Leptogenesis with small violation of B-L Nuria Rius IFIC, Universidad de Valencia-CSIC with Juan Racker and Manuel Pe˜ na, arXiv:1205.1948 to appear in JCAP What is ν ?, GGI Workshop, Firenze 2012
OUTLINE 1. Introduction 2. Leptogenesis in models with small violation of B-L 3. Boltzmann Equations 4. Results and conclusions What is ν ?, GGI Workshop, Firenze 2012 1
1. Introduction • Neutrino masses and baryon asymmetry of the Universe (BAU) naturally explained by the seesaw mechanism → generically no testable → Davidson-Ibarra bound on M 1 for hierarchical heavy SM singlets: ∼ 10 9 GeV ( 10 8 GeV if M 2 /M 1 < M 1 > ∼ 10 ). → tension between thermal leptogenesis and gravitino bound on the reheating temperature T RH in SUSY seesaw scenarios with hierarchical RHN: ∼ 10 5 − 10 7 GeV ( 10 9 − 10 10 GeV) for m 3 / 2 ∼ - Unstable gravitino → T RH < 100 GeV - 1 TeV ( > ∼ 10 TeV) ∼ 10 9 GeV can - Gravitino is the LSP: bounds depend on the NLSP, but T RH > be obtained for m 3 / 2 > ∼ 10 GeV Kawasaki et al. (2008) What is ν ?, GGI Workshop, Firenze 2012 2
• Global lepton number U (1) L slightly broken by small parameters µ , λ ′ , protected from radiative corrections. h † P R N i ℓ α − 1 L L = − λ αi � ′′ c i N ′ j − 1 i N ′′ 2 M ij N c 2 M ij N j + h.c. h † P R N ′ h † P R N ′′ αj � αj � ′ c / = − λ ′ j ℓ α − λ ′ j ℓ α − 1 i N k − 1 2 µ ′ j N ′ 2 µ ik N c L L jk N k + h.c. λ αi can be large, because they do not vanish in the B − L conserved limit → in the absence of µ, µ ′ and λ ′ αi , a perturbatively conserved lepton number can be defined: L N = 1 L N ′ = − 1 L N ′′ = 0 L ℓ α = 1 for the SM leptons. Example: Inverse seesaw → Only ( N i , N ′ i ) per generation, with µ ik = λ ′ αj = 0 Mohapatra, Valle (1986) What is ν ?, GGI Workshop, Firenze 2012 3
• Rich phenomenology: - Large neutrino Yukawa couplings and heavy neutrino masses at the TeV scale - Flavour and CP violating effects not suppressed by light neutrino masses Bernabeu et al. (1987); NR, Valle (1990); Gonz´ alez-Garc´ ıa, Valle (1992);Gavela et al. (2009) - Heavy neutrinos may be at LHC reach Han, Zhang (2006); F. del Aguila et al. (2007); Kersten, Smirnov (2007) - Two strongly degenerate RH neutrinos (quasi-Dirac fermion) → resonant leptogenesis at T ∼ O (1 TeV) Pilaftsis, Underwood (2005); Asaka, Blanchet (2008); Blanchet et al. (2010) What is ν ?, GGI Workshop, Firenze 2012 4
Low energy effective Lagrangian: � c d =6 Λ LN O d =5 + L = L SM + c d =5 F L O d =6 i + . . . , Λ 2 i i where Λ F L can be O (TeV) and Λ LN ≫ Λ F L If B − L is approximately conserved: i) N i is a Majorana neutrino with small Yukawa couplings λ ′ αi , λ ′ αi λ ′ ( c d =5 ) αβ βi M = . Λ LN M i λ ′ αi λ ′∗ ( c d =6 ) αβ βi M = . Λ 2 M 2 i F L What is ν ?, GGI Workshop, Firenze 2012 5
ii) The N i is a Dirac or quasi Dirac neutrino with four degrees of freedom → there are two Majorana neutrinos N ih and N il with masses M i + µ i and M i − µ i respectively. √ If B − L is conserved, µ i = 0 and N i = ( N ih + iN il ) / 2 is a Dirac fermion. Yukawa interactions: h † P R h † P R L Y Ni = − λ αi � N ih + iN il αi � N ih − iN il ℓ α − λ ′ ℓ α + h.c., √ √ 2 2 Contribution of a quasi Dirac heavy neutrino to the Weinberg operator at leading order: ( c d =5 QD ) αβ αi − µ i βi − µ i = ( λ ′ M i λ αi ) 1 M i λ βi + λ αi 1 M i ( λ ′ M i λ βi ) + . . . Λ LN ( c d =6 λ αi λ ∗ QD ) αβ βi = . Λ 2 M 2 i F L What is ν ?, GGI Workshop, Firenze 2012 6
2. Leptogenesis in models with small violation of B-L Sakharov’s conditions for generating the BAU are naturally satisfied in the seesaw framework for neutrino masses → Leptogenesis: • Out of equilibrium decay of heavy Majorana neutrinos ( L violation) • CP asymmetry → lepton asymmetry Y L • ( B + L ) -violating non-perturbative sphaleron interactions partially convert Y L into a baryon asymmetry Y B . What can be different regarding leptogenesis in different models with approximately conserved B-L ? N 1 → heavy neutrino which generates the lepton asymmetry N 2 → heavy neutrino which makes the most important (non resonant) contribution to asymmetry in N 1 decay. What is ν ?, GGI Workshop, Firenze 2012 7
I. Both N 1 and N 2 are type i) Majorana fermions → nothing different from the standard seesaw II. N 1 is type i) and N 2 type ii) (quasi Dirac) CP asymmetry produced in the decay of N 1 into leptons of flavour α : Γ( N 1 → ℓ α h ) − Γ( N 1 → ¯ ℓ α ¯ h ) = ǫ L / α 1 + ǫ L � ǫ α 1 ≡ α 1 Γ( N 1 → ℓ α h ) + Γ( N 1 → ¯ ℓ α ¯ h ) α with � � ǫ L / ǫ L f ( a j )Im λ ∗ αj λ α 1 ( λ † λ ) j 1 g ( a j )Im λ ∗ αj λ α 1 ( λ † λ ) 1 j α 1 = α 1 = j =2 h, 2 l j =2 h, 2 l Covi et al. (1996) • a j ≡ M 2 j /M 2 1 • to lowest order in µ 2 , f ( a 2 h ) = f ( a 2 l ) and g ( a 2 h ) = g ( a 2 l ) . What is ν ?, GGI Workshop, Firenze 2012 8
λ α 2 l = iλ α 2 h , therefore when µ 2 → 0 : ǫ L / α 2 h λ α 1 ( λ † λ ) 2 h 1 (1 + i ∗ 2 ) = 0 → f ( a 2 h )Im λ ∗ α 1 − � � � � ǫ L λ ∗ α 2 h λ α 1 ( λ † λ ) 1 2 h (1+ | i | 2 ) = 2 g ( a 2 h )Im λ ∗ α 2 h λ α 1 ( λ † λ ) 1 2 h α 1 − → g ( a 2 h )Im . ǫ L α 1 are related to the lepton number conserving d=6 operators → escape the DI bound because they are not linked to neutrino masses (LNV d=5 Weinberg operator) Antusch et al. (2010) However, ǫ 1 ≡ � α ǫ α 1 ∝ µ 2 because � α ǫ L α 1 = 0 → flavour effects mandatory for successful leptogenesis Barbieri et al., (2000); Endoh et al. (2004); Abada et al. (2006); Nardi et al. (2006) What is ν ?, GGI Workshop, Firenze 2012 9
III. N 1 is quasi Dirac and N 2 type i): resonant enhancement of the CP asymmetry for degenerate neutrinos N 1 h , N 1 l when µ 1 > ∼ Γ N 1 , with ( λ † λ ) ii ( λ † λ ) ii Γ N ih = M i + µ i ≈ M i − µ i = Γ N il ≡ Γ N i 8 π 2 8 π 2 Covi and Roulet, (1997); Pilaftsis (2005); Anisinov et al. (2006) λ ′ √ Resonant contribution suppressed by ( λ † λ ) 11 → the CP asymmetry can not α 1 reach the maximum value 1/2. Successful leptogenesis with M = 10 6 GeV (1 TeV), for ǫ ≡ λ ′ /λ ∼ 10 − 3 and ǫ M ≡ µ 1 /M 1 ∼ 10 − 8 ( 10 − 11 ) → no observable low energy effects Asaka et al. (2008) µ 1 ≪ Γ N 1 : observable µ → eγ Blanchet et al. (2010) Bolztmann picture breaks down De Simone, Riotto (2007); Garny et al. (2010); Garbrecht, Herranen (2012); Garny et al. (2012) What is ν ?, GGI Workshop, Firenze 2012 10
IV. Both N 1 and N 2 are type ii) quasi Dirac neutrinos: → both, resonant contributions from N 1 l, 1 h and large contribution of N 2 to ǫ α 1 l and ǫ α 1 h This work: • We do NOT consider resonant contributions, widely studied • We focus on ǫ L , not bounded by neutrino masses and large in models which approximately conserve B-L • Exhaustive analysis of parameter space What is ν ?, GGI Workshop, Firenze 2012 11
4. Boltzmann equations Scenario for leptogenesis involving three fermion singlets N 1 , N 2 l , N 2 h with masses M 1 , M 2 − µ 2 , M 2 + µ 2 and Yukawa couplings given by the Lagrangian h † P R N 1 ℓ α − λ α 2 � h † P R N 2 h + iN 2 l L Y = − λ α 1 � ℓ α + h.c. . √ 2 with λ α 1 ≪ λ α 2 • We neglect N 2 LNV Yukawa couplings, λ ′ α 2 ≪ λ α 2 (checked that they have negligible effects) • We consider two flavours for simplicity (3 flavours discussed later) • Include decays and inverse decays of N 1 , N 2 and rapid L -conserving but L α -violating flavour changing interaction (FCI): ℓ β ¯ h → ℓ α ¯ h ¯ h → ℓ α ¯ ℓ β h → ℓ α h h and ℓ β • Neglected spectator processes and ∆ L = 1 scatterings: few 10% What is ν ?, GGI Workshop, Firenze 2012 12
Relevant parameters: • M 1 : for fixed CP asymmetry, lower M 1 → stronger washout (slower Universe expansion rate) • M 2 /M 1 : ǫ α 1 ∝ ( M 1 /M 2 ) 2 for M 1 ≪ M 2 γ F CI ( T ) ∝ ( M 1 /M 2 ) 4 for T ∼ M 1 ≪ M 2 If M 2 /M 1 < ∼ 20 → include real N 2 in BE m 1 ≡ ( λ † λ ) 11 v 2 /M 1 � m ∗ ≃ 10 − 3 eV • ( λ † λ ) 11 : Effective mass ˜ Γ N 1 H ( T = M 1 ) = ˜ m 1 with m ∗ defined by m ∗ • ( λ † λ ) 22 : ǫ α 1 ∝ ( λ † λ ) 22 , but - FCI washout processes increase with ( λ † λ ) 22 - N 2 interactions should be slower than τ Yukawa interactions Blanchet et al. (2007) What is ν ?, GGI Workshop, Firenze 2012 13
M 2 = 10 7 GeV , ( λ ✝ λ ) 22 = 10 -4 1e+10 1e+08 γ τ / γ N 2 , γ τ / | γ Σ FCI | 1e+06 10000 100 1 0.01 10000 100000 1e+06 1e+07 1e+08 1e+09 T [GeV] What is ν ?, GGI Workshop, Firenze 2012 14
• Flavour projectors: K αi ≡ λ αi λ ∗ αi ( λ † λ ) ii For two flavours, only two independent projectors, we take K µ 1 , K µ 2 • µ 2 : discrete parameter → Y B takes different values for µ 2 ≫ Γ N 2 and µ 2 ≪ Γ N 2 Notation: Y X ≡ n X /s X ) /Y eq y X ≡ ( Y X − Y ¯ X Reaction densities: γ a,b,... c,d,... ≡ γ ( a, b, . . . → c, d, . . . ) z ≡ M 1 /T What is ν ?, GGI Workshop, Firenze 2012 15
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