5), the fact that asymmetry: the coherent oscillation and interference
O | |
the LH leptons scatter at different rates into the sterile of different sterile neutrino states in L L scat-
α β
→
sector results in a non-vanishing total lepton asymmetry tering; the magnitudes of the Yukawa couplings which
at (F 6). determine the rates of processes generating the lepton
O | |
The asymmetry generated at early times can be de- asymmetry; and the presence of differences in scatter-
stroyed later when the different lepton flavours establish ing rates of individual LH lepton flavours into sterile
chemical equilibrium with the sterile neutrinos. This oc- neutrinos. We now elaborate on each, as they have
curs when the transition rate exceeds the Hubble rate, implicationsforwhatpartsofparameterspacemaximize
Γ(Lα N†) (FF†)ααT (cid:38) H. At this time, the par- the baryon asymmetry in the minimal model, and how
→ ∼
ticularleptonflavourLα reachchemicalequilibriumwith new interactions can enhance the asymmetry.
the sterile neutrinos, but not yet with the other SM lep-
tons since lepton flavour is still conserved to a good ap- Sterile neutrino mass splitting: In the absence
proximation. Suppose, for concreteness, that Lτ comes of a sterile-neutrino mass splitting, the sterile-neutrino
intoequilibriumwiththesterileneutrinos,butLµandLe masses and couplings can be simultaneously diagonal-
remainoutofequilibrium. Then,beforeLτ comesinto ized, and there is no coherent oscillation/interference as
equilibrium, we can use the approximate conservation required in Fig. 1. The size of the mass splitting dic-
of ∆Ltot ∆N =0 to write tates the time scale at which the phases of the coher-
−
Y x, Y y, Y z, ently evolving sterile neutrino eigenstates become sub-
∆Le ≡ ∆Lµ ≡ ∆Lτ ≡ stantially different. From Eq. (4) and the Hubble scale
in a radiation-dominated universe [16],
Y =Y =x+y+z. (7)
∆N ∆Ltot
Once L comes into equilibrium with the sterile neutri- 1.66√g∗T2
τ H = (9)
nos, the Yukawa coupling F L HN leads to the equi- M
τI τ I Pl
librium chemical potential relation2 µ = µ . Be-
Lτ − N (g is the number of relativistic degrees of freedom), we
cause the fields L and N both have two components ∗
τ
have an (1) phase from oscillation at
andopposite-signchemicalpotentials, theirasymmetries
O
are therefore equal and opposite: Y = Y . The
∆Lτ − ∆N 2/3
flavour asymmetries ∆L and ∆L remain unchanged 3√M
µ e t Pl . (10)
btheecafuascet tthhaeyt tahree∆noLt yet i∆nNequil0ibraitumhi.ghTtoegmetpheerratwuirteh, osc ≈(cid:32)1.66g∗1/4√2(MN23−MN22)(cid:33)
tot
− ≈
this implies that after Lτ equlibration: The oscillation time is later for smaller mass splittings.
x+y At later times, the rates of scattering between active-
Y =Y = . (8)
∆N ∆Ltot 2 sterile neutrinos is larger relative to the Hubble scale,
Therefore, at the point of equilibration of the τ flavour, and the asymmetry is consequently larger. Therefore,
thetotalleptonasymmetryrapidlychangesfromx+y+z small but non-zero sterile neutrino mass splittings
to (x+y)/2, and the ∆L asymmetry rapidly changes enhance the size of the baryon asymmetry [1, 2], as
τ
from z to (x+y)/2. We see that nothing remains of long as active-sterile neutrino scattering is not so rapid
the origina−l ∆L flavour asymmetry z by the time the as to decohere the sterile neutrinos prior to coherent
τ
tau flavour comes fully into equilibrium. Both the to- oscillation. At even later times, t tosc, the oscillation
(cid:29)
tal lepton asymmetry and L flavour asymmetry change rate is rapid compared to the Hubble scale, and sterile
τ
dramaticallywhenequilibriumisreached,withtheasym- neutrinos produced at different times have different
metry after equilibrium being fixed by the remaining phases; averaging over the entire ensemble results in
asymmetries in the flavours L and L . It is also ap- a cancellation of the asymmetry production from each
e µ
parent that if all of the flavours come into equilibrium, sterile neutrino. Therefore, the lepton flavour asymme-
∆Le = ∆Lµ = ∆Lτ = ∆N = 0. Clearly, sphaleron in- tries are dominated by production at t∼tosc.
teractions must decouple before this time is reached or
no baryon asymmetry results from sterile neutrino oscil- Magnitude of Yukawa couplings: The rate of pro-
lation. duction of individual lepton flavour asymmetries in (4)
is (F 4); therefore, increasing the magnitude of the
O | |
Yukawacouplingsgivesasubstantialenhancementtothe
2 Strictlyspeaking,therelationisµLτ+µΦ=−µN;forillustrative individualflavourasymmetries. Also,largerYukawacou-
purposes,andbecauseµΦ istypicallysmallcomparedtotheLH plingsgiveamorerapidtransferratefromtheindividual
SMleptonflavourasymmetries,wesetµΦ=0here. leptonflavourasymmetriesintoatotalleptonasymmetry
5
atorder (F 6), furtherenhancingthebaryonasymme- Temperature(cid:72)GeV(cid:76)
O | | 106 105 104 103 102
try. However, increase in the Yukawas also enhances the
washout processes. The characteristic time scale associ- 10(cid:45)7 RegimeI
ated with the washout of lepton flavour α is (cid:200) Tau
Y(cid:68)L 10(cid:45)9 Muon
(cid:200) Electron
1 1 y
tαwashout ∼ Γ(L N†) ∼ (FF†) T. (11) metr10(cid:45)11
α αα m
→ y
as10(cid:45)13
If the Yukawa coupling is too large, then washout on
pt
occurs before the electroweak phase transition, and all le10(cid:45)15
lepton flavour asymmetries are driven to zero in the Total
equilibriumlimit. Thefinalasymmetryismaximalwhen 10(cid:45)1170(cid:45)20 10(cid:45)18 10(cid:45)16 10(cid:45)14 10(cid:45)12 10(cid:45)10
the Yukawa couplings are large enough to equilibrate time(cid:72)seconds(cid:76)
all lepton species immediately after the electroweak Temperature(cid:72)GeV(cid:76)
106 105 104 103 102
phase transition, but not larger. Since sphalerons
decouple at T , the baryon asymmetry is frozen in even 10(cid:45)7 RegimeII
W
though the lepton asymmetry is rapidly damped away (cid:200) Tau
shortly after the phase transition. This is true, provided Y(cid:68)L 10(cid:45)9 Muon
(cid:200) Electron
y
that the Yukawa couplings are not so large that the metr10(cid:45)11
sterile neutrinos decohere before the time of asymmetry m
y
generation, which can occur if the oscillation time is as10(cid:45)13
n
o
very late (∆MN/MN (cid:28)1). lept10(cid:45)15
Total
Lepton flavour dependence in scattering rates:
10(cid:45)17
10(cid:45)20 10(cid:45)18 10(cid:45)16 10(cid:45)14 10(cid:45)12 10(cid:45)10
As discussed in Section IIC, the generation of individ-
time(cid:72)seconds(cid:76)
ual flavour asymmetries at order (F 4) due to sterile Temperature(cid:72)GeV(cid:76)
neutrino oscillations is insufficientO. |A|total SM lepton 106 105 104 103 102
asymmetry is generated only at (F 6) due to flavour- 10(cid:45)7 RegimeIII
O | |
dependent scattering rates according to Eq. (6). In the (cid:200) Tau
absence of lepton flavour-dependent effects, the individ- Y(cid:68)L 10(cid:45)9 Muon
ualscatteringratesareallequal,Γ(L N†)=Γ(L (cid:200)y Electron
N†) = Γ(L N†), and the totalel→epton asymmµet→ry metr10(cid:45)11
remains zeroτ e→ven at higher orders in F. Therefore, dif- asym10(cid:45)13
ferences in lepton flavour rates are crucial to generate on
pt
a baryon asymmetry. Fortunately, there is already evi- le10(cid:45)15
Total
dence for lepton flavour dependence in interactions with
10(cid:45)17
neutrinos. First,thestructureoftheLHneutrinomasses 10(cid:45)20 10(cid:45)18 10(cid:45)16 10(cid:45)14 10(cid:45)12 10(cid:45)10
and mixing angles tells us that their Yukawa couplings time(cid:72)seconds(cid:76)
are non-universal, and proportional to the mixing angles
FIG.2. Timeevolutionoftheindividualleptonflavourasym-
θ . Second, the CP phases appear in very particular
ij metriesandthetotalleptonasymmetryindifferentparameter
terms in the interaction: for instance, the Dirac phase δ
regimes. The dashed vertical line indicates the electroweak
appears only in terms proportional to sinθ . Changing
13 phase transition. (Top) Regime I: The Yukawa couplings
the phase can lead to constructive or destructive inter-
are small enough that washout effects are always irrelevant;
ference of rates involving a specific lepton flavour. The (Centre)RegimeII:TheYukawacouplingsarelargeenough
totalleptonasymmetryismaximizedinregionsofparam- that equilibration of the sterile neutrinos occurs at the elec-
eterspacethataccentuatethedifferencesbetweenlepton troweak phase transition; (Bottom) Regime III: Large lep-
flavour interaction rates. The importance of flavour ef- ton flavour dependence in the L → N† rates, such that
fects was recently emphasized in [10]. Γ(Le → N†) (cid:28) Γ(Lµ → N†), Γ(Lτ → N†). The Yukawa
couplings are even larger than in Regime II, such that L
Considering all of these effects, we identify regimes of τ
and L equilibrate completely with the sterile neutrinos at
parameter space depending on the relative time scales of µ
T (cid:29)T , and L comes into equilibrium at T ∼T .
