XIV International Workshop on Telescopes Venice, 15 -18 March 2011

LeptogenesisLeptogenesis andand neutrinoneutrino massesmasses

PasqualePasquale DiDi BariBari TheThe doubledouble sideside ofof LeptogenesisLeptogenesis

Cosmology Neutrino , (early Universe) New Physics • Cosmological Puzzles : 1. Dark Leptogenesis complements 2. Matter - antimatter asymmetry low energy neutrino 3. Inflation experiments testing the 4. Accelerating Universe • New stage in early Universe history : high energy parameters

14 < 10 GeV Inflation ⇒ It provides a Leptogenesis precious information on the BSM physics 100 GeV EWSSB on the BSM physics TT 100 GeV EWSSB responsible for neutrino 0.1 - 1 MeV BBN masses and mixing: a model builders compass 0.1 - 1 eV Recombination PrimordialPrimordial mattermatter --antimatterantimatter asymmetryasymmetry

 Symmetric Universe with matter - anti matter domains ? Excluded by CMB + cosmic rays n -n ⇒⇒ ηCMB B B -10 ηB == ng == (6.2(6.2 ±± 0.15)0.15) xx 1010

 Pre -existing ? It conflicts with inflation ! (Dolgov ‘97) ⇒⇒ dynamicaldynamical generationgeneration ()(baryogenesis) (Sakharov ’67) NeutrinoNeutrino masses:masses: mm 11 << mm 22 << mm 33

Tritium b decay :m e<2.3 eV (Mainz 95% CL)

bb 0n:m bb < 0.3 – 1.0 eV (Heidelberg-Moscow 90% CL, CUORICINO ) using the flat prior ( W0=1): Cosmology:

S mi < (0.2-0.6) eV (90% CL) , (Melchiorri, Lisi talk) MinimalMinimal scenarioscenario •Type I seesaw

⇒ t •Thermal production of the RH TRH Mi AnAn impossibleimpossible tasktask ??

Is it possible to reconstruct m D and M just from low energy neutrino experiments measuring m i and U PMNS ?

(in the basis where charged and Majorana mass matrices are diagonal)

However, hand neutrino experiments give information only on the 9 parameters contained in The 6 parameters in the orthogonal matrix Ω [it encodes the 3 life times and the 3 total CP asymmetries of the RH neutrinos and it is an invariant

(King ‘‘07) ] + the 3 masses M i escape the conventional investigation ! Leptogenesis is important to obtain information on the high ener gy parameters complementing the low energy neutrino experiments TheThe simplestsimplest description:description: vanillavanilla leptogenesisleptogenesis

1) Flavor composition of final is neglected

Total CP asymmetries

If εi ≠ 0 a lepton asymmetry is generated from N i decays and partly converted into a asymmetry by t (Kuzmin,Rubakov,Shaposhnikov, ’85) processes if T reh 100 GeV ! baryon-to -photon number ratio > efficiency factors # of N i decaying out -of -equilibrium CMB -10 Successful leptogenesis : ηB = ηB =(6.2 ± 0.15) x 10 The total CP asymmetries can be calculated from : (Flanz,Paschos,Sarkar’95; Covi,Roulet,Vissani’96; Buchmüller,Plümacher’98)

It does not depend on U !

2) Strongly hierarchical heavy RH neutrino spectrum

3) N3 does not interfere with N 2-decays: under the last two assumptions

CMB Imposing ηB=ηB ,one obtains a N1-dominated scenario :

efficiency factor 4) Barring fine-tuned mass cancellations

⇒ Upper bound on ε1 (Davidson, Ibarra ’02)

5) Classical Kinetic equations integrated on momenta

⇒ NeutrinoNeutrino massmass boundsbounds (Davidson,Ibarra ‘02 ;Buchm üller,PDB,Pl ümacher ’02, ’03, ’04; Giudice et al. ‘04)

N1 – dominated scenario

Vanilla Imposing: leptogenesis is not compatible with quasi -deg. neutrinos No constraints on the on the These large leptonic temperatures mixing in gravity mediated matrix U ! SUSY models matrix U ! suffer from the gravitino problem An encouraging coincidence The early Universe „knows “ neutrino masses ... (Buchm üller,PDB,Pl ümacher ’04)

decay parameter

wash -out of

3 a pre -existing p,final p,initial π f,N1 N = N e− 8 K1  N asymmetry B−L B−L B−L BeyondBeyond vanillavanilla LeptogenesisLeptogenesis

The degenerate Non minimal Leptogenesis limit (in type II seesaw, non thermal, ….)

