Nonequilibrium QFT approach to leptogenesis

Tibor Frossard

MPIK-Heidelberg

In collaboration with M. Garny, A. Hohenegger, A. Kartavtsev, and D. Mitrouskas

Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 1 / 11 1 B + L violating sphalerons + L violating Majorana mass

2 Yukawa coupling hαi

3 Expanding Universe + Ni decay

Leptogenesis SM + 3 heavy Majorana neutrinos: Dynamical generation of the 1
L =LSM + N¯i i∂/ − Mi Ni baryon asymmetry: 2 ¯ ˜ † ¯ ˜† → 3 Sakharov conditions: − hαi`αφPRNi − hiαNiφ PL`α 1 Baryon number violation 2 C and CP violation 3 Out of equilibrium dynamics ⇒ Need physic beyond the SM

Leptogenesis Baryogenesis via Leptogenesis

Matter-antimatter asymmetry in the Universe CMB + BBN ⇒ n − n ¯ η = B B = (6.2 ± 0.2) 10−10 nγ

Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 2 / 11 1 B + L violating sphalerons + L violating Majorana mass

2 Yukawa coupling hαi

3 Expanding Universe + Ni decay

Leptogenesis SM + 3 heavy Majorana neutrinos: 1 ¯ /
L =LSM + 2 Ni i∂ − Mi Ni ¯ ˜ † ¯ ˜† − hαi`αφPRNi − hiαNiφ PL`α

⇒ Need physic beyond the SM

Leptogenesis Baryogenesis via Leptogenesis

Matter-antimatter asymmetry in the Universe CMB + BBN ⇒ n − n ¯ η = B B = (6.2 ± 0.2) 10−10 nγ

Dynamical generation of the baryon asymmetry: → 3 Sakharov conditions: 1 Baryon number violation 2 C and CP violation 3 Out of equilibrium dynamics

Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 2 / 11 1 B + L violating sphalerons + L violating Majorana mass

2 Yukawa coupling hαi

3 Expanding Universe + Ni decay

Leptogenesis Baryogenesis via Leptogenesis

Matter-antimatter asymmetry in the Universe CMB + BBN ⇒ n − n ¯ η = B B = (6.2 ± 0.2) 10−10 Leptogenesis nγ SM + 3 heavy Majorana neutrinos: Dynamical generation of the 1
L =LSM + N¯i i∂/ − Mi Ni baryon asymmetry: 2 ¯ ˜ † ¯ ˜† → 3 Sakharov conditions: − hαi`αφPRNi − hiαNiφ PL`α 1 Baryon number violation 2 C and CP violation 3 Out of equilibrium dynamics ⇒ Need physic beyond the SM

Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 2 / 11 2 Yukawa coupling hαi

3 Expanding Universe + Ni decay

Leptogenesis Baryogenesis via Leptogenesis

Matter-antimatter asymmetry in the Universe CMB + BBN ⇒ n − n ¯ η = B B = (6.2 ± 0.2) 10−10 Leptogenesis nγ SM + 3 heavy Majorana neutrinos: Dynamical generation of the 1
L =LSM + N¯i i∂/ − Mi Ni baryon asymmetry: 2 ¯ ˜ † ¯ ˜† → 3 Sakharov conditions: − hαi`αφPRNi − hiαNiφ PL`α 1 Baryon number violation 1 B + L violating sphalerons+ 2 C and CP violation L violating Majorana mass 3 Out of equilibrium dynamics ⇒ Need physic beyond the SM

Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 2 / 11 3 Expanding Universe + Ni decay

Leptogenesis Baryogenesis via Leptogenesis

Matter-antimatter asymmetry in the Universe CMB + BBN ⇒ n − n ¯ η = B B = (6.2 ± 0.2) 10−10 Leptogenesis nγ SM + 3 heavy Majorana neutrinos: Dynamical generation of the 1
L =LSM + N¯i i∂/ − Mi Ni baryon asymmetry: 2 ¯ ˜ † ¯ ˜† → 3 Sakharov conditions: − hαi`αφPRNi − hiαNiφ PL`α 1 Baryon number violation 1 B + L violating sphalerons + 2 C and CP violation L violating Majorana mass 3 Out of equilibrium dynamics 2 Yukawa coupling hαi ⇒ Need physic beyond the SM

Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 2 / 11 Leptogenesis Baryogenesis via Leptogenesis

Matter-antimatter asymmetry in the Universe CMB + BBN ⇒ n − n ¯ η = B B = (6.2 ± 0.2) 10−10 Leptogenesis nγ SM + 3 heavy Majorana neutrinos: Dynamical generation of the 1
L =LSM + N¯i i∂/ − Mi Ni baryon asymmetry: 2 ¯ ˜ † ¯ ˜† → 3 Sakharov conditions: − hαi`αφPRNi − hiαNiφ PL`α 1 Baryon number violation 1 B + L violating sphalerons + 2 C and CP violation L violating Majorana mass 3 Out of equilibrium dynamics 2 Yukawa coupling hαi ⇒ Need physic beyond the 3 Expanding Universe + N SM i decay

Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 2 / 11 ⇒ into the Boltzmann equation for the lepton number!

Leptogenesis Canonical approach Generation of an asymmetry: production washout 4
2
Ni (inverse)decay O h O h ∆L = 2 scattering O h6
O h4
2 4
2 2
top scattering O λt h O λt h gauge scattering O g2h4
O g2h2

Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 3 / 11 ⇒ into the Boltzmann equation for the lepton number!

Leptogenesis Canonical approach Generation of an asymmetry: production washout 4
2
Ni (inverse)decay O h O h ∆L = 2 scattering O h6
O h4
2 4
2 2
top scattering O λt h O λt h gauge scattering O g2h4
O g2h2

Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 3 / 11 Leptogenesis Canonical approach Generation of an asymmetry: production washout 4
2
Ni (inverse)decay O h O h ∆L = 2 scattering O h6
O h4
2 4
2 2
top scattering O λt h O λt h gauge scattering O g2h4
O g2h2

⇒ into the Boltzmann equation for the lepton number!

Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 3 / 11 Vacuum amplitudes in the hot Universe → thermal masses? → thermal decay width? → thermal amplitudes?

Nonequilibrium Quantum Field Theory approach is needed!!

Leptogenesis Canonical approach

This approach suffers from several problems: Double counting problem → can be solved if one neglects the quantum statistical factors ⇒ Real intermediate state substraction

Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 4 / 11 Nonequilibrium Quantum Field Theory approach is needed!!

Leptogenesis Canonical approach

This approach suffers from several problems: Double counting problem → can be solved if one neglects the quantum statistical factors ⇒ Real intermediate state substraction Vacuum amplitudes in the hot Universe → thermal masses? → thermal decay width? → thermal amplitudes?

Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 4 / 11 Leptogenesis Canonical approach

This approach suffers from several problems: Double counting problem → can be solved if one neglects the quantum statistical factors ⇒ Real intermediate state substraction Vacuum amplitudes in the hot Universe → thermal masses? → thermal decay width? → thermal amplitudes?

Nonequilibrium Quantum Field Theory approach is needed!!

Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 4 / 11 Schwinger-Dyson equation

ˆ−1 ˆ−1 ˆ αβ δiΓ2 S (x, y) = S0 (x, y) − Σ(x, y) , Σij (x, y) = − βα δSji (y, x)

2PI effective Γ = = 2 action

ˆ ˆ i 0 0 ˆ → doubling of the d.o.f.: S(x, y) = SF (x, y) − 2 signC(x − y ) Sρ(x, y) | {z } | {z } statistical spectral

Nonequilibrium QFT approach Basics of nonequilibrium QFT

Right-handed neutrinos propagator on CTP: αβ α ¯ β Sij (x, y) =< TCNi (x)Nj (y) >

Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 5 / 11 ˆ ˆ i 0 0 ˆ → doubling of the d.o.f.: S(x, y) = SF (x, y) − 2 signC(x − y ) Sρ(x, y) | {z } | {z } statistical spectral

Nonequilibrium QFT approach Basics of nonequilibrium QFT

Right-handed neutrinos propagator on CTP: αβ α ¯ β Sij (x, y) =< TCNi (x)Nj (y) >

Schwinger-Dyson equation

ˆ−1 ˆ−1 ˆ αβ δiΓ2 S (x, y) = S0 (x, y) − Σ(x, y) , Σij (x, y) = − βα δSji (y, x)

2PI effective Γ = = 2 action

Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 5 / 11 Nonequilibrium QFT approach Basics of nonequilibrium QFT

Right-handed neutrinos propagator on CTP: αβ α ¯ β Sij (x, y) =< TCNi (x)Nj (y) >

Schwinger-Dyson equation

ˆ−1 ˆ−1 ˆ αβ δiΓ2 S (x, y) = S0 (x, y) − Σ(x, y) , Σij (x, y) = − βα δSji (y, x)

2PI effective Γ = = 2 action

ˆ ˆ i 0 0 ˆ → doubling of the d.o.f.: S(x, y) = SF (x, y) − 2 signC(x − y ) Sρ(x, y) | {z } | {z } statistical spectral

Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 5 / 11 Quantum Boltzmann equation

Z 4 d d p h p p
i (n`(t)−n`¯(t)) = gw 4 tr Σ`<(t, p)(1 − f` ) + Σ`>(t, p)f` S`ρ(t, p) dt (2π) | {z } | {z } gain term loss term

Σ` =

Nonequilibrium QFT approach Quantum Boltzmann equation

Wigner transform ↔ Fourrier transform wrt relative coordinates Gradient expansion ↔ Expansion in slow relative to fast time-scales (H/T ) Quasiparticle approximation ↔ Narrow width approximation Kadanoff-Baym ansatz ↔ One-particle distribution function

Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 6 / 11 Nonequilibrium QFT approach Quantum Boltzmann equation

Wigner transform ↔ Fourrier transform wrt relative coordinates Gradient expansion ↔ Expansion in slow relative to fast time-scales (H/T ) Quasiparticle approximation ↔ Narrow width approximation Kadanoff-Baym ansatz ↔ One-particle distribution function

Quantum Boltzmann equation

Z 4 d d p h p p
i (n`(t)−n`¯(t)) = gw 4 tr Σ`<(t, p)(1 − f` ) + Σ`>(t, p)f` S`ρ(t, p) dt (2π) | {z } | {z } gain term loss term

Σ` =

Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 6 / 11 - neglect the off-diagonal components

The amplitudes incorporate thermal masses and thermal widths but, → Self-energy contribution to the decay rate is missing! → s × s and t × t channels of the scattering processes are missing! ⇒ need to go beyond the free propagator approximation!

Nonequilibrium QFT approach Lepton asymmetry Free (thermal-) propagators into the lepton self-energy: 1 p 0 2 2 1 k 0 2 2 S`F = ( 2 − f` )PLp/sign(p )(2π)δ(p − m` ) ∆φF = ( 2 + fφ )sgn(k )(2π)δ(k − mφ) ij ij 1 q 0 2 2 S = δ ( − f )(q + Mi)sign(q )(2π)δ(q − M ) NF 2 Ni / i

Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 7 / 11 - neglect the off-diagonal components

The amplitudes incorporate thermal masses and thermal widths but, → Self-energy contribution to the decay rate is missing! → s × s and t × t channels of the scattering processes are missing! ⇒ need to go beyond the free propagator approximation!

Nonequilibrium QFT approach Lepton asymmetry Free (thermal-) propagators into the lepton self-energy: 1 p 0 2 2 1 k 0 2 2 S`F = ( 2 − f` )PLp/sign(p )(2π)δ(p − m` ) ∆φF = ( 2 + fφ )sgn(k )(2π)δ(k − mφ) ij ij 1 q 0 2 2 S = δ ( − f )(q + Mi)sign(q )(2π)δ(q − M ) NF 2 Ni / i

Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 7 / 11 - neglect the off-diagonal components

The amplitudes incorporate thermal masses and thermal widths but, → Self-energy contribution to the decay rate is missing! → s × s and t × t channels of the scattering processes are missing! ⇒ need to go beyond the free propagator approximation!

Nonequilibrium QFT approach Lepton asymmetry Free (thermal-) propagators into the lepton self-energy: 1 p 0 2 2 1 k 0 2 2 S`F = ( 2 − f` )PLp/sign(p )(2π)δ(p − m` ) ∆φF = ( 2 + fφ )sgn(k )(2π)δ(k − mφ) ij ij 1 q 0 2 2 S = δ ( − f )(q + Mi)sign(q )(2π)δ(q − M ) NF 2 Ni / i

Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 7 / 11 - neglect the off-diagonal components

The amplitudes incorporate thermal masses and thermal widths but, → Self-energy contribution to the decay rate is missing! → s × s and t × t channels of the scattering processes are missing! ⇒ need to go beyond the free propagator approximation!

Nonequilibrium QFT approach Lepton asymmetry Free (thermal-) propagators into the lepton self-energy: 1 p 0 2 2 1 k 0 2 2 S`F = ( 2 − f` )PLp/sign(p )(2π)δ(p − m` ) ∆φF = ( 2 + fφ )sgn(k )(2π)δ(k − mφ) ij ij 1 q 0 2 2 S = δ ( − f )(q + Mi)sign(q )(2π)δ(q − M ) NF 2 Ni / i

Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 7 / 11 - neglect the off-diagonal components

Nonequilibrium QFT approach Lepton asymmetry Free (thermal-) propagators into the lepton self-energy: 1 p 0 2 2 1 k 0 2 2 S`F = ( 2 − f` )PLp/sign(p )(2π)δ(p − m` ) ∆φF = ( 2 + fφ )sgn(k )(2π)δ(k − mφ) ij ij 1 q 0 2 2 S = δ ( − f )(q + Mi)sign(q )(2π)δ(q − M ) NF 2 Ni / i

The amplitudes incorporate thermal masses and thermal widths but, → Self-energy contribution to the decay rate is missing! → s × s and t × t channels of the scattering processes are missing! ⇒ need to go beyond the free propagator approximation!

Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 7 / 11 Nonequilibrium QFT approach Lepton asymmetry Free (thermal-) propagators into the lepton self-energy: 1 p 0 2 2 1 k 0 2 2 S`F = ( 2 − f` )PLp/sign(p )(2π)δ(p − m` ) ∆φF = ( 2 + fφ )sgn(k )(2π)δ(k − mφ) ij ij 1 q 0 2 2 S = δ ( − f )(q + Mi)sign(q )(2π)δ(q − M ) NF 2 Ni / i - neglect the off-diagonal components

The amplitudes incorporate thermal masses and thermal widths but, → Self-energy contribution to the decay rate is missing! → s × s and t × t channels of the scattering processes are missing! ⇒ need to go beyond the free propagator approximation!

Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 7 / 11 The pole contribution describes quasiparticles and contribute to the (inverse-)decay: → QP approximation and KB ansatz S˜ii = ( 1 − f q )(q + M )sign(q0)(2π)δ(q2 − M 2) F 2 Ni / i i The off-shell contributions describe scatterings.

→ expand SˆF around its poles ↔ extended QP approximation: ˆ ˜ˆ 1 ˆ ˆ d ˆ ˆ ˆ d ˆ
SF = SF − 2 SRΣF SR + SAΣF SA ⇔

→ assume to be valid out of equilibrium

Nonequilibrium QFT approach Resummation and extended QP approximation

In equilibrium the full statistical propagator SˆF can be expressed in terms of the flavor diagonal propagators Sˆ and flavor off-diagonal self-energy Σˆ od:  −1h i −1 ˆ 1 ˆ ˆ od ˆ ˆ ˆ od ˆ 1 ˆ od ˆ SF = + SRΣR SF − SRΣF SA + ΣA SA

Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 8 / 11 The pole contribution describes quasiparticles and contribute to the (inverse-)decay: → QP approximation and KB ansatz S˜ii = ( 1 − f q )(q + M )sign(q0)(2π)δ(q2 − M 2) F 2 Ni / i i The off-shell contributions describe scatterings.

→ assume to be valid out of equilibrium

Nonequilibrium QFT approach Resummation and extended QP approximation

In equilibrium the full statistical propagator SˆF can be expressed in terms of the flavor diagonal propagators Sˆ and flavor off-diagonal self-energy Σˆ od:  −1h i −1 ˆ 1 ˆ ˆ od ˆ ˆ ˆ od ˆ 1 ˆ od ˆ SF = + SRΣR SF − SRΣF SA + ΣA SA

→ expand SˆF around its poles ↔ extended QP approximation: ˆ ˜ˆ 1 ˆ ˆ d ˆ ˆ ˆ d ˆ
SF = SF − 2 SRΣF SR + SAΣF SA ⇔

Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 8 / 11 The off-shell contributions describe scatterings.

→ assume to be valid out of equilibrium

Nonequilibrium QFT approach Resummation and extended QP approximation

In equilibrium the full statistical propagator SˆF can be expressed in terms of the flavor diagonal propagators Sˆ and flavor off-diagonal self-energy Σˆ od:  −1h i −1 ˆ 1 ˆ ˆ od ˆ ˆ ˆ od ˆ 1 ˆ od ˆ SF = + SRΣR SF − SRΣF SA + ΣA SA

→ expand SˆF around its poles ↔ extended QP approximation: ˆ ˜ˆ 1 ˆ ˆ d ˆ ˆ ˆ d ˆ
SF = SF − 2 SRΣF SR + SAΣF SA ⇔ The pole contribution describes quasiparticles and contribute to the (inverse-)decay: → QP approximation and KB ansatz S˜ii = ( 1 − f q )(q + M )sign(q0)(2π)δ(q2 − M 2) F 2 Ni / i i

Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 8 / 11 → assume to be valid out of equilibrium

Nonequilibrium QFT approach Resummation and extended QP approximation

In equilibrium the full statistical propagator SˆF can be expressed in terms of the flavor diagonal propagators Sˆ and flavor off-diagonal self-energy Σˆ od:  −1h i −1 ˆ 1 ˆ ˆ od ˆ ˆ ˆ od ˆ 1 ˆ od ˆ SF = + SRΣR SF − SRΣF SA + ΣA SA

→ expand SˆF around its poles ↔ extended QP approximation: ˆ ˜ˆ 1 ˆ ˆ d ˆ ˆ ˆ d ˆ
SF = SF − 2 SRΣF SR + SAΣF SA ⇔ The pole contribution describes quasiparticles and contribute to the (inverse-)decay: → QP approximation and KB ansatz S˜ii = ( 1 − f q )(q + M )sign(q0)(2π)δ(q2 − M 2) F 2 Ni / i i The off-shell contributions describe scatterings.

Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 8 / 11 Nonequilibrium QFT approach Resummation and extended QP approximation

In equilibrium the full statistical propagator SˆF can be expressed in terms of the flavor diagonal propagators Sˆ and flavor off-diagonal self-energy Σˆ od:  −1h i −1 ˆ 1 ˆ ˆ od ˆ ˆ ˆ od ˆ 1 ˆ od ˆ SF = + SRΣR SF − SRΣF SA + ΣA SA

→ expand SˆF around its poles ↔ extended QP approximation: ˆ ˜ˆ 1 ˆ ˆ d ˆ ˆ ˆ d ˆ
SF = SF − 2 SRΣF SR + SAΣF SA ⇔ The pole contribution describes quasiparticles and contribute to the (inverse-)decay: → QP approximation and KB ansatz S˜ii = ( 1 − f q )(q + M )sign(q0)(2π)δ(q2 − M 2) F 2 Ni / i i The off-shell contributions describe scatterings.

→ assume to be valid out of equilibrium

Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 8 / 11 The internal Majorana propagators are automatically RIS-subtracted ⇒ don’t need to neglect the quantum statistical factors!

Nonequilibrium QFT approach Resummation and extended QP approximation

→ insert the resummed Majorana propagator into the one-loop lepton self-energy:

→ The insertion of the resummed propagator reproduces the missing terms!

Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 9 / 11 Nonequilibrium QFT approach Resummation and extended QP approximation

→ insert the resummed Majorana propagator into the one-loop lepton self-energy:

→ The insertion of the resummed propagator reproduces the missing terms! The internal Majorana propagators are automatically RIS-subtracted ⇒ don’t need to neglect the quantum statistical factors!

Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 9 / 11 Nonequilibrium QFT approach Numerical comparison with canonical approach

√ M2 = 10M1 Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 10 / 11 Nonequilibrium QFT approach Numerical comparison with canonical approach

√ M2 = 10M1

Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 10 / 11 include thermal corrections

is free of double counting problem

same procedure can be applied to the top scattering Thank you!

Conclusion Conclusion

Nonequilibrium quantum field theory approach to leptogenesis:

reproduce all vacuum diagrams

Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 11 / 11 is free of double counting problem

same procedure can be applied to the top scattering Thank you!

Conclusion Conclusion

Nonequilibrium quantum field theory approach to leptogenesis:

reproduce all vacuum diagrams

include thermal corrections

Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 11 / 11 same procedure can be applied to the top scattering Thank you!

Conclusion Conclusion

Nonequilibrium quantum field theory approach to leptogenesis:

reproduce all vacuum diagrams

include thermal corrections

is free of double counting problem

Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 11 / 11 Thank you!

Conclusion Conclusion

Nonequilibrium quantum field theory approach to leptogenesis:

reproduce all vacuum diagrams

include thermal corrections

is free of double counting problem

same procedure can be applied to the top scattering

Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 11 / 11 Conclusion Conclusion

Nonequilibrium quantum field theory approach to leptogenesis:

reproduce all vacuum diagrams

include thermal corrections

is free of double counting problem

same procedure can be applied to the top scattering Thank you!

Tibor Frossard (MPIK-HD) Nonequilibrium QFT in leptogenesis 27.09.2012 11 / 11