oscillation, sterile neutrino equilibration, and sphaleron W e W
decoupling. Becausethesearetheonlytimescalesinthe
problem, this results in three regimes that completely
characterizetheminimalmodel. Regime Iisdefinedby because the small Yukawa couplings implied by the late
t (cid:46)t t andwasconsideredintheoriginalworks washoutalsosuppresstheratesinEq.(6). Toaccountfor
osc W eq
(cid:28)
on baryogenesis from neutrino oscillations [1, 2]. Equili- theobservedbaryonasymmetry,theoscillationtimescale
brationoccurslongaftertheelectroweakphasetransition must be made as large as possible and thus close to the
and so washout effects are entirely irrelevant. However, electroweak scale, t (cid:46) t . This coincidence of scales
osc W
the total baryon asymmetry is also generally too small is achieved through a fine-tuning of the sterile-neutrino
6
masssplitting∆M M M (cid:46)(10−6 10−8)M .
In Regime II, thNe s≡terilNe3n−eutrNin2os come i−nto equiliNb- and sin22θ =0.096 0.013.
13
±
rium around the weak scale, and the sphaleron processes
The data are consistent with one of the LH neutrinos
freeze out “just in time” to avoid washing out the en-
being massless, and one of the sterile neutrinos being
tire asymmetry t t t . This alleviates some of
osc eq W
(cid:28) ∼ largely decoupled from the SM. This decoupled sterile
thetuningnecessaryinthemasssplitting∆M ,butthe
N
neutrino, which we take for concreteness to be N , is
coincidence between the equilibration scale and the elec- 1
a possible dark matter candidate, but does not play a
troweak scale requires some tuning in the Yukawa cou-
plings. Finally, in Regime III, first identified in [10], role in leptogenesis. Therefore, we consider an effective
theory with only N and N as the two sterile neutrinos.
the Yukawa couplings are made even larger. This gener- 2 3
The assumption of one massless LH neutrino fixes
ically results in the equilibration time, t , being even
eq
the other LH neutrino masses, up to a discrete choice
earlierthantheweakscale. Thiswouldhavebeenaphe-
of mass hierarchy. We consider the normal hierarchy,
nomenological disaster with the entire asymmetry being
where m = 0, m = m2 9 meV, and m =
washed-outtooearly,exceptthatdestructiveinterference ν1 ν2 sol ≈ ν3
inthescatteringrateforoneoftheleptonflavoursmakes ∆m2 49meV.Itisalsopossiblefortheset-upto
| atm|≈ (cid:112)
itmuchsmallerthantheothers. Theresultisatimescale berealizedintheinvertedhierarchy(m3 =0,m2 m1),
(cid:112) ∼
ordering t t t t . In this regime, even but the qualitative dependence of the baryon asymme-
osc eq,α eq,β W
(cid:28) (cid:28) ≈
lesstuningisnecessaryinthemasssplitting,buttheun- try on model parameters is similar to the normal hier-
naturally early equilibration time for two of the flavours archy, while the value of the baryon asymmetry can be
requires a substantial tuning in the Yukawa couplings. somewhat larger with an inverted hierarchy [8]. We fo-
We quantify this combined tuning below in section VD. cus exclusively on the normal hierarchy as a benchmark,
In Fig. 2, we show the time evolution of the individual since our results also qualitatively hold in the inverted
lepton flavour asymmetries and total lepton asymmetry hierarchy and easily generalize to that scenario.
for each of the regimes discussed above. In Regime II, The magnitudesof theYukawacouplings Fα2, Fα3 are
the downward spikes in the total lepton asymmetry and crucialforsuccessfulbaryogenesis. Tobetterunderstand
the L asymmetry are the result of a change in sign of theconnectionbetweentheYukawacouplingsandphysi-
µ
the asymmetry, as shown in Eqs. (7)-(8) and the related calparameters,wedecomposetheYukawacouplingswith
discussion. The spike indicates the time of the equili- the Casas-Ibarra parameterization [18],
bration of L around T = 200 GeV T . In Regime
III, these spµikes occur due to Lµ and∼LτWequilibration F = i Uν√mνR∗ MN. (13)
around T (cid:38) 3 TeV. Indeed, the hallmark of Regime III Φ
(cid:104) (cid:105)
(cid:112)
is that the equilibration temperatures of L and L are
µ τ HereU istheMNSmatrixcontainingmixinganglesand
ν
more than an order of magnitude larger than that of L
e CP phasesfromtheLHneutrinomixing[19],m isadi-
atT (cid:46)100GeV,whereasinRegimeII,allflavourscome ν
agonalmatrixofLHneutrinomasses,andM isadiago-
N
into equilibrium near the same scale.
nalmatrixofsterileneutrinomasses. ThematrixRisan
orthogonal matrix specifying the mixing between sterile
neutrino mass and interaction eigenstates; in the case of
III. MODEL PARAMETERIZATION a normal neutrino hierarchy with two sterile neutrinos,
R has the form
The model we consider is the νMSM [2], with the La-
0 0
grangiangiveninEq.(1). AtlowtemperaturesT M ,
(cid:28) I R = cosω sinω , (14)
the SM neutrinos acquire a mass in the effective theory, αI −
sinω cosω
(m ) = Φ 2 FM−1FT . (12)
ν αβ (cid:104) (cid:105) N αβ where ω is a complex angle parameterizing the misalign-
This is the usual see-saw sup(cid:0)pression of(cid:1)the SM neutrino mentbetweensterileneutrinomassandinteractioneigen-
states. WereferthereadertoAppendixAfortheexplicit
masses, and its parametric scaling was discussed in rela-
form of the decomposition in Eq. (13). We note that
tion with Eq. (2) . The observed masses and mixings of
when ω = 0, the Yukawa interactions can be diagonal-
the SM neutrinos are [17],
ized in the sterile neutrino mass basis, and the interfer-
∆m2 =2.35+0.12 10−3 eV2, ence/oscillation necessary for leptogenesis is absent.
• | atm| −0.09×
The parameters that emerge out of the decomposition
m2 =7.58+0.22 10−5 eV2, of the Yukawa couplings can be grouped as follows:
sol −0.26×
and m (cid:46) eV. Parameters currently fixed by experiment:
i νi •
– Three LH neutrino masses, (m ) ;
sin2θ(cid:80) =0.312+0.018, ν i
• 12 −0.015 – Three LH neutrino mixing angles from the
sin2θ23 =0.42+−00..0083, MNS matrix: θ12, θ13, θ23.
7
Parameters constrained by experiment: to F (1+(cid:15))F but otherwise left unchanged. In
22 22
• →
the simplest case with θ = Reω = 0, the LH neutrino
αI
– The VEV Φ . If Φ is the SM Higgs, it has
masses have a simple analytic form and the eigenvalue
(cid:104) (cid:105)
the value Φ = 174 GeV; otherwise, it is un-
m changes according to
(cid:104) (cid:105) 2
determinedbutconstrainedbyboundsonnew
dlogm
sourcesofelectroweaksymmetrybreaking(see 2 =1+cosh(2Imω). (16)
Section VIC). d(cid:15)
With non-zero θ and Reω, the change in LH neutrino
masses from (cid:15) is apportioned among the different
mass eigenstates, but the overall shift is of the order
Unconstrained parameters: of eq. (16). We have also verified this numerically.
•
Therefore, while it is possible to exponentially enhance
– Two phases from LH neutrino mixing: a Ma-
the rates relevant for baryogenesis in the minimal sterile
jorana phase, η, and a Dirac phase, δ;
neutrino model, it necessarily implies an exponential
– Two sterile neutrino masses, (MN)I; tuning of the Yukawa couplings to obtain the observed
– A complex mixing angle, ω. LHneutrinomasses. Sincemuchoftheviableparameter
space in the baryogenesis studies of [5, 8, 10] requires
These parameters, in turn, influence the dynamics of Imω 1, baryogenesis in these set-ups is unnatural in
| |(cid:29)
baryogenesis in three key ways: by setting the magni- the sense of ref. [21].
tude of the Yukawa couplings, controlling lepton flavour
dependence in scattering rates, and providing the CP Lepton flavour-dependent scattering rates: Asdis-
violation necessary for baryogenesis. We consider each cussed in Section II, lepton flavours must have different
in turn. scattering rates into sterile neutrinos in order to convert
the individual lepton flavour asymmetries into a total
Magnitude of Yukawa couplings: The scaling of the lepton asymmetry. There are several parameters in F
Yukawa magnitude is generally fixed by the see-saw re- that contribute differently to the asymmetry generation
lation mν FFT Φ 2/mN. In particular, increasing and scattering rates for each flavour. The LH neutrino
∼ (cid:104) (cid:105)
either the sterile or LH neutrino masses enhances the mass hierarchy (m m m ) and hierarchy among
1 2 3
(cid:28) (cid:28)
Yukawa couplings, as does decreasing Φ . The pres- mixing angles (θ θ θ ) provides some differen-
(cid:104) (cid:105) 13 (cid:28) 12 ∼ 23
ence of the complex parameter ω, however, allows for an tiation among lepton flavours; this typically suppresses
enhancement of the Yukawas well beyond the na¨ıve see- ratesofelectronscatteringversusthecorrespondingrates
saw value. The reason is that, while FFT is fixed by for muons and taus.
the LH neutrino masses, physical rates depend on the The CP phases δ and η also play an important role
quantities FF† and F†F. It is therefore possible that in distinguishing lepton flavours. Since they appear in
squaring the complex terms in ω leads to large cancella- different entries of the MNS matrix, these phases can
tionsamongthetermsinF suchthatFFT FF†. The lead to constructive or destructive interference among
(cid:28)
Yukawacouplingsareactuallyenhancedexponentially by processesinvolvingspecificleptonflavours. Forinstance,
the imaginary part of ω as when δ + η = π/2, there is destructive interference
−
F 2 mνMN cosh(2Imω) (15) rbeestuwlteienng itnheΓ(LLe →NN†)2† anΓd(LLe →N†)N3†Γa(mLplituNde†s),
| | ∝ Φ 2 e → (cid:28) µ → ∼ τ →
(cid:104) (cid:105) [10, 22]. Such interference effects can substantially alter
when Imω islarge. TheparameterImωdoesnotother- the rates of asymmetry creation or washout, modifying
| |
wise have any impact on experimentally observed quan- the total baryon asymmetry. We emphasize that this
tities, and it can be thought of as a dial to enhance the effect is independent of the role of the phases in CP
rates of sterile neutrino production and scattering. violation. In fact, over many regions of parameter space,
When Imω 1, the see-saw relation only holds due this is the dominant contribution of the Majorana and
| |(cid:29)
to a precise cancellation of parameters, and there is a Dirac CP phases to the baryon asymmetry.
very specific alignment of the Yukawa couplings associ-
ated with this enhancement. While the Yukawa matrix CP violation: There are three sources of CP violation
is stable under radiative corrections, and therefore tech- in the theory: the Majorana phase η, the Dirac phase
nically natural in the sense of ’t Hooft [20], the phys- δ, and the complex sterile neutrino mixing angle ω. Its
ically observed parameters (such as the LH neutrino imaginary part, Imω, gives rise to two phases that are
masses) change significantly under small perturbations related to one another,
of the Yukawa matrix entries. This results in tuning in
tanφ tan(argcosω)= tan(Reω)tanh(Imω)(,17)
1
the sense of Barbieri and Giudice [21], which is quanti- ≡ −
tanφ tan(argsinω)=cot(Reω)tanh(Imω). (18)
fied by observing how the physical masses mν change 2 ≡
under perturbations of the Yukawa coupling, F. For AccordingtotheSakharovconditions,asourceofCP vi-
concreteness, consider a Yukawa matrix decomposed ac- olation is required to generate a baryon asymmetry. Ex-
cording to eq. (13), which is then perturbed according cept where δ, η, and Imω all vanish, there is generically
8
an (1) phase contributing to baryogenesis coming from oscillations among the sterile neutrino, the density ma-
O
a combination of these individual phases. Indeed, even trix formalism is well-suited for tracking the evolution of
in the regions where the phases are aligned to give con- abundances and coherences between states. Such an ap-
structive or destructive interference in scattering rates proach was used in [1], with [2] being the first analysis
(as discussed above), the combination of the LH neu- toincludealloftherelevanttermsintheevolutionequa-
trino phases δ +η is generally non-zero. For instance, tions, with various factors corrected in subsequent work
with Imω 1, constructive (destructive) interference in [7, 9]. More sophisticated approaches have also been
Γ(L N(cid:29)†) occurs when δ+η is π/2 ( π/2), which is taken, such as separately evolving different momentum
e
→ −
also the phase alignment that gives maximal CP viola- modes in the density matrix [9] or using non-equilibrium
tion in the LH lepton sector. Therefore, CP violation quantum field theory [10]. These results are very similar
is generally (1) over the entire parameter space of the to those using thermally averaged density matrix evolu-
O
minimal model. The vanishing of one phase (such as the tion; for example, ref. [9] find enhancements of the total
experimentally accessible Dirac phase δ) does not neces- baryon asymmetry of factors of 10-40% when computing
sarily constrain the other phases relevant for leptogene- theasymmetriesseparatelyineachmomentummoderel-
sis [6] or determine the relative lepton flavour scattering ativetothermalaveraging. Becauseofthecomputational
rates. simplicity of thermal averaging, and the small changes
Finally, wenotethattherehavebeensomestatements to the total asymmetry when using more sophisticated
in the literature implying that the limit of large Imω methods, we employ thermal averaging in the current
| |
somehow leads to more CP violation. As shown in work.
Eqs. (17)-(18), the CP violating phases φ1 and φ2 de- Asinearlierworks,wefollowtheevolutionofthester-
pend on tanh(Imω) and quickly saturate with increased ile neutrino and anti-neutrino density matrices, ρ , and
N
Imω. Instead, as discussed above, the main effect of ρN¯, as well as the LH lepton asymmetry density matrix,
|Imω|(cid:29)1isanexponentialenhancementoftheYukawa ρL−L¯. Here, L refers to a single component of the SU(2)
couplings,notthepresenceofanadditionalsourceofCP lepton doublet, and is a matrix in lepton flavour space.
violation. The diagonal elements of the density matrices are equal
totheabundancesofthecorrespondingfieldsnormalized
by the equilibrium abundance,
IV. ASYMMETRY EVOLUTION EQUATIONS
Y (t)
i
In this section, we review the formalism for comput- ρii(t)= Yeq(t), (19)
ing the baryon asymmetry from sterile neutrino oscilla- i
tions. Along the way we discuss corrections we made
to the existing formalism to account for the contribu- and the off-diagonal terms correspond to the coherences
tion of equilibrium processes to the evolution of the lep- between the fields.
ton asymmetries. Because of the central role of coherent The evolution equations have the general form [2],
dρ 1 1
dtN =−i[H(t), ρN]− 2{Γ(L† →N)2×2, ρN −ρeL¯qI2×2}− 2γavT F†ρL−L¯F, (20)
ddρtN¯ =−i[H(t), ρN¯]− 12{Γ(L→N†)2×2, ρN¯ −ρeLqI2×2}+ 21γavT FTρL−L¯F∗, (21)
dρdLt−L¯ =−14{Γ(L→N†)3×3, ρL−L¯}+ 12γavT FρN¯F†−F∗ρNFT . (22)
(cid:0) (cid:1)
Here ( , ) [,] denotes a matrix (anti-) commutator. ponents are non-zero.
{ }
The evolution equations satisfy the relation Tr(ρN−N¯ − 2. The terms proportional to Γ(L N†) lead to
2ρL−L¯)=0, whichreflectstheconservationoftheglobal bothproductionofthesterileneu→trinoabundance
L N charge. The interested reader can find a de-
− (through the terms ρeq) and destruction (or
tailed account of these equations in Appendix B. Here ∝ L
washout) of the asymmetries in both N and L.
we instead concentrate on a qualitative understanding
These interactions drive the fields towards their
of the significance of the different terms that appear in
equilibrium distributions and erase any asymme-
Eqs. (20)-(22).
tries. These scattering rates are proportional to
thetemperatureT andcanoftenbefactorizedinto
1. The Hamiltonian terms induce coherent oscil-
the form
lations between diagonal and off-digonal compo-
nents of the density matrix. Such a term is not Γ(L N†)=γav(T)T F 2. (23)
included for the LH leptons because these rapidly → | |
decohere in the thermal bath of the early universe, The factor γav(T) parameterizes the rate after fac-
andthereforeonlyon-diagonaldensitymatrixcom- toringouttheYukawacouplingsandhasbeencom-
9
asymmetry ρL−L¯(tW). At the weak scale, chemical po-
tential relations relate the size of the baryon asymmetry
NI† Lα to the totNaIl†LH lepton asymmetry [24]:
Lα 28
Y (t )= Y (t ). (24)
∆B W −79 ∆Lα W
α
(cid:88)
Sincesphaleronsdecoupleattheelectroweakphasetran-
Φ V H
∗ sition, the fi∗nal baryon asymmetry is frozen at this time
NI† Lα NI† and is simply Y∆B(tW) as given in Eq. (24).
Theaboveapproachtosolvingforthelate-timebaryon
L
α asymmetryisincorrect, asitneglectstheeffectsofinter-
NI† Lα NI† actions between SM fields, which are rapid compared to
the active-sterile neutrino scatterings. Because of these
Lα Φ∗ t†L V tHcR ∗ scatterings,theasymmetriesinindividualleptonflavours
created by sterile neutrino oscillations are rapidly dis-
tributed among all SM fields. Since the asymmetries are
Φ V Φ destroyed only through interactions of the LH leptons
∗ ∗
with sterile neutrinos, this modifies the relative rates of
asymmetry creation and destruction.
t†L Lα tcR NI† ofTanheaseyffmecmtsetorfyeqhuavileibbrieuemn usncadtetresrtionogds oanndthceorerveocltuetdioinn
adifferentcontext,namelyofbaryogenesisthroughweak-
t†L tcR scale dark matter scatterings [25]. Our approach here is
similar: we include in our density matrices only quan-
tities that are preserved by the equilibrium SM interac-
tions, justifying the absence of such rapid interactions
Lα NI† in the evolution equations. Specifically, we exchange the
anomalous asymmetries in individual lepton flavours L
α
in (22) for asymmetries of B 3L , which are exactly
α
Lα FcrIeGat.i3o.nRtehprroeusgehnta(NTtiovI†epF)e1yn→ma2ndpiraogcreasmsess;fo(rCsteenritlreen)eugtaruingoe cmoantsreirxveedqubaytioSnMs:scbaetctaeurisnegEs.q−.T(h2i2s)mnoodwifireesptrheseendtesnstihtye
boson 2→2 scattering; (Bottom) quark 2→2 scattering.
evolution of ∆(B 3L ), instead of ∆L , all terms are
α α
−
a factor of 2 3 larger than for the individual LH lep-
− ×
ton species, due to the re-definition of the charge and
putedcompletelyatleadingorder[23]. Representa-
an SU(2) factor from summing over charged and neutral
tivediagramscontributingtotheprocessareshown
components of L . The last term in Eq. (22), which cre-
α
in Fig. 3. The rates from ref. [23] sum over both
ates the individual ∆L asymmetries, is otherwise un-
α
lepton doublet components. modified, since the coherent scattering N† L gen-
α
→
erating the asymmetry is not sensitive to the details of
3. The final terms in each line produce the asym-
SM thermal scatterings. By contrast, the first term in
metries in each sector. For ρL−L¯, the coherent Eq. (22), which destroys the individual lepton flavour
oscillations of the sterile neutrinos (encoded in the
asymmetries,isproportionalonly totheindividualasym-
off-diagonal terms of ρ ) create an asymmetry in
N metry∆L ,not thetotal∆(B 3L )asymmetry3. This
individual LH lepton flavours at (F 4). For the α − α
sterileneutrinos,anexcessofLα oOver|L¯|α translates disivbideceadubseytthheeaesqyumilimbreiturmy isncafitetledrsincgasrrayminogngBm−an3yLαSMis
into an excess of scattering into N† vs. N, sourc-
fields,suchasthequarksandRHchargedleptons,which
ing an asymmetry in the sterile sector as well at
havenodirectcouplingtosterileneutrinos. Towritethe
(F 6). The rate of transfer of SM lepton flavour
O | | correctevolutionequation,werelatetheLHleptonabun-
asymmetries into sterile neutrino asymmetries de-
dancesfromEq.(22)totheB 3L chargesthroughthe
pends on the asymmetry in the flavour Lα and the chemical potential constraints−of thαe SM interactions:
rateL N†; thematrixρ appearsbetweenthe
Flep†taonnαdfl→aFvomuIraterfficeecsts.becauseNof this sensitivity to (ρL−L¯)α =−21133 [221δαβ −16(1−δαβ)] ρB−3Lα.
β
(cid:88)
(25)
The initial conditions are such that there are initially
no sterile neutrinos ρN(0) = ρN¯(0) = 0 and no primor-
dialleptonasymmetryρL−L¯(0)=0. Inexistingworksin
the literature, these initial conditions are used to evolve 3 Thesameistrueforthetermsinthekineticequationsgenerating
Eqs. (20)-(22) down to the weak scale to determine the thesterileneutrinoasymmetries.
10
We see immediately that the rates are suppressed by at example, Reω = π/2 gives a comparable baryon asym-
least 6 221/2133 0.6 due to the fact that much of metry, with somewhat larger values than Reω =π/4 for
≈ × ≈
the asymmetry leaks out of the LH leptons, and possi- Imω 0.5 3, andsmallervalueselsewhere(seeFig.7).
≈ −
bly more due to the other negative terms in the rotation
matrix. Having made this substitution, we express the
densitymatrixevolutionequationspurelyintermsofρN, A. Regime I
ρN¯,andρB−3Lα. Thesequantitiesarethenevolveddown
totheweakscale,andinsteadofEq.(24)thefinalbaryon
The generation of a baryon asymmetry in this regime
asymmetry is
is maximized when the rates of LH lepton scattering
28 into sterile neutrinos are very different for different lep-
Y (t )= Y (t ). (26)
∆B W 237 ∆(B−3Lα) W ton flavours. However, the Yukawa couplings here are
(cid:88)α set to their small, natural values expected from the see-
The equations are actually easiest to solve in a different saw relation, Eq. (2). Therefore, even with large differ-
basis: we provide the complete evolution equations in encesinleptonflavourscatteringrates,thebaryonasym-
this basis and a few details on numerical integration in metry in Regime I is not large enough to account for
Appendix B. In that same appendix, we also include the the observed asymmetry unless the coherent oscillation
effects arising from a phase-space suppression of certain time is maximal, which requires a strong degeneracy be-
diagramscontributingtothe∆Lαdestructionrates;such tween the masses of N2 and N3. A mass degeneracy of
modifications were first discussed in [9]. ∆M /M (10−6 10−8) is needed, depending on
N N
∼ O −
Imω.
To illustrate this, we show in Fig. 4 the baryon asym-
V. BARYON ASYMMETRY IN THE νMSM metry for a set of parameters where the magnitude of
the Yukawa coupling F is held fixed, while the relative
This section is devoted to a quantitative confirmation rates of L N†, L N†, and L N† are al-
e µ τ
→ → →
of the qualitative features discussed above. We demon- lowed to vary. We see that the asymmetry is largest
stratethecontinuoustransitionbetweenthedifferentpa- when flavour dependence on the scattering rates is sig-
rameter regimes discussed in section II and depicted in nificant, i.e. the rates of Γ(L N†) are very different,
α
→
Fig. 2. We also show that, over the entire parameter especiallytheratesofL andL . Inthisregime,thecon-
µ τ
space of the minimal model, the observed baryon asym- tribution from the L flavour asymmetry and scattering
e
metryrequiresaveryspecificalignmentofthemodelpa- rate is subdominant, due to the smallness of θ and the
13
rameterstowithinatleast (10−5). Wedonotperforma factthatm hasthelargestYukawacouplingtothester-
O 3
comprehensivescanofpossibleparametersintheνMSM, ile sector. Over the parameters scanned in Fig. 4, the
assuchascanwascarriedoutin[5,8]forRegimesIand rate Γ(L N†) is typically (cid:46) 3% of the correspond-
e
II with kinetic equations similar to those we use here, ing L and→L , and the asymmetry in L is (cid:46) 10% of
µ τ e
and our results are in qualitative agreement with theirs. the L and L asymmetries4. The asymmetry vanishes
µ τ
However, our qualitative picture provides a simple un- when Γ(L N†)/Γ(L N†) 0.8, which is when
µ τ
derstanding for the parameters maximizing the baryon the differenc→e in L N†→and L≈ N† rates exactly
µ τ
→ →
asymmetry found in ref. [5, 8]. Finally, with regime III compensates for the difference in flavour asymmetries.
we explore a new region of parameter space whose exis- This behaviour is easily understood with the qualita-
tence was first demonstrated in ref. [10] with a couple of tive lessons learned in previous sections. In this regime,
representative points in a model with three sterile neu- the Yukawa couplings are sufficiently small that equili-
trinos (instead of the two we consider here). We expand bration of the active and sterile neutrinos is irrelevant
upon ref. [10] with a comprehensive scan of the parame- as it occurs long after the electroweak time scale. As
ter space and demonstrate the emergence of Regime III a result, the asymmetries in individual lepton flavours
as a continuous part of the other regimes. remain unchanged from their generation at the time of
TheobservedbaryonasymmetryoftheUniverseis[26] coherent sterile neutrino oscillation to the sphaleron de-
coupling at T . The baryon asymmetry is determined
Y 8.6 10−11. (27) W
∆B ≈ × bytheslowtransferofasymmetryfromindividuallepton
Throughout this section, we take T = 140 GeV as flavoursintothesterilesector,Eq.(6),whichgeneratesa
W
thetemperatureoftheelectroweakphasetransition,cor- total lepton asymmetry. It is therefore maximized with
responding to a SM Higgs mass of 126 GeV [5]. For large differences in rates associated with different lepton
the numerical solutions of the evolution equations in flavours.
all regimes, we take values consistent with Section III:
m = 0, m = 9 meV, m = 49 meV, sinθ = 0.55,
1 2 3 12
sinθ = 0.63, and sinθ = 0.16. For concreteness, we
23 13
useReω =π/4throughoutouranalyses, asthisgivesan 4 This is true except in a small window around Γ(Lµ →
appreciableasymmetryforallvaluesofImω. Theasym- N†)/Γ(Lτ → N†) ≈ 0.7, where the Lτ asymmetry becomes
metry does not change substantially with this angle: for verysmallandtheLe asymmetryisimportant.