Improved Vanilla Kinetic description Leptogenesis (momentum dependence, quantum kinetic effects,finite temperature effects, …… .)

Flavour Effects (heavy flavour effects, light flavour effects, light+heavy flavour effects) Improved kinetic description • Momentum dependence in Boltzmann equations (Hannestad ’’ 06; Hahn -Woernle, M. Pl ümacher, Y.Wong ’’09; Pastor, Vives ’’09) • Kadanoff -Baym equations (Buchm üller,Fredenhagen ‘‘01; De Simone,Riotto ‘‘07; Garny,Hohenegger, Kartavtsev,Lindner ’’09; Anisimov,Buchm üller,Drewes,Mendizibal ’’09; Beneke, Garbrecht, Herranen, Schwaller ‘‘10)

The asymmetry is directly calculated in terms of Green functions instead than in terms of number densities and they account for off - shell , memory and medium effects in a systematic way All studies confirm what also happens for other effects (e.g. inclusion of scatterings) and that is expected: large theoretical uncertainties in the weak wash -out regime, limited O(1) corrections in the strong wash -out regime where the asymmetry is produced in a narrow range of temperatures for T << M i (Buchm üller,PDB,Pl ümacher) Light neutrino flavour effects (Nardi,Nir,Roulet,Racker ’06;Abada,Davidson,Losada,Josse-Michaux,Riotto’06; Blanchet, PDB, Raffelt ‘06; Riotto, De Simone ‘06) Flavor composition of lepton quantum states:

t 12 • interactions are flavour blind for Mi 10 GeV d 12 • But for Mi 10 GeV ï t-Yukawa interactions are fast enough to break the coherent evolution of and ï they become an incoherent mixture of a t and of m+e d 9 If M1 10 GeV then also m- Yukawas in equilibrium î 3-flavor regime

heavy neutrino lepton flavor index flavor index UNFLAVOURED

~ 10 12 GeV Transition regions M i 2 Flavour regime (τ, e+ µ)

~ 10 9 GeV

3 Flavour regime (e, µ, τ ) Fully two -flavored regime

Let us first insist with a N 1-dominated scenario: The bounds get relaxed (Abada et al. ’ 07 Blanchet,PDB ’08) PMNS phases off 10 12 10 12 (GeV) (GeV) 1 1 (GeV) 1 M M M 10 9 10 9 m (eV) 1 m1(eV)

10 12 (GeV) imposing a condition of 1 validity of Boltzmann M equations 10 9 m (eV) K1 1 0.1 Heavy neutrino flavour effects:N 2-dominated scenario ( PDB ’05 )

If lepton flavour effects are neglected the asymmetry from the n ext -to -lightest (N 2) RH neutrinos is typically negligible: 3 2 π 1 f,N − 8 K1 f,N NB−L = ε2κ(K2) e  NB−L = ε1 κ(K1)

...except for a special choice of W=R 23 when K 1= m 1/m * << 1 and ε1=0:

The lower bound on M 1 disappears and is replaced by a lower bound on M 2 …

that however still implies a lower bound on T reh ! N2-flavored leptogenesis ( Vives ’05; Blanchet, PDB ’06; Blanchet, PDB ’08) Combining together lepton and heavy neutrino flavour effects on e has

A two stage process:

N - Asymmetry Production M 2 2 in the unflavoured regime... ~ 10 12 GeV ...or in the 2 flavour regime

~ 10 9 GeV N - washout in the 3 fl. regime M 1 1

Notice that the presence of the heaviest RH neutrino N 3 is necessary for the CP asymmetries of N 2 not to be negligible ! N2-flavored leptogenesis (Vives ’05; Blanchet, PDB ’06; Blanchet, PDB ’08) t 12 ⇒ If (for definiteness) M 2 10 GeV

3π 3π 3π f 0 − 8 K1e 0 − 8 K1µ 0 − 8 K1τ NB−L(N2) = P2e ε2 κ(K2) e +P2µ ε2 κ(K2) e +P2τ ε2 κ(K2) e

Notice that K1 = K1e + K1µ + K1τ Wash -out is neglected

CMB Wash -out and flavor effects ηB are both taken into account ηB Unflavored case

M2

Thanks to flavor effects the domain of applicability extends much beyond the particular choice W=R 23 ! More generally one has to distinguish 10 different RH neutrino mass patterns (Bertuzzo,PDB,Marzola ‘10)

Heavy flavored scenario

For each pattern a specific set of kinetic equations has to be considered Heavy flavored scenario (Engelhard, Nir, Nardi ‘08 , Bertuzzo,PDB,Marzola ‘10)

Assume M i+1 > 3M i (i=1,2) The heavy neutrino flavours basis is not orthogonal in general and this complicates the calculation of the final asymmetry

Some deviation from orthogonality (it is realized in form domina nce models discussed in King ’s talk) is typically necessary since otherwise (e.g. with tri -bimaximal mixing) one would have vanishing CP asymmetries and therefore no asymmetry produced from leptogenesi s (Antusch, King, Riotto ’08; Aristizabal,Bazzocchi,Merlo,Morisi ‘09) Heavy flavoured scenario in models with A4 discrete flavour symmetry (Manohar, Jenkins ’08;Bertuzzo,PDB,Feruglio,Nardi ’09;Hagedorn,Molinaro,Petcov ’09)

17 NORMAL HIERARCHY INVERTED HIERARCHY 10 GeV 10 17 GeV M 1 M 3 16 y=1 y=1 10 GeV 16 M 10 GeV 2

15 15 M

10 GeV 2

10 GeV M 3 M 1 14 10 GeV 10 14 GeV

m m m m 13 sol atm sol atm 10 GeV 10 13 GeV m1(eV) m (eV) imposing 1 successful leptogenesis η Symmetry Breaking parameter

The different lines correspond to values of y between 0.3 and 3 Heavy flavored scenario

For each pattern a specific set of kinetic equations has to be considered BaryogenesisBaryogenesis andand thethe earlyearly UniverseUniverse historyhistory

TRH = ? Inflation Affleck -Dine (at preheating) Gravitational baryogenesis GUT baryogenesis TT Leptogenesis (minimal) 10 8 GeV

100 GeV EWBG

0.1 - 1 MeV BBN

0.1 - 1 eV Recombination The problem of the initial conditions in flavoured leptogenesis (Bertuzzo,PDB,Marzola ‘10)

Residual “pre -existing ” asymmetry possibly Asymmetry generated generated by some from leptogenesis external mechanism

……………… ………………

The wash -out of a pre -existing asymmetry is

guaranteed only in a N 2-dominated scenario where the final asymmetry is dominantly in the tauon flavour (loophole:in supersymmetric models (Antusch,King,Riotto ‘‘06) 2 t also in N 1 dominated scenarios with tan b 20)

This mass pattern is particularly interesting because it is just that one realized in SO(10) inspired models SO(10)-inspired leptogenesis ( Branco et al. ’02; Nezri, Orloff ’02; Akhmedov, Frigerio, Smirnov ‘03)

Expressing the neutrino Dirac mass matrix mD (in the basis where the Majorana mass and charged lepton mass matrices are diagonal): † mD = VL DmD UR (bi -unitary parametrization)

where DmD = diag{λD1, λD2, λD3} and assuming: 1) λD1 = α1 mu , λD2 = α2 mc , λD3 = α3 mt , (αi = O(1))

2) VL  VCKM  I

one typically obtains (barring fine -tuned exceptions):

2 5 2 10 2 15 M1 ∼ α1 10 GeV,M2 ∼ α2 10 GeV,M3 ∼ α3 10 GeV

9 ⇒⇒ CMB sincesince MM 1 <<<< 1010 GeVGeV ηηB(N(N 1)) <<<< ηηB !! ⇒⇒ ffailureailure ofof thethe NN 1--dominateddominated scenarioscenario !! YES:YES: thethe NN 2--dominateddominated scenarioscenario rescuesrescues SO(10)SO(10) inspiredinspired modelsmodels !! (PDB, Riotto ’08, ’10)

Independent of !

a =5 a =4 a =3 2 2 2 VVL== II NormalNormal orderingordering

Θ13

lower bound on Θ13 VanishingVanishing initialinitial NN 2 abundanceabundance The model yields constraints on all low energy neutrino observab les !

VVLL== II

NORMAL ORDERING (PDB, Riotto ‘10)

10 -3 10 -2 10 -1 16 16

14 14

/GeV) 12 12 RH

10 10

8 8 /GeV),Log(T i 6 6 Log(M 4 4

10 -3 10 -2 10 -1 The reheating m (eV) 1 Yellow points : a2=5 temperature lower bound is ~ 4x10 10 GeV Green points: a2=4 problem in SUSY

Red star : a2=3 (PDB, Riotto ‘10)

correlation between Θ13 and Θ23

Low values of the atmospheric angle are strongly favoured and Yellow points : a2=5 maximal mixing is very Green points : a2=4 marginally allowed and Red star : a2=3 º excluded for Θ13 < 6 The model does not seem to predict necessa rily CP violation in neutrino oscillati ons On the other hand the Majorana phases play a crucial role Effective Majorana mass small but non vanishing and unambiguosly related to m 1 A third encouraging coincidence ! (PDB, Riotto ‘10) º The scenario seems to like Θ < 10 ! -5 -4 13 -3 -2 -1 0 10 10 10 10 10 10 60 60

50 50 Blue points : a2=4 and mixing º angles let free in (0,60 º) 40 40

30 30

a Green points : a2=4 and θ 13 current experimental constraints 20 20

imposed on mixing angles 10 10

0 0 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 m (eV) 1 -3 For the solution with m1 ~ 3x10 eV the asymmetry is 2 dominantly produced in the tauon flavour since e2τ,µ,e ∝ (m t,c,u )

e2τ

e2µ

e2e

For these solutions all conditions for a full independence of the initial confitions are fullfilled ! The model yields constraints on all low energy neutrino observab les !

VVLL== II

INVERTED ORDERING

a2=5

a2=4.7 II << VV L << VV CKM M i Θ13 Θ23

m1(eV) m1(eV)

NORMAL ORDERING

a2=5

a2=4

a2=1 NORMAL ORDERING a =5 a =4 II << VV L << VV CKM 2 2

a2=3.7 m1 < 0.01 eV

Θ 23 Θ12

Θ13 Θ13 Θ13 a2=1 m1 > 0.01 eV

Θ23 Θ12 II << VV L << VV CKM Θ23

m1(eV)

INVERTED ORDERING

a2=5

a2=4

a2=1.5 ConclusionsConclusions Leptogenesis is an important way to complement low energy neutrino experiments to test the see -saw mechanism since the high energy parameters are involved as well.

Leptogenesis+low energy neutrino experiments are still not sufficient to over -constraint the see -saw parameter space in a general case and ine has i) either to look for additional phenomenologies (LFV processes ? EDM ’s ?, collider physics ?) or ii) Restrict the parameter space imposing some assumption

For example SO(10) -inspired models are potentially predictive. They

Are ruled out in a traditional N 1-dom scenario but when production from N2 neutrinos is taken into account they are viable and produce inte resting constraints on the light neutrino mass matrix parameters Additional contribution to CP violation: depends on U !

N 1) 1

N¯ { 1 N1 2) ++ t m N1 t m Low energy phases as the only source of CP violation (Nardi et al; Blanchet,PDB , ’06; Pascoli , Petcov , Riotto ; Anisimov , Blanchet, PDB ’08) The whole CP violation can stems just from low energy phases (D irac, Majorana phases) and still it is possible to have successful leptogen esis !

independent of initial N 1 abundance initial thermal N 1 abundance Green points: (Blanchet, only Dirac phase PDB ’09) with sin θ13 = 0.2

Red points: only Majorana phases

However, in general, we cannot constraint the low energy phases with leptogenesis and viceversa we cannot test leptogenesis just meas uring CP violation at low energies: we need to add some further conditio n ! R (Blanchet, PDB ’06) FULLY TWO-FLAVORED REGIME

Lτ Φ LtµR

l1 N1 Φ Φ Φ

N1 N1 06) 06) ’ ’ (Blanchet, PDB PDB (Blanchet, (Blanchet, PDB PDB (Blanchet, Φ l l Φ NO NO FLAVOR NO NO FLAVOR PuzzlesPuzzles ofof ModernModern CosmologyCosmology

1.1. DarkDark mattermatter

2.2. MatterMatter -- antimatterantimatter asymmetryasymmetry

3.3. Inflation Inflation Baryogenesis 4.4. AcceleratingAccelerating UniverseUniverse

5.5. ExtraExtra ultrarelativisticultrarelativistic degreesdegrees ofof eff freedomfreedom ?? N ν = 4.3 ± 0.85 (WMAP 7 ‘10) ï clash between the SM and LCDM ! The total CP asymmetries can be calculated from : (Flanz,Paschos,Sarkar’95; Covi,Roulet,Vissani’96; Buchmüller,Plümacher’98)

It does not depend on U !

It holds if:

Hierarchical RH neutrino spectrum

N3 does not interfere with N 2-decays: (PDB ’05) under these two conditions Leptogenesis “conspiracy“ (2)

m 10 −5 eV m 10 eV atm = matm = 05.0 eV atm =

Green points: Unflavored Red points: Flavored Leptogenesis and discrete flavour symmetries: A4 (Ma ’04; Altarelli , Feruglio ’05; Bertuzzo , Di Bari, Feruglio , Nardi ’09;Hagedorn,Molinaro,Petcov ‘09 )

NORMAL HIERARCHY INVERTED HIERARCHY 10 17 GeV 10 17 GeV M M 1 3 y=1 y=1 10 16 GeV 10 16 GeV M 2

15 M 15 2

10 GeV 10 GeV M M 3 1 14 10 14 GeV 10 GeV

m m m sol atm m atm 13 10 13 GeV sol 10 GeV

0 0 0 0 10 eV 10 eV 10 eV 10 eV

-1 -1 -1 -1 10 eV 10 eV 10 eV m 10 eV m 3 m 3 atm matm

m 2

-2 m -2 2 10 eV -2 -2 10 eVm 10 eV sol 10 eV m m 1 1 m -3 m atm -3 m m sol -3 sol -3 10 eV 10 eV atm 10 eV -3 -2 -1 0 10 eV 10 eV 10 eV 10 eV 10 eV -3 -2 -1 0 m 10 eV 10 eV 10 eV 10 eV 1 m 1 The situation is less attractive than in SO(10) inspired models because the RH neutrino mass spectrum first requires very high temperatu res, second it does not allow a wash -out of a pre -existing asymmetry Leptogenesis in A4 models (Ma ’04; Altarelli , Feruglio ’05; Bertuzzo , Di Bari, Feruglio , Nardi ‘09) However successful leptogenesis seems to be possible (better in the normal hierarchical case) just for the best values of the symmetry breaking parameters

Normal ordering Inverted ordering

Symmetry Breaking parameter

The different lines correspond to values of y between 0.3 and 3 Flavoured Boltzmann equations

Let us introduce the projectors (Barbieri,Creminelli,Strumia,Tetradis’01) :

These 2 terms correspond to 2 different flavour effects : 1. In each inverse decay the Higgs interacts now with incoherent flavour eigenstates ! ï the wash-out is reduced and

2. In general and this produces an additional CP violating contribution to the flavoured CP asymmetries:

Interestingly one has that now this additional contribution depends on U ! TheThe doubledouble sideside ofof LeptogenesisLeptogenesis Cosmology (early Universe) Neutrino Physics, New Physics • Cosmological Puzzles :

1. Dark matter Leptogenesis complements 2. Matter - antimatter asymmetry low energy neutrino experiments 3. Inflation testing the high energy parameters 4. Accelerating Universe of the seesaw mechanism • New stage in early Universe history : ⇒ It provides a 14 Inflation < 10 GeV Inflation precious guidance Leptogenesis to try to understand what kind of new physics is 100 GeV EWSSB kind of new physics is TT responsible for the neutrino 0.1 - 1 MeV BBN masses and mixing 0.1 - 1 eV Recombination BeyondBeyond thethe typetype II seesawseesaw It is motivated typically by two reasons: - Again avoid the reheating temperature lower bound - In order to get new phenomenological tests ….the most typical motivation in this respect is quite obviously whether we can test the seesaw and leptogenesis at the LHC Typically lowering the RH neutrino scale at TeV , the RH neutrinos decouple and they cannot be efficiently produced in colliders Many different proposals to circumvent the problem:

•• additional gauged U(1) B-L (King,Yanagida ’04) • leptogenesis with Higgs triplet (type II seesaw mechanism) (Ma,Sarkar ’’00 ; Hambye,Senjanovic ’’03; Rodejohann ’’04; Hambye,Strumia ’’05; Antusch ‘‘07) •• leptogenesis with three body decays (Hambye ‘01) • see -saw with vector fields ( Losada,Nardi ‘07) • inverse seesaw mechanism and leptogenesis (talk by R. Mohapatra ) Non minimal leptogenesis NonNon thermalthermal leptogenesisleptogenesis The RH neutrino production is non -thermal and typically associated to inflation. They are often motivated in order to obtain successful leptogenesis with low reheating temperature.

- RH neutrino production from inflaton decays (Shafi,Lazarides ’’ 91)

- Leptogenesis from RH sneutrinos decays ( Murayama,Yanagida ’’ 93)

- RH neutrinos can also be produced at the end of inflation during the pre -heating stage (Giudice,Peloso,Riotto,Tkachev99 )

-The connections with low energy neutrino experiments become even looser in these scenarios, while they can be made with properties of CMBR anisotropies (Asaka , Hamaguchi,Yanagida ‘‘99 ) Improved kinetic description • Momentum dependence in Boltzmann equations (Hannestad ’’ 06; Hahn -Woernle , M. Pl ümacher , Y.Wong ’’09; Pastor, Vives ’’09)

• Kadanoff -Baym equations (Buchm üller,Fredenhagen ‘‘01; De Simone,Riotto ‘‘07; Garny,Hohenegger , Kartavtsev,Lindner ’’09; Anisimov,Buchm üller,Drewes,Mendizibal ’’09; Beneke , Garbrecht , Herranen , Schwaller ‘‘10) The asymmetry is directly calculated in terms of Green functions instead than in terms of number densities and they account for off - shell , memory and medium effects in a systematic way At the moment all these analyses confirm what also happens for other effects (e.g. inclusion of scatterings) and that is expected : large theoretical uncertainties in the weak wash -out regime, limited corrections ( O (1) ) in the strong wash -out regime where the asymmetry is produced in a narrow range of temperatures for T << M i (Buchm üller,PDB,Pl ümacher) The degenerate limit (Covi,Roulet , Vissani ‘96; Pilaftsis ‘ 97; Blanchet,PDB ‘06)

M  3 M DifferentDifferent possibilities, possibilities, for for example: example: : 3 2

• partial hierarchy: M 3 >> M 2 , M 1

M 2 M - M δ ≡ 2 1 2 M CP asymmetries get enhanced ∝∝∝ 1/ δδδ } 1 2 M 1

d For d2 0.01 ( degenerate limit ) :

The reheating temperature lower bound is relaxed

The required tiny value of d2 can be obtained e.g. in radiative leptogenesis (Branco , Gonzalez, Joaquim , Nobre ’04, ’05) Flavor effects do not spoil the conspiracy

Green points: Unflavored Red points: Flavored

m 10 −5 eV m 10 eV atm = matm = 05.0 eV atm =

….but they yield two interesting results: A pictorial representation Let us give a pictorial description focusing on the dominant Higgs asymmetry and disregarding the asymmetries in quarks and charged lepton singlets d Assume K2α 1 while K2β >> 1

N † N2 N2α Φ β

N2 N2

N¯ NΦ N¯ 2α 2β

This β-asymmetry is induced by the “thermal contact ” with the α-leptons via the Higgs Production stage

We have to solve :

dNN2 eq = −D2 (NN2 − NN2 ) , dz2 dN∆ γ = ε D (N − N eq ) − P 0 W C(2) N , dz 2γ 2 N2 N2 2γ 2 γα ∆α 2 α=γ,τ  dN ∆τ = ε ∆ (N − N eq ) − P 0 W C(2) N . dz 2τ 2 N2 N2 2τ 2 τα ∆α 2 α=γ,τ  P 0 C(2) P 0 C(2) P 0 ≡ 2γ γγ 2γ γτ Defining U as the matrix that diagonalizes : 2 0 (2) 0 (2)  P2τ Cτγ P2τ Cττ  0 −1 0 0 UP2 U = diag(P2γ ,P2τ  )

The asymmetry at T ~ M 2 is then given by :

T ∼M2 −1 −1 N = U  [U  ε + U  ε ] κ(K ) + U  [U  ε + U  ε ] κ(K ) , ∆γ γγ γ γ 2γ γ τ 2τ 2γ γτ τ γ 2γ τ τ 2τ 2τ T ∼M2 −1 −1 N = U  [U  ε + U  ε ] κ(K ) + U  [U  ε + U  ε ] κ(K ) , ∆τ τγ γ γ 2γ γ τ 2τ 2γ ττ τ γ 2γ τ τ 2τ 2τ N T ∼M2 = N T ∼M2 + N T ∼M2 . B−L ∆γ ∆τ Flavour coupling in the N 2-dom.scenario (Antusch , PDB, Jones, King ‘10) Flavor coupling does not relevantly affect the final asymmetry

In N 1 –leptogenesis (Abada, Josse -Michaux ‘07) but a strong enhancement is possible in N 2 –leptogenesis because here now there are three stages to be taken into account: 12 t 9 1) Production at 10 GeV >> T ∼ M2 10 GeV (2 -flavour regime):

dNN2 eq = −D2 (NN2 − NN2 ) , dz2 dN∆ γ = ε D (N − N eq ) − P 0 W C(2) N , (γ ≡ e + µ) dz 2γ 2 N2 N2 2γ 2 γα ∆α 2 α=γ,τ  dN ∆τ = ε ∆ (N − N eq ) − P 0 W C(2) N . dz 2τ 2 N2 N2 2τ 2 τα ∆α 2 α=γ,τ T∼M2 T∼M2 T∼M2 9 N splits intoN and N 2) Decoherence at T ∼ 10 GeV : ∆γ ∆µ ∆e

9 3) Lightest RH neutrino wash -out at T ∼ M1 << 10 GeV (3 -fl. regime):

dN∆ (3) α P 0 C W N , α, β e, µ, τ dz1 = − 1α β αβ 1 ∆β ( = )  Lightest RH neutrino wash-out We have to solve

dN∆ (3) α P 0 C W N , α, β e, µ, τ dz1 = − 1α β αβ 1 ∆β ( = )  in T ∼M2 using as initial conditions N∆β = N∆β

If we first neglect the flavour coupling using the approximation C(3) = I, then

dN∆ α P 0 W N , α, β e, µ, τ dz1 = − 1α 1 ∆α ( = )

This can be straightforwardly solved finding:

T TM 3π T M 3π T M 3π f ∼ 2 8 K1e 2 8 K1µ 2 8 K1τ NB L = N∆e e− +N∆∼µ e− +N∆∼τ e− − Flavor swap scenario (Antusch , PDB, Jones, King ‘10) Suppose that at the production the e+ µ (γ) flavour component of the asymmetry is weakly washed -out while the τ component is strongly washed -out . Then the latter can be considerably enhanced by flavor coupling: N T ∼M2  ε κ(K ) − C(2) ε κ(K )  C(2) ε κ(K ) , ∆τ 2τ 2τ τγ 2γ 2γ τγ 2γ 2γ N T ∼M2  ε κ(K ) , ∆γ 2γ 2γ At the production the total asymmetry does not relevantly change (Abada, Josse -Michaux ‘07) but … a “flavor -swap ” can be

induced at the N 1 wash -out if K1e,K1µ 1 ,K1τ  1

T TM 3π T M 3π T M 3π T M ⇒ f ∼ 2 8 K1e 2 8 K1µ 2 8 K1τ 2 (2) NB L = N∆e e− +N∆∼µ e− +N∆∼τ e− N∆∼τ Cτγ ε2γ κ(K2γ) −  

In this way the strong enhancement of the τ-asymmetry at the production translates into a strong enhancement of the final asy mmetry Flavour coupling at the N 1 wash-out

Let us now take into account flavour coupling at the N 1-wash -out as well: dN∆ (3) α P 0 C W N , α, β e, µ, τ dz1 = − 1α β αβ 1 ∆β ( = )

in T ∼M2 using as initial conditions N∆β = N∆β

We can repeat the same trick as before, i.e. introducing a matri x V that diagonalizes :

0 (3) 0 (3) 0 (3) P1e Cee P1e Ceµ P1e Ceτ P 0  P 0 C(3) P 0 C(3) P 0 C(3)  1 ≡ 1µ µe 1µ µµ 1µ µτ  0 (3) 0 (3) 0 (3)   P Cτe P Cτµ P Cττ   1τ 1τ 1τ    One finally finds the general solution :

3π f −1 − 8 K1α T ∼M2 f f N = V  e Vαβ N ,N = N ∆α αα  ∆β  B−L ∆α  α α β    > with K 1α’’’’ K1α Circumventing the N 1 wash-out

Because of flavour coupling at the N 1 wash -out there is another interesting effect. Let us “unpack ” the previous general expression for example for the τ-asymmetry:

3π f −1 T ∼M2 − 8 K1e N  V  Veβ N e ∆τ τe  ∆β  β   3π −1 T ∼M2 − 8 K1µ + V  Vµβ N e τµ  ∆β  β   3π −1 T ∼M2 − 8 K1τ + V  Vτ β N e ττ  ∆β  β    Now even though one has K1τ >> 1 , there is still a final τ asymmetry that manages to escape the N 1 wash -out. Why ? Again because of the Higgs asymmetry present in the thermal bath that is not exactly that one needed for a complete wash -out of the τ asymmetry ⇒ the lightest RH neutrino wash -out becomes less efficient ! Phantom terms We have now to answer: how at the decoherence , at T ∼ 10 9 GeV , NT∼M2 splits intoNT∼M2 and NT∼M2 ? ∆γ ∆µ ∆e

First stage: T t M (decays) N First stage: T M2 (decays) τ N2γ Assume an initial thermal t µ e N2-abundance N¯ N¯ { τ { 2γ N2

t m e N∆γ

Second stage: T ~ M (N – washout) Second stage: T ~ M 2 (N 2 – washout) µ e

The N 2 wash -out can only suppress the γ-asymmetry but it cannot change the m e ¯ T ∼M2 flavour compositions of 2γ and2γ N ∆γ Phantom terms

9 t Third stage: 10 GeV T’ >> M 1 (3 -flavour regime) µ e 2 ¯ ¯ 2 |α|2γ| +|α|2γ | f2α ≡ 2 m e NT∼M2 N T ∼M2 ∆µ ∆e

f2µ T∼M2  f2e T ∼M2 N (T ) = p + N N (T ) = pe + N , ∆µ µ f2 +f2 ∆γ ∆e f2e+f2µ ∆γ e µ

 κ(K2γ) ε2γ (neglecting flavour coupling !)

f2e pe = ε e − ε 2 f2e+f2µ 2 Phantom terms f2µ pµ = ε µ − ε = −pe 2 f2e+f2µ 2

Notice that phantom terms are not suppressed by N 2 wash -out ! Phantom Leptogenesis

We can have then a situation where K2γγγ, K 2τττ>> 1 so that at the End of the N 2 washout the total asymmetry is negligible: t m e T ~ M 2 t m e T∼M2 T∼M2 T∼M2 NB−L = N∆τ + N∆γ  0 ! 9 t 10 GeV T >> M 1 T∼M2 T∼M2 T∼M2 T∼M2 NB−L = N∆τ + N∆e + N∆µ  0 ! d > Assume K 1e 1 and K1µµµ >> 1 T M1 f T∼M2 NB−L = N∆e  pe !

The N 1 wash -out un -reveal the phantom term and effectively it create a N B-L asymmetry ! There is nothing esoteric but there is a... Drawback of phantom Leptogenesis

We assumed an initial N 2 thermal abundance but if we were assuming An initial vanishing N 2 abundance the phantom terms were just zero ! Therefore, more generally :

f2e in pe = ε e − ε N 2 f2e+f2µ 2 N2   The reason is that phantom terms with opposite sign would be cre ated during the N 2 production by inverse decays and exactly cancelling with the contribution generated from decays ! More generally

In conclusion ....phantom leptogenesis is more a problem for the

N2 dominated scenario since it introduces a strong dependence on the initial conditions !!