Quantum and Medium Effects in Resonant Leptogenesis Andreas - PowerPoint PPT Presentation

Quantum and Medium Effects in Resonant Leptogenesis Andreas Hohenegger with Tibor Frossard Mathias Garny Alexander Kartavtsev (arXiv: 0909.1559, 0911.4122, 1002.0331, 1112.642, in preparation ) École Polytechnique Fédérale de Lausanne PASCOS Mérida, 2012-6-7 Andreas Hohenegger, Mérida 2012

Leptogenesis s L s L t L CP CP CP CP CP CP CP c L b L d L b L ν τ d L ν µ u L ν e L = L SM + 1 2 ¯ � i / � N i ∂ − M i N i − h α i ¯ ℓ α ˜ φ P R N i − h † i α ¯ N i ˜ φ † P L ℓ α Andreas Hohenegger, Mérida 2012

Leptogenesis kinetic equations needed, usually generalized Boltzmann equations (BEs) C ℓ + . ↔ i + . k µ D µ f ℓ ( X , k ) = � [ . f ℓ . ]( X , k ) k interactions of ℓ 2 ↔ 1 collision term � � C ℓ φ ↔ N i ( k ) = 1 � p d Π N i d Π φ q ( 2 π ) 4 δ ( k + p − q ) − 2 where 2 � � p ) f N i ( 1 ± f ℓ k )( 1 ± f φ � � = � � q � � 2 � � k f φ p ( 1 ± f N i f ℓ � � = q ) � � � � 2 ↔ 1 collision term antiparticles � � C ℓ φ ↔ N i ( k ) = 1 � d Π φ p d Π N i q ( 2 π ) 4 δ ( k + p − q ) − 2 Andreas Hohenegger, Mérida 2012

Leptogenesis one-loop vertex and self-energy contribution ℓ = + + N j N i φ N i N i N j parametrize matrix elements by CP -violating parameter 2 2 = 1 = 1 � � � � 2 ( 1 + ǫ i ) |M| 2 2 ( 1 − ǫ i ) |M| 2 � � � � N i , � � � � N i � � � � |M| 2 N i → ℓφ − |M| 2 N i → ¯ ℓ ¯ φ ǫ i = |M| 2 N i → ℓφ + |M| 2 N i → ¯ ℓ ¯ φ ( h † h ) 2 Im � � � � � � � ∗ ∗ ij ǫ vac = 2 Im + 2Im × × i ( h † h ) ii M i self-energy contrib. completed by finite width (Paschos et. al., Pilaftsis et. al., Plümacher et. al., Covi et. al. . . . ) Im ( h † h ) 2 M 2 j − M 2 � � R ij i ǫ vac with R ≡ = − R 2 + A 2 , i ( h † h ) ii ( h † h ) jj M i Γ j Andreas Hohenegger, Mérida 2012

Resonant Leptogenesis self-energy contribution is resonantly enhanced R/ ( R 2 + A 2 ) 1 A = M 1 /M 2 , A = Γ 1 / Γ 2 − M 2 /M 1 A = Γ 1 / Γ 2 + M 2 /M 1 0 . 1 0 . 01 0 . 01 0 . 1 1 10 10 2 R ∼ 1 if Im � ( h † h ) 2 � resonance conditions: ǫ vac ( h † h ) ii ( h † h ) jj ∼ 1 and | R | ≃ A ij i require | M j − M i | ∼ Γ i , j Andreas Hohenegger, Mérida 2012

Nonequilibrium Quantum Field Theory repeat nonequilibrium quantum field analysis of resonant “leptogenesis” within toy-model 0911.4122 for SM+3 ν R � � � � N i → ℓ φ , N i → ℓ φ ψ i → b b , ψ i → b b = ⇒ see also: Buchmüller, Anisimov, Drewes, Mendizabal and Garbrecht, Herannen, Schwaller, et. al. divergence of lepton current (flavour independent) L ( x ) = dn L ∂ µ j µ dt + 3 Hn L j µ ¯ � αα ℓ a α ( x ) γ µ ℓ a Tr [ γ µ S ℓ L ( x ) = �� α ( x ) � = − aa ( x , x )] α, a α, a Kadanoff-Baym equations � x 0 � y 0 d 4 z Σ ℓ d 4 z Σ ℓ αβ αγ γβ αγ γβ i / ∂ x S ℓ F ( x , y ) = ρ ( x , z ) S ℓ F ( z , y ) − F ( x , z ) S ℓ ρ ( z , y ) 0 0 � x 0 y 0 d 4 z Σ ℓ αβ αγ γβ i / ∂ x S ℓ ρ ( x , y ) = ρ ( x , z ) S ℓ ρ ( z , y ) Andreas Hohenegger, Mérida 2012

Nonequilibrium Quantum Field Theory 1-loop self-energy ∆ − → S N S gradient expansion, Wigner transformation . . . ∞ dk 0 d 3 k � � ∂ µ j µ L ( t ) = 2 ( 2 π ) 3 Tr αβ βα αβ βα �� < ( t , / > ( t , / > ( t , / < ( t , / � Σ ℓ k ) S ℓ k ) − Σ ℓ k ) S ℓ k ) 2 π 0 βα k ) ¯ αβ βα k ) ¯ αβ � ¯ k ) − ¯ < ( t , / > ( t , / > ( t , / < ( t , / �� − Σ ℓ S ℓ Σ ℓ S ℓ k ) compare with conventional Boltzmann equations d 3 k � � � ∂ µ j µ C ℓ φ ↔ N i ( k ) − C ℓ φ ↔ N i ( k ) L ( x ) = ( 2 π ) 3 E ℓ k Andreas Hohenegger, Mérida 2012

Nonequilibrium Quantum Field Theory Kadanoff-Baym ansatz and quasi-particle approximation for leptons and Higgs (lepton flavour-diagonal) k ) = δ αβ P L / kP R ( 2 π ) sign ( k 0 ) δ ( k 2 ) αβ ρ ( t , / S ℓ ∆ φ ρ ( t , p ) = ( 2 π ) sign ( p 0 ) δ ( p 2 ) quasi-particle approximation insufficient for Majorana neutrinos overlap due to finite width neglected diagonal approximation neglects cross-correlations ij ρ ( t , q ) = S N δ ij ( / q + M i ) ( 2 π ) sign ( q 0 ) δ ( q 2 − M 2 i ) extended QP-approximation required Andreas Hohenegger, Mérida 2012

Nonequilibrium Quantum Field Theory equilibrium solution for ˆ S with off-diagonal components elegant analytic solution with off-shell dynamics possible (neglecting back-reaction and expansion) 1112.6428 rewrite full ˆ S in terms of diagonal ˆ S and off-diagonal ˆ Σ ′ S − 1 ( x , y ) = ˆ ˆ S − 1 ( x , y ) − ˆ Σ ′ ( x , y ) solve for ˆ S � ˆ S F ( ρ ) = ˆ ˆ S R ˆ � ˆ S F ( ρ ) − ˆ Σ ′ F ( ρ ) ˆ Θ R S A Θ A � − 1 and ˆ � − 1 I + ˆ I + ˆ where ˆ A ˆ S R ˆ � Σ ′ � Σ ′ Θ A = S A Θ R = R insert ˆ S in self-energy analyse the pole structure of ˆ S (extended) quasi-particle approximation for diagonal propagator Andreas Hohenegger, Mérida 2012

Nonequilibrium Quantum Field Theory consider 2RHN, spectral functions for R = 100, ( M 1 , M 2 ≪ M 3 ) T = 0 . 1 M 1 , M 1 M 1 = 1 · 10 3 GeV S 11 ρ S 22 � ( h † h ) 11 ( h † h ) 12 � ρ h † h = S 11 ρ ( h † h ) 21 ( h † h ) 22 S 22 ρ S 12 1 . 7 0 . 14 + 0 . 2 i ρ � � = 10 − 12 0 . 14 − 0 . 2 i 0 . 33 M 2 > M 1 , ( h † h ) 11 > ( h † h ) 22 q 0 q 0 I J Andreas Hohenegger, Mérida 2012

Nonequilibrium Quantum Field Theory spectral functions for R = 100, T = 0 . 1 M 1 , M 1 S 11 ρ S 22 ρ S 11 ρ S 22 ρ S 12 ρ S 11 ρ S 22 ρ S 11 ρ S 22 ρ S 12 ρ q 0 q 0 I J q 0 q 0 I J Andreas Hohenegger, Mérida 2012

Nonequilibrium Quantum Field Theory spectral functions for R = 10, T = 0 . 1 M 1 , M 1 S 11 ρ S 22 ρ S 11 ρ S 22 ρ S 12 ρ q 0 q 0 I J Andreas Hohenegger, Mérida 2012

Nonequilibrium Quantum Field Theory spectral functions for R = 10, T = 0 . 1 M 1 , M 1 S 11 ρ S 22 ρ S 11 ρ S 22 ρ S 12 ρ S 11 ρ S 22 ρ S 11 ρ S 22 ρ S 12 ρ q 0 q 0 I J q 0 q 0 I J N 1 , N 2 undergo level-crossing Andreas Hohenegger, Mérida 2012

Nonequilibrium Quantum Field Theory quantum-corrected rate equation ∂ µ j µ L = � � � � p d Π N i d Π ℓ k d Π φ q ( 2 π ) 4 δ ( k + p − q ) |M| 2 � − N i → ℓφ i � � � − |M| 2 − N i → ¯ ℓ ¯ φ corrected CP -violating parameter in R ≫ 1 limit ( h † h ) 2 ( M 2 i − M 2 ǫ i = Im � � j ) M i Γ j k · L ρ ij � 2 + � 2 ( h † h ) ii ( h † h ) jj k · q M 2 � � i − M j Γ j / M j q · L ρ medium effects � d Π ℓ k d Π φ p ( 2 π ) 4 δ ( q − p − k ) / q ) = 16 π k ( 1 + f φ − f ℓ ) L ρ ( t , / retrieve conventional amplitudes in R → ∞ , T → 0 limit Andreas Hohenegger, Mérida 2012

Nonequilibrium Quantum Field Theory L h ( ρ ) capture medium effects (two components for each sufficient) m ℓ = m φ = 0, T = M 1 ( − ) L ρ ( h ) q 0 4 L 0 ρ 3 . 5 L 3 ρ L 0 h 3 L 3 h 2 . 5 2 1 . 5 1 0 . 5 0 0 5 10 15 20 q M 1 Andreas Hohenegger, Mérida 2012

Rate Equations rate equations � γ D N i � dY N i ( Y N i − Y eq = − κ i z N i ) n eq 2 Γ vac dz i N i � � ǫ i γ D � � γ W N i � dY L � � N i ( Y N i − Y eq N i ) − ( 1 + c φℓ ) c ℓ dz = κ i z Y L n eq 2 n eq 2 Γ vac 2 Γ vac i N i i ℓ i medium corrected reaction densities (compare Garbrecht et. al., Wong et. al., Hannestad et. al., Pastor et. al.) � d Π N i q d Π ℓ p ( 2 π ) 4 δ ( q − k − k ) φ � γ D N i � ≡ k d Π × |M| 2 N i ( 1 + f φ − f ℓ ) f N i � d Π N i q d Π ℓ p ( 2 π ) 4 δ ( q − k − k ) φ ǫ i γ D � � ≡ k d Π N i × ǫ i |M| 2 N i ( 1 + f φ − f ℓ ) f N i � p ( 2 π ) 4 δ ( q − p − k ) d Π N i q d Π ℓ φ � γ W N i � ≡ k d Π × |M| 2 N i ( 1 + f φ − f ℓ )( 1 − f N i ) f N i Andreas Hohenegger, Mérida 2012

Preliminary Results averaged CP -violating parameter � ǫ 1 � − 10 − 6 R = 1 � ǫ vac R = 10 1 − 10 − 5 R = 100 R = 1 � R = 10 � ǫ 1 � − 10 − 4 R = 100 | R | < 10 − 10 − 3 − 0 . 01 − 0 . 1 − 1 0 . 1 1 10 z Andreas Hohenegger, Mérida 2012

Thermal Masses and Modified Dispersion Relations L h ( ρ ) depend on the kinematics in the decay thermal masses of leptons and Higgs m ℓ ∼ 0 . 2 T , m φ ∼ 0 . 4 T , φ → ℓ + N i for m φ > m ℓ + M i = + + ǫ i for Higgs ( 1 + f φ − f ℓ ) − → ( f φ + f ℓ ) leptons have modified dispersion relations (Weldon; Giudice et.al.; Plümacher, Kießig) d 3 k dk 0 d 3 k dk 0 � � ( 2 π ) 3 δ ( k 2 − m 2 d Π ℓ q = ℓ ) − → ( 2 π ) 3 δ ( δ ℓ ( k )) collinear enhancement can enable Majorana decay with enhanced rate at high T (Bödecker et. al., Laine et. al.) Andreas Hohenegger, Mérida 2012

Conclusions Nonequilibrium quantum field theory can put Leptogenesis on solid ground improved rate equations for quasi-degenerate case quantitative corrections can be significant need quantum analysis with off-shell dynamics in degenerate case need to include further medium effects Andreas Hohenegger, Mérida 2012