Kadanoff-Baym Equations and Baryogenesis
Mathias Garny (DESY Hamburg)
KBE, Kiel, 13.10.11
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Nonequilibrium dynamics at high energy
Early universe
Reheating after Inflation
Dark matter freeze-out
Baryogenesis
. . .
Heavy Ion Collisions
LHC: ALICE
RHIC
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Outline
Matter-Antimatter asymmetry
Baryogenesis and Leptogenesis
Relativistic quantum kinetic theory for leptogenesis
Results in Marcovian and Thermal-bath limits
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Standard Model of Cosmology
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
No antimatter
Ratio of cosmic ray fluxes observed at Earth
Φ(anti-proton)
Φ(proton)∼ 10−5 − 10−3
kinetic energy [GeV]
110 1 10 210
/pp
610
510
410
310
BESS 2000 (Y. Asaoka et al.)
BESS 1999 (Y. Asaoka et al.)
BESSpolar 2004 (K. Abe et al.)
CAPRICE 1994 (M. Boezio et al.)
CAPRICE 1998 (M. Boezio et al.)
HEATpbar 2000 (A. S. Beach et al.)
PAMELA
PAMELA Phys.Rev.Lett. 105 (2010) 121101
Antinuclei (e.g. anti-helium) < 10−6A. Alcaraz et al., Phys. Lett. B 461, 387 (1999)
Consistent with secondary production p + N → p + p + . . .
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
No antimatter
Galaxy: Antistars would accrete interstellar gas, leading toannihilation gamma-rays. Observations of discrete Galacticgamma-ray sources limit the fraction of antistars in the Galaxy to< 10−4
Galaxy clusters (106 − 107 lyr): annihilations would producehigh-energy gamma ray flux
N + N → π0, π± → γ Eγ & 100MeV
Non-observation yields a limit
anti-matter
matter. 10−6 − 10−9
G. Steigman JCAP 0810 (2008) 001
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
No antimatter
Particle physics: symmetry between particles and anti-particles(opposite charge, same mass, (almost) same interaction strength)
Paul Dirac (Nobel lecture 1933) proposed a symmetric universe(50% of the stars made out of anti-nuclei and positrons)
Why is there a matter/antimatter asymmetry?
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Matter-Antimatter asymmetry
Asymmetry parameter
η =nb − nb
nγ
nb = baryon density
nb = anti-baryon density
nγ = photon density
Consistent value inferred from Big Bang Nucleosynthesis (T ∼ keV) andthe cosmic microwave background (T ∼ eV)
η =nb − nb
nγ= (6.21± 0.16) · 10−10
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Matter-Antimatter asymmetry
(1) Initial baryon asymmetry after Big Bang
Problem:
Diluted by inflationWashed out by ∆B 6= 0 processes at high energy
(2) Dynamical creation: Baryogenesis
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Matter-Antimatter asymmetry
(1) Initial baryon asymmetry after Big Bang
Problem:
Diluted by inflationWashed out by ∆B 6= 0 processes at high energy
(2) Dynamical creation: Baryogenesis
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Matter-Antimatter asymmetry
(1) Initial baryon asymmetry after Big Bang
Problem:
Diluted by inflationWashed out by ∆B 6= 0 processes at high energy
(2) Dynamical creation: Baryogenesis
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Baryogenesis
Sakharov conditions Sakharov 1967
baryon number violation: 〈B〉 6= const.
C,CP violation: γ(i → f ) 6= γ(i → f )
deviation from thermal equilibrium: γ(i → f ) 6= γ(f → i)
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Baryogenesis within the Standard Model of particles ?
SU(3)c × SU(2)L × U(1)Y
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Baryogenesis within the Standard Model of particles ?
B-violation at quantum level t Hooft 76
∂µjµ =
g2
32π2Fµν F
µν
Exponentially suppressed for T < TEW ∼ 100GeV
Γproton ∝ e−16π2/g 2
= 10−165
Unsupressed for T > TEW (sphaleron) Klinkhamer, Manton 84; Kuzmin, Rubakov,
Shaposhnikov 85
∆B = ∆L
CP-violation in quark mixing
→ K 0/K 0 decay, Bd,s decay
Electroweak phase-transition: first-order for mH < 60− 80GeV
LEP limit mH > 114GeV ⇒ Third Sakharov condition is not fulfilled
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Baryogenesis
Baryogenesis in models beyond the Standard Model of particles
Electroweak baryogenesis
GUT baryogenesis
Affleck-Dine baryogenesis
Leptogenesis
. . .
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Motivation: Observation of massive neutrinos
Neutrino oscillation (νe ↔ νµ ↔ ντ )
m2ν1−m2
ν2= 7.6...8.6 · 10−5eV2
|m2ν1−m2
ν3| = 1.9...3 · 10−3eV2
Direct search (3H→3 He + e− + νe)
mνe < 2.2eV 95%C .L.
⇒ At least two massive neutrinos
⇒ Tiny mass
Standard Model: neutrinos massless
Superkamiokande
KATRIN (0.2eV)
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
SM + Right-handed neutrinos (νR)e,µ,τ
↔
Neutrino masses
Majorana mass term
L = LSM−y νRh†`L−MR νRν
cR
See-saw mechanism
mνL' y2〈h〉2
MR' meV - eV
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
SM + Right-handed neutrinos (νR)e,µ,τ
↔ ↔
Neutrino masses
Majorana mass term
L = LSM−y νRh†`L−MR νRν
cR
See-saw mechanism
mνL' y2〈h〉2
MR' meV - eV
Matter-Antimatter asymmetry
B- via L-violation MR νRνcR
→ 0νββ: (A, Z)→ (A, Z + 2) + 2e−
(Gerda, Nemo, Exo, . . . )
CP-violation in ν-mixing→ ν-oscillation(Double Chooz, Daya Bay, . . . )
Expanding universe H = a/a
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Leptogenesis Fukugita, Yanagida
MνR,i→`L,αh† =yiα + . . .
MνR,i→`cL,αh =
y∗iα + . . .
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Leptogenesis Fukugita, Yanagida
MνR,i→`L,αh† =yiα +
y∗iβ
yjβ
yjα
+ . . .
MνR,i→`cL,αh =
y∗iα +yiβ
y∗jβ
y∗jα
+ . . .
Matter-antimatter (CP) asymmetry
⇔ interference of tree and loop processes
Γ(νR,i → `L,αh†)− Γ(νR,i → `cL,αh) ∼ Im(yiαyiβy∗jαy∗jβ) · Im
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Leptogenesis
B via L violation
νR,i → `h† νR,i → `ch
CP violation
εi =Γ(νR,i→`h†)−Γ(νR,i→`c h)Γ(νR,i→`h†)+Γ(νR,i→`c h)
∝ Im
(+
)
Deviation from equilibrium ?
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Leptogenesis
thermal plasma `, h, q,W ,Z , γ, gauge interactions ‘fast’
g4T � H ∼ T 2/Mpl
right-handed neutrinos νR,i ≡ Ni ‘slow’
ΓNi =(y†y)iiMNi
8π� g4T
. . . produced when T � MNi
. . . equilibrate if ΓNi e−MNi
/T > H
. . . decay when T < MNi and t > τNi
⇒ deviation from thermal equilibrium
K ≡ (ΓNi/H)|T =MNi
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Leptogenesis
thermal plasma `, h, q,W ,Z , γ, gauge interactions ‘fast’
g4T � H ∼ T 2/Mpl
right-handed neutrinos νR,i ≡ Ni ‘slow’
ΓNi =(y†y)iiMNi
8π� g4T
. . . produced when T � MNi
. . . equilibrate if ΓNi e−MNi
/T > H
. . . decay when T < MNi and t > τNi
⇒ deviation from thermal equilibrium
K ≡ (ΓNi/H)|T =MNi
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Leptogenesis
thermal plasma `, h, q,W ,Z , γ, gauge interactions ‘fast’
g4T � H ∼ T 2/Mpl
right-handed neutrinos νR,i ≡ Ni ‘slow’
ΓNi =(y†y)iiMNi
8π� g4T
. . . produced when T � MNi
. . . equilibrate if ΓNi e−MNi
/T > H
. . . decay when T < MNi and t > τNi
⇒ deviation from thermal equilibrium
K ≡ (ΓNi/H)|T =MNi
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Leptogenesis
10−2
10−2
10−1
10−1
100
100
101
101
102
102
10−11
10−11
10−9
10−9
10−7
10−7
10−5
10−5
10−3
10−3
10−1
10−1
101
101
z=M1/T
eq
zeq
NN1
|NB−L|
K=10−2
NN1
10−2
10−2
10−1
10−1
100
100
101
101
102
102
10−11
10−11
10−9
10−9
10−7
10−7
10−5
10−5
10−3
10−3
10−1
10−1
101
101
z=M1/T
eq
NN1
|NB−L|
K=100
NN1
zeq
Effective rate equations Buchmuller, Di Bari, Plumacher,. . .
dNNi
dt= −(Di + Si )(NNi − Neq
Ni)
dNB−L
dt=
∑
i
εiDi (NNi − NeqNi
)
︸ ︷︷ ︸source term
− WNB−L
︸ ︷︷ ︸washout term
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Leptogenesis
final asymmetry
η =NB−L
Nγ= −3
4ε1κf
K
κf
[N1-dominated, D+ID, unflavoured] Buchmuller, Di Bari, Plumacher
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Leptogenesis
m1 (eV)
M1(G
eV)
[N1-dominated, D+ID, unflavoured] Buchmuller, Di Bari, Plumacher
lower bound on M1 Davidson, Ibarra
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Leptogenesis
10-4
10-3
10-2
10-1
100
m~1(eV)
108
109
1010
1011
1012
M1(G
eV)
MEG
PRISM/PRIME
Probing supersymmetric leptogenesis with µ→ eγ Ibarra, Simonetto 09
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Leptogenesis
Baryogenesis via Leptogenesis
CP violation in decay described by loop process
deviation from thermal equilibrium
Quantum nonequilibrium effects ?
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Kinetic theory: standard Boltzmann approach
NL(t) =
∫d3x√|g |∫
d3p
(2π)3
∑`
[f`(t, x, p)− f ¯(t, x, p)]
pµDµf`(t, x, p) =∑
i
∫dΠNi dΠh
× (2π)4δ(p` + ph − pNi )
×[|M|2Ni→`h† fNi (1− f`)(1 + fh) + . . .
− |M|2`h†→Nif`fh(1− fNi ) + . . .
]
fψ(t, x, p) : distribution function of on-shell particles
|M|2 : matrix elements computed in vacuum, off-shell effects
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Kinetic theory: standard Boltzmann approach
NL(t) =
∫d3x√|g |∫
d3p
(2π)3
∑`
[f`(t, x, p)− f ¯(t, x, p)]
pµDµf`(t, x, p) =∑
i
∫dΠNi dΠh
× (2π)4δ(p` + ph − pNi )
×[|M|2Ni→`h† fNi (1− f`)(1 + fh) + . . .
− |M|2`h†→Nif`fh(1− fNi ) + . . .
]
fψ(t, x, p) : distribution function of on-shell particles
|M|2 : matrix elements computed in vacuum, off-shell effects
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Corrections within Boltzmann framework
Bose-enhancement, Pauli-Blocking; kinetic (non-)equilibrium
quantum statistical factors 1± fknon-integrated Boltzmann equations
Hannsestad, Basbøll 06; Garayoa, Pastor, Pinto, Rius, Vives 09; Hahn-Woernle, Plumacher, Wong 09
Thermal corrections via thermal QFT
medium correction to CP-violating parameter ε = εvac + δεth
thermal masses, decay width
Covi, Rius, Roulet, Vissani 98; Giudice, Notari, Raidal, Riotto, Stumia 04; Besak, Bodeker 10
Kiessig, Thoma, Plumacher 10; . . .
Flavour effectsNardi, Nir, Roulet, Racker 06; Adaba, Davidson, Ibarra, Josse-Micheaux, Losada, Riotto 06; Blanchet,
diBari 06; . . .
Spectator processes, scatterings, N2, . . .
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Double Counting Problem
Naive contribution from decay/inverse decay
|M|2Ni→`h†= |M0|2(1 + εi ) |M|2`h†→Ni
= |M0|2(1− εi )
|M|2Ni→`c h = |M0|2(1− εi ) |M|2`c h→Ni= |M0|2(1 + εi )
dNB−L
dt∝ (|M|2Ni→`h† − |M|
2Ni→`c h)NNi
− (|M|2`h†→Ni− |M|2`c h→Ni
)NeqNi
∝ εi (NNi +NeqNi
)
⇒ spurious generation of asymmetry even in equilibrium
Origin: Double Counting Problem ↔ +
→ need real intermediate state subtraction
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Double Counting Problem
Naive contribution from decay/inverse decay
|M|2Ni→`h†= |M0|2(1 + εi ) |M|2`h†→Ni
= |M0|2(1− εi )
|M|2Ni→`c h = |M0|2(1− εi ) |M|2`c h→Ni= |M0|2(1 + εi )
dNB−L
dt∝ (|M|2Ni→`h† − |M|
2Ni→`c h)NNi
− (|M|2`h†→Ni− |M|2`c h→Ni
)NeqNi
∝ εi (NNi +NeqNi
)
⇒ spurious generation of asymmetry even in equilibrium
Origin: Double Counting Problem ↔ +
→ need real intermediate state subtraction
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Kinetic theory for leptogenesis
Goalderivation of kinetic equations starting from first principles
on-/off-shell treated in a unified way (avoid double-counting)
thermal medium corrections, . . . , resonant leptogenesis, coherent flavortransitions
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
CTP/Kadanoff-Baym approach and leptogenesis
Resolve double counting problems occuring in the Boltzmann approachassociated to real intermediate states Buchmuller, Fredenhagen 00, De Simone, Riotto 05,
MG, Kartavtsev, Hohenegger, Lindner 09,10, Beneke, Garbrecht, Herranen, Schwaller 10
Medium corrections to decay/inverse decay and scatteringsMG, Kartavtsev, Hohenegger Lindner 09,10; Beneke, Garbrecht, Herranen, Schwaller 10; Garbrecht 10
see also Bodecker, Besak, Anisimov 10; Kiessig, Thoma, Plumacher 10; Salvio Lodone Strumia 11
Finite width of lepton, Higgs, and non-equilibrium Majorana neutrinoevolution Anisimov, Buchmuller, Drewes, Mendizabal 08,10
Flavored leptogenesis: transition between flavored/unflavored regimesBeneke, Garbrecht, Fidler, Herranen 10
Systematic inclusion of higher-order effects (e.g. gradient corrections)MG, Kartavtsev, Hohenegger 10
Resonant leptogenesis
self-consistent resummation scheme for mixing particlesnon-equilibrium propagators for unstable/off-shell particlesinclusion of coherent N1—N2 flavor transitions (oscillations)
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Kinetic theory for leptogenesis
First Step: Bosonic toy model
L =1
2∂µNi∂
µNi + ∂µ`†∂µ`− 1
2Mi Ni Ni − yi Ni ``− y∗i Ni `
†`† − λ
4[`†`]2 + . . .
MNi→`` = + + + . . .
MNi→`†`† = + + + . . .
CP-violating parameter (vacuum, non-degenerate) xj = M2j /M
2i
εvaci =
Γ(Ni → ``)− Γ(Ni → `†`†)
Γ(Ni → ``) + Γ(Ni → `†`†)= −
∑j
|yj |28πM2
j
Im
(yi y∗j
y∗i yj
)[xj ln
(1 + xj
xj
)+
1
2(xj − 1)
]
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Kinetic theory for leptogenesis
Schwinger-Keldysh propagator: x , y ∈ C
D(x , y) = 〈TC`(x)`†(y)〉
Schwinger-Dyson equation
i(�x + m2)D(x , y) = δC(x − y) +
∫C
d4zΣ(x , z)D(z, y)
Integration over closed time path∫C d4z =
∫d4z+ −
∫d4z−
tinit ≤ z0 ≤ max(x0, y0)
Kadanoff-Baym equations
(�x + m2)D≷(x , y) = −i
∫ x0
tinit
d4z (Σ> − Σ<)(x , z)D≷(z, y)
+ i
∫ y0
tinit
d4z Σ≷(x , z)(D> − D<)(z, y)
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Kinetic theory for leptogenesis
B-L current
jµ(x) = 2i⟨
[Dµ`(x)] `†(x)− `(x)Dµ`†(x)⟩
= (nB−L,~jB−L)
dNB−L
dt=
∫d3x
√|g | Dµjµ
= 2i
∫d3x
√|g |⟨
[DµDµ`(x)] `†(x)− `(x)DµDµ`†(x)⟩
= 2i
∫d3x
√|g | �x [D>(x , y)− D>(y , x)]|x=y
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Kinetic theory for leptogenesis
B-L current
jµ(x) = 2i⟨
[Dµ`(x)] `†(x)− `(x)Dµ`†(x)⟩
= (nB−L,~jB−L)
dNB−L
dt=
∫d3x
√|g | Dµjµ
= 2i
∫d3x
√|g |⟨
[DµDµ`(x)] `†(x)− `(x)DµDµ`†(x)⟩
= 2i
∫d3x
√|g | �x [D>(x , y)− D>(y , x)]|x=y
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Kinetic theory for leptogenesis
B-L current
jµ(x) = 2i⟨
[Dµ`(x)] `†(x)− `(x)Dµ`†(x)⟩
= (nB−L,~jB−L)
dNB−L
dt=
∫d3x
√|g | Dµjµ
= 2i
∫d3x
√|g |⟨
[DµDµ`(x)] `†(x)− `(x)DµDµ`†(x)⟩
= 2i
∫d3x
√|g | �x [D>(x , y)− D>(y , x)]|x=y
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Kinetic theory for leptogenesis
dNB−L
dt= −
∫d3x
√|g |∫ t
tinit
d4z√|g |[Σ<(x , z)D>(z, x)− Σ>(x , z)D<(z, x)
+Σc<(x , z)Dc
>(z, x)− Σc>(x , z)Dc
<(z, x)]
x0=t
Σ≷(x , y) = + + . . .
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Kinetic theory for leptogenesis
N ↔ ``N ↔ `†`† |tree|2 tree × vertex-corr. tree × wave-corr.
``↔ `†`† s × t, s × u, t × u s × s, t × t, u × u
unified description of CP-violating decay, inverse decay, scattering
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Kinetic theory for leptogenesis
How to solve the equations?
Solve two-time equations numerically
Marcovian limit (zero-width limit, one-time)
Buchmuller, Fredenhagen 00
MG, Kartavtsev, Hohenegger, Lindner 09,10
Beneke, Garbrecht, Herranen, Schwaller 10
Thermal bath limit (finite-width, two-time, neglect back-reaction)
Anisimov, Buchmuller, Drewes, Mendizabal 08,10
MG, Kartavtsev, Hohenegger, 11
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Kinetic theory for leptogenesis
How to solve the equations?
Solve two-time equations numerically
Marcovian limit (zero-width limit, one-time)
Buchmuller, Fredenhagen 00
MG, Kartavtsev, Hohenegger, Lindner 09,10
Beneke, Garbrecht, Herranen, Schwaller 10
Thermal bath limit (finite-width, two-time, neglect back-reaction)
Anisimov, Buchmuller, Drewes, Mendizabal 08,10
MG, Kartavtsev, Hohenegger, 11
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Marcovian limit
Separation of fast/short and slow/large scales
∆xinteraction, λde−Broglie � λmfp, lhorizon
1/M, 1/T � 1/Γ, 1/y 2T , 1/H
Wigner transformation k ↔ s = x − y , X = (x + y)/2
D(X , k) =
∫d4s e iksD(X + s/2,X − s/2)
x
y
X
s
Gradient expansion ∂X∂k ∼ slowfast
∼ Γ,H,y2TM,T∫
d4z Σ(x , z)D(z , y)→ Σ(X , k)D(X , k) +i
2
(∂Σ∂X
∂D∂k− ∂Σ
∂k∂D∂X
)
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Marcovian limit
Separation of fast/short and slow/large scales
∆xinteraction, λde−Broglie � λmfp, lhorizon
1/M, 1/T � 1/Γ, 1/y 2T , 1/H
Wigner transformation k ↔ s = x − y , X = (x + y)/2
D(X , k) =
∫d4s e iksD(X + s/2,X − s/2)
x
y
X
s
Gradient expansion ∂X∂k ∼ slowfast
∼ Γ,H,y2TM,T∫
d4z Σ(x , z)D(z , y)→ Σ(X , k)D(X , k) +i
2
(∂Σ∂X
∂D∂k− ∂Σ
∂k∂D∂X
)
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Marcovian limit
Separation of fast/short and slow/large scales
∆xinteraction, λde−Broglie � λmfp, lhorizon
1/M, 1/T � 1/Γ, 1/y 2T , 1/H
Wigner transformation k ↔ s = x − y , X = (x + y)/2
D(X , k) =
∫d4s e iksD(X + s/2,X − s/2)
x
y
X
s
Gradient expansion ∂X∂k ∼ slowfast
∼ Γ,H,y2TM,T∫
d4z Σ(x , z)D(z , y)→ Σ(X , k)D(X , k) +i
2
(∂Σ∂X
∂D∂k− ∂Σ
∂k∂D∂X
)Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Marcovian limit
Marcovian evolution equation for NB−L
dNB−L
dt= −
∫d3X
√|g |∫
d4k
(2π)4
[Σ<(X , k)D>(X , k)− Σ>(X , k)D<(X , k)
+Σc<(X , k)Dc
>(X , k)− Σc>(X , k)Dc
<(X , k)]
In equilibrium: Kubo-Martin-Schwinger relations (β = 1/T )
Deq> = eβk0
Deq< Σeq
> = eβk0
Σeq<
⇒ vanishes automatically in equilibrium⇒ consistent equations free of double-counting problems
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Marcovian limit
Marcovian evolution equation for NB−L
dNB−L
dt= −
∫d3X
√|g |∫
d4k
(2π)4
[Σ<(X , k)D>(X , k)− Σ>(X , k)D<(X , k)
+Σc<(X , k)Dc
>(X , k)− Σc>(X , k)Dc
<(X , k)]
In equilibrium: Kubo-Martin-Schwinger relations (β = 1/T )
Deq> = eβk0
Deq< Σeq
> = eβk0
Σeq<
⇒ vanishes automatically in equilibrium⇒ consistent equations free of double-counting problems
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Quantum corrections
dNB−L
dt∝ Dµjµ ≈ 2 |y1|2
∫dΠpdΠk dΠq Θ(p0)(2π)4δ(k − p − q)
×(
DN1> (X , k)D`
<(X , p)Dh<(X , q)− DN1
< (X , k)D`>(X , p)Dh
>(X , q))
× ε1(X , k, p, q)
εvertexi (X , k, p, q) =
∑j
|yj |2Im
(yi y∗j
y∗i yj
)∫dΠk1
dΠk2dΠk3
× (2π)4δ(p + k1 + k2)(2π)4δ(k2 − k3 + q)
× [D`ρ(X , k1)Dh
F (X , k2)DNj
h (X , k3) + {k1 ↔ k2}
+ D`h (X , k1)Dh
F (X , k2)DNjρ (X , k3) + {k1 ↔ k2}
+ D`ρ(X , k1)Dh
h (X , k2)DNj
F (X , k3)− {k1 ↔ k2}]
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Quantum corrections
dNB−L
dt∝ Dµjµ ≈ 2 |y1|2
∫dΠpdΠk dΠq Θ(p0)(2π)4δ(k − p − q)
×(
DN1> (X , k)D`
<(X , p)Dh<(X , q)− DN1
< (X , k)D`>(X , p)Dh
>(X , q))
× ε1(X , k, p, q)
εvertexi (X , k, p, q) =
∑j
|yj |2Im
(yi y∗j
y∗i yj
)∫dΠk1
dΠk2dