Quantum Mechanics of Leptogenesis

Introduction Non-Equilibrium Dynamics of QM Systems Solutions Thermal Leptogenesis

QUANTUM MECHANICSOF LEPTOGENESIS

Sebastián Mendizabal Cofré DESY

based on Annals Phys.324:1234-1260,2009 Phys.Rev.Lett.104:121102,2010.

International School of Nuclear Physics Erice-Sicily September 16-24, 2010

QUANTUM MECHANICSOF LEPTOGENESIS 1 / 28 Introduction Non-Equilibrium Dynamics of QM Systems Solutions Thermal Leptogenesis Outline

Introduction

Non-equilibrium dynamics - Boltzmann equations - Kadanoff-Baym

Solution to the Kadanoff-Baym

Thermal leptogenesis - Boltzmann equations - Kadanoff-Baym

Conclusions

QUANTUM MECHANICSOF LEPTOGENESIS 2 / 28 Introduction Non-Equilibrium Dynamics of QM Systems Solutions Thermal Leptogenesis Matter-antimater asymmetry

BBN : nB = (5.1 − 6.5) × 10−10 nγ

WMAP: nB = (6.19 ± 0.15) × 10−10 nγ

Sakharov’s conditions Baryon number violation C, CP violation Departure from thermal equilibrium

QUANTUM MECHANICSOF LEPTOGENESIS 3 / 28 Not enough asymmetry is generated!

Extension to the SM is required

Introduction Non-Equilibrium Dynamics of QM Systems Solutions Thermal Leptogenesis Asymmetry in the SM

The Sakharov’s conditions are satisﬁed in the SM Baryon number violation via the triangle anomaly

CP violation with the Kobayashi-Maskawa mechanism

Out-of-equilibrium in the electro-weak phase transition

QUANTUM MECHANICSOF LEPTOGENESIS 4 / 28 Introduction Non-Equilibrium Dynamics of QM Systems Solutions Thermal Leptogenesis Asymmetry in the SM

The Sakharov’s conditions are satisﬁed in the SM Baryon number violation via the triangle anomaly

CP violation with the Kobayashi-Maskawa mechanism

Out-of-equilibrium in the electro-weak phase transition

Not enough asymmetry is generated!

Extension to the SM is required

QUANTUM MECHANICSOF LEPTOGENESIS 4 / 28 ⇒the study of the non-equilibrium dynamics of the Majorana particle is fundamental to understand leptogenesis

Introduction Non-Equilibrium Dynamics of QM Systems Solutions Thermal Leptogenesis See-saw mechanism and Thermal Leptogenesis

Insertion of right-handed neutrinos νR 1 L = L + iν¯ 6∂ν −¯l Φ˜λν − ν¯c Mν + h.c. SM R R L R 2 R R

see-saw mechanism explains small neutrino masses c mass term of N ≈ νR + νR violates lepton number mass and ﬂavour eigenstates are not identical complex phases appear out-of-equilibrium decay of N generally violates CP and generates the asymmetry

QUANTUM MECHANICSOF LEPTOGENESIS 5 / 28 Introduction Non-Equilibrium Dynamics of QM Systems Solutions Thermal Leptogenesis See-saw mechanism and Thermal Leptogenesis

Insertion of right-handed neutrinos νR 1 L = L + iν¯ 6∂ν −¯l Φ˜λν − ν¯c Mν + h.c. SM R R L R 2 R R

⇒the study of the non-equilibrium dynamics of the Majorana particle is fundamental to understand leptogenesis QUANTUM MECHANICSOF LEPTOGENESIS 5 / 28 Introduction Non-Equilibrium Dynamics of QM Systems Solutions Thermal Leptogenesis

Non-Equilibrium Dynamics of Quantum Systems

QUANTUM MECHANICSOF LEPTOGENESIS 6 / 28 Introduction Non-Equilibrium Dynamics of QM Systems Solutions Thermal Leptogenesis Methods

Boltzmann equations (BE)

(full) quantum Boltzmann equations (QBE)

kinetic equations for matrices of density

Kadanoff-Baym equations (KBE)

QUANTUM MECHANICSOF LEPTOGENESIS 7 / 28 Introduction Non-Equilibrium Dynamics of QM Systems Solutions Thermal Leptogenesis Quantum Boltzmann Equations

n˙ (t) + 3H(t)n(t) + Γ(t)(n(t) − neq) = 0

describe time evolution of classical particle numbers cross sections are imported from quantum field theory BE are known to work well in some examples, e.g. photon decoupling big bang nucleosynthesis but note that BE cannot describe coherent oscillations BE assume particles move freely between scatterings BE are Markovian , they are local in time classical particle number is not well defined in interacting quantum field theory

QUANTUM MECHANICSOF LEPTOGENESIS 8 / 28 A full quantum treatment needs to be performed

Introduction Non-Equilibrium Dynamics of QM Systems Solutions Thermal Leptogenesis Full Quantum Boltzmann Equations

First order solution to a gradient expansion of the Kadanoff-Baym equations assumes close to equilibrium condition slow variation with respect to the center of mass coordinate many quantum-interferences are lost ill-deﬁned quantities such particle densities are used

QUANTUM MECHANICSOF LEPTOGENESIS 9 / 28 Introduction Non-Equilibrium Dynamics of QM Systems Solutions Thermal Leptogenesis Full Quantum Boltzmann Equations

A full quantum treatment needs to be performed

QUANTUM MECHANICSOF LEPTOGENESIS 9 / 28 Introduction Non-Equilibrium Dynamics of QM Systems Solutions Thermal Leptogenesis

Is a Quantum Treatment possible? spacial homogeneity weak coupling ⇒ perturbative background plasma is in equilibrium backreaction can be neglected

’Weak coupling to a thermal bath’

QUANTUM MECHANICSOF LEPTOGENESIS 10 / 28 Introduction Non-Equilibrium Dynamics of QM Systems Solutions Thermal Leptogenesis Boltzmann vs Kadanoff-Baym Equations

initial value problem for density matrix ρ(t)...... or for correlation functions h...i = tr(ρ ...) KBE contain full quantum mechanics

particle numbers ⇔ correlation functions collision term ⇔ self energies

Π, Π encode information about all decay and scattering processes QUANTUM MECHANICSOF LEPTOGENESIS 11 / 28 Introduction Non-Equilibrium Dynamics of QM Systems Solutions Thermal Leptogenesis

KBE Formalism

QUANTUM MECHANICSOF LEPTOGENESIS 12 / 28 Introduction Non-Equilibrium Dynamics of QM Systems Solutions Thermal Leptogenesis Kadanoff-Baym Equations

Z Z t1 2 2 − 3 0 0 − 0 − 0 (∂1 + m )∆ (x1, x2) = − d x dt Π (x1, x )∆ (x , x2) t2 Z Z t1 2 2 + 3 0 0 − 0 + 0 (∂1 + m )∆ (x1, x2) = − d x dt Π (x1, x )∆ (x , x2) ti Z Z t2 3 0 0 + 0 − 0 + d x dt Π (x1, x )∆ (x , x2) ti

Z Z t1 − 3 0 0 − 0 − 0 − (i6∂1 − m)S (x1, x2) = − d x dt Π (x1, x )S (x , x2) t2 Z Z t1 + 3 0 0 − 0 + 0 −(i6∂1 − m)S (x1, x2) = − d x dt Π (x1, x )S (x , x2) ti Z Z t2 3 0 0 + 0 − 0 + d x dt Π (x1, x )S (x , x2) ti QUANTUM MECHANICSOF LEPTOGENESIS 13 / 28 For such systems spectral propagators ∆−, S−, G− are time translation invariant KBE are equivalent to a stochastic Langevin equation KBE can be solved analytically up to a memory integral

Introduction Non-Equilibrium Dynamics of QM Systems Solutions Thermal Leptogenesis Weak Coupling to a thermal Bath

QUANTUM MECHANICSOF LEPTOGENESIS 14 / 28 Introduction Non-Equilibrium Dynamics of QM Systems Solutions Thermal Leptogenesis Weak Coupling to a thermal Bath

consider fields that are weakly coupled to a large bath in equilibrium assume interaction mainly with bath fields X then self energies are computed with equilibrium propagators in practice realised by using couplings that are linear in the field of interest, e.g. gφO[X ], gΨO[X ], at leading order in g For such systems spectral propagators ∆−, S−, G− are time translation invariant KBE are equivalent to a stochastic Langevin equation KBE can be solved analytically up to a memory integral

QUANTUM MECHANICSOF LEPTOGENESIS 14 / 28 Introduction Non-Equilibrium Dynamics of QM Systems Solutions Thermal Leptogenesis Kadanoff-Baym Equations

Z t1 (∂2 + ω2)∆−(t − t )= − dt0Π−(t − t0)∆−(t0 − t ) t1 q q 1 2 q 1 q 2 t2 Z t1 (∂2 + ω2)∆+(t , t ) = − dt0Π−(t − t0)∆+(t0, t ) t1 q q 1 2 q 1 q 2 ti Z t2 0 + 0 − 0 + dt Πq (t1 − t )∆q (t − t2) ti

Z t2 − 0 − 0 − 0 (iγ0∂t1 − qγ − m)Sq (t1 − t2) = − dt Πq (t1 − t )Sq (t − t2) t1 Z t1 + 0 − 0 + 0 (iγ0∂t1 − qγ − m)Sq (t1, t2) = − dt Πq (t1 − t )Sq (t , t2) ti Z t2 0 + 0 − 0 + dt Πq (t1 − t )Sq (t − t2) ti QUANTUM MECHANICSOF LEPTOGENESIS 15 / 28 Introduction Non-Equilibrium Dynamics of QM Systems Solutions Thermal Leptogenesis

Solutions

QUANTUM MECHANICSOF LEPTOGENESIS 16 / 28 particle behaviour⇒ Boltzmann equations

R ImΠq (Ωq) ⇒ ωq → Ωq, Γq ≈ − Ωq single resonance kinematically behaves like quasiparticle but total energy receives vacuum contribution

particle interpretation and Boltzmann equations break down, at large T possibly even in a weakly coupled theory

Introduction Non-Equilibrium Dynamics of QM Systems Solutions Thermal Leptogenesis Properties of the Solutions

R R R retarded self-energy Π = Π |T =0 + δΠ (T ) is the decisive quantity ReΠR gives thermal mass ImΠR decay width Γ to resonance

Three regimes

1 2 |ReΠ|, |ImΠ| ωq

2 2 2 |ReΠ| ≈ ωq, |ImΠ| ωq

3 2 |ReΠ|, |ImΠ| ≈ ωq

QUANTUM MECHANICSOF LEPTOGENESIS 17 / 28 R ImΠq (Ωq) ⇒ ωq → Ωq, Γq ≈ − Ωq single resonance kinematically behaves like quasiparticle but total energy receives vacuum contribution

particle interpretation and Boltzmann equations break down, at large T possibly even in a weakly coupled theory

Introduction Non-Equilibrium Dynamics of QM Systems Solutions Thermal Leptogenesis Properties of the Solutions

R R R retarded self-energy Π = Π |T =0 + δΠ (T ) is the decisive quantity ReΠR gives thermal mass ImΠR decay width Γ to resonance

Three regimes

1 2 |ReΠ|, |ImΠ| ωq particle behaviour⇒ Boltzmann equations

2 2 2 |ReΠ| ≈ ωq, |ImΠ| ωq

3 2 |ReΠ|, |ImΠ| ≈ ωq

QUANTUM MECHANICSOF LEPTOGENESIS 17 / 28 single resonance kinematically behaves like quasiparticle but total energy receives vacuum contribution

particle interpretation and Boltzmann equations break down, at large T possibly even in a weakly coupled theory

Introduction Non-Equilibrium Dynamics of QM Systems Solutions Thermal Leptogenesis Properties of the Solutions

R R R retarded self-energy Π = Π |T =0 + δΠ (T ) is the decisive quantity ReΠR gives thermal mass ImΠR decay width Γ to resonance

Three regimes

1 2 |ReΠ|, |ImΠ| ωq particle behaviour⇒ Boltzmann equations

R 2 2 ImΠq (Ωq) 2 |ReΠ| ≈ ω , |ImΠ| ω ⇒ ωq → Ωq, Γq ≈ − q q Ωq

3 2 |ReΠ|, |ImΠ| ≈ ωq

QUANTUM MECHANICSOF LEPTOGENESIS 17 / 28 particle interpretation and Boltzmann equations break down, at large T possibly even in a weakly coupled theory

Introduction Non-Equilibrium Dynamics of QM Systems Solutions Thermal Leptogenesis Properties of the Solutions

R R R retarded self-energy Π = Π |T =0 + δΠ (T ) is the decisive quantity ReΠR gives thermal mass ImΠR decay width Γ to resonance

Three regimes

1 2 |ReΠ|, |ImΠ| ωq particle behaviour⇒ Boltzmann equations

R 2 2 ImΠq (Ωq) 2 |ReΠ| ≈ ω , |ImΠ| ω ⇒ ωq → Ωq, Γq ≈ − q q Ωq single resonance kinematically behaves like quasiparticle but total energy receives vacuum contribution

3 2 |ReΠ|, |ImΠ| ≈ ωq

QUANTUM MECHANICSOF LEPTOGENESIS 17 / 28 Introduction Non-Equilibrium Dynamics of QM Systems Solutions Thermal Leptogenesis Properties of the Solutions

R R R retarded self-energy Π = Π |T =0 + δΠ (T ) is the decisive quantity ReΠR gives thermal mass ImΠR decay width Γ to resonance

Three regimes

1 2 |ReΠ|, |ImΠ| ωq particle behaviour⇒ Boltzmann equations

R 2 2 ImΠq (Ωq) 2 |ReΠ| ≈ ω , |ImΠ| ω ⇒ ωq → Ωq, Γq ≈ − q q Ωq single resonance kinematically behaves like quasiparticle but total energy receives vacuum contribution

3 2 |ReΠ|, |ImΠ| ≈ ωq particle interpretation and Boltzmann equations break down, at large T possibly even in a weakly coupled theory

QUANTUM MECHANICSOF LEPTOGENESIS 17 / 28 Introduction Non-Equilibrium Dynamics of QM Systems Solutions Thermal Leptogenesis The Spectral Function

Damped oscillatory behaviour Breit-Wigner breaks down at high temperatures

QUANTUM MECHANICSOF LEPTOGENESIS 18 / 28 Introduction Non-Equilibrium Dynamics of QM Systems Solutions Thermal Leptogenesis The Statistical Propagator

depends on two time arguments equilibrates independent of initial conditions after characteristic time τ ∼ 1/Γ oscillates with plasma frequency

QUANTUM MECHANICSOF LEPTOGENESIS 19 / 28 Introduction Non-Equilibrium Dynamics of QM Systems Solutions Thermal Leptogenesis

Application to Leptogenesis

QUANTUM MECHANICSOF LEPTOGENESIS 20 / 28 Introduction Non-Equilibrium Dynamics of QM Systems Solutions Thermal Leptogenesis Boltzmann approach for asymmetry The coupled differential equations in a static universe are given by

∂f N = C[f ] for Majorana ∂t N ∂fl−¯l = C[f ¯, f ] for lepton asymmetry ∂t l−l N C[f ]: collision term Inserting the solution for the Majoranas to the asymmetry equation, and without wash-out terms Z 1 4 4 fl−¯l = − CP (2π) δ (k + q − p)p · k k q,p 1 × f f eq (1 − e−Γt ) lφ N Γ

QUANTUM MECHANICSOF LEPTOGENESIS 21 / 28 Introduction Non-Equilibrium Dynamics of QM Systems Solutions Thermal Leptogenesis KBE approach for the Asymmetry

How to calculate the asymmetry without reference to particle number or distribution function? deﬁne lepton number matrix 0 + Lkij (t1, t2) = −tr[γ Skij (t1, t2)].

Lkii (t, t) gives leptonic charge in ﬂavour i at time t CP-violation comes from interference between LO and NLO terms (tree level and 1-loop level) Considering an stationary universe and neglecting wash-out terms

QUANTUM MECHANICSOF LEPTOGENESIS 22 / 28 Introduction Non-Equilibrium Dynamics of QM Systems Solutions Thermal Leptogenesis Lepton Self-Energy

+ QUANTUM MECHANICSOF LEPTOGENESIS 23 / 28 Z 16π 0 eq fLi (t, k) = −ii k · k flφ(k, q)fN (ω) |k| q,p,q0,k0 1 × (2π)4δ4(k + q − p)(2π)4δ4(k 0 + q0 − p) Γ × 1 − e−Γt

Introduction Non-Equilibrium Dynamics of QM Systems Solutions Thermal Leptogenesis Solution of the Asymmetry

Z 0 16π k · k eq 0 0 Lkij (t, t) = −ij 0 flφ(k, q)fN (ω)flφ(k , q ) |k| q,q0 |k |ω 1 Γ × 4 2 Γ2 0 0 2 Γ2 ((ω − |k| − |q|) + 4 )((ω − |k | − |q |) + 4 ) × cos[(|k| + |q| − |k0| − |q0|)t] + e−Γt

0 0 − Γt −(cos[(ω − |k| − |q|)t] + cos[(ω − |k | − |q |)t])e 2 ,

QUANTUM MECHANICSOF LEPTOGENESIS 24 / 28 Introduction Non-Equilibrium Dynamics of QM Systems Solutions Thermal Leptogenesis Solution of the Asymmetry

0 0 − Γt −(cos[(ω − |k| − |q|)t] + cos[(ω − |k | − |q |)t])e 2 ,

Z 16π 0 eq fLi (t, k) = −ii k · k flφ(k, q)fN (ω) |k| q,p,q0,k0 1 × (2π)4δ4(k + q − p)(2π)4δ4(k 0 + q0 − p) Γ × 1 − e−Γt

QUANTUM MECHANICSOF LEPTOGENESIS 24 / 28 Introduction Non-Equilibrium Dynamics of QM Systems Solutions Thermal Leptogenesis On-Shell Approximation (unjustiﬁed!)

Z os 16π 0 eq 0 0 Lkij (t, t) = −ij k · k flφ(k, q)fN (ω)flφ(k , q ) k q,q0,p,k0 1 × (2π)4δ4(k + q − p)(2π)4δ4(k 0 + q0 − p) Γ 2 − Γt × 1 − e 2

Z 16π 0 eq fLi (t, k) = −ii k · k flφ(k, q)fN (ω) |k| q,p,q0,k0 1 × (2π)4δ4(k + q − p)(2π)4δ4(k 0 + q0 − p) Γ × 1 − e−Γt

QUANTUM MECHANICSOF LEPTOGENESIS 25 / 28 Introduction Non-Equilibrium Dynamics of QM Systems Solutions Thermal Leptogenesis Inclusion of SM widths

QUANTUM MECHANICSOF LEPTOGENESIS 26 / 28 Introduction Non-Equilibrium Dynamics of QM Systems Solutions Thermal Leptogenesis Inclusion of SM widths

Z 0 ˜ 16π k · k eq 0 0 Lkij (t, t) = −ij 0 flφ(k, q)fN (ω)flφ(k , q ) |k| q,q0 |k |ω 1 0 1 ΓlφΓ × 4 lφ Γ 2 1 2 0 0 2 1 02 ((ω − k − q) + 4 Γlφ)((ω − k − q ) + 4 Γlφ) 1 − e−Γt

Z 16π 0 eq fLi (t, k) = −ii k · k flφ(k, q)fN (ω) |k| q,p,q0,k0 1 × (2π)4δ4(k + q − p)(2π)4δ4(k 0 + q0 − p) Γ × 1 − e−Γt BUT: This is not yet a consistent treatment of gauge interactions!!!

QUANTUM MECHANICSOF LEPTOGENESIS 27 / 28 Introduction Non-Equilibrium Dynamics of QM Systems Solutions Thermal Leptogenesis Conclusions

We computed the generated lepton asymmetry for hierarchical heavy neutrino masses and a constant (or very slowly changing) temperature without semi-classical approximations. Quantum and non-Markovian effects can be crucial for leptogenesis. We ﬁnd signiﬁcant deviations from Boltzmann equations due to off-shell effects, memory effects and temperature dependent corrections. The consistent inclusion of all SM corrections remains an issue.

QUANTUM MECHANICSOF LEPTOGENESIS 28 / 28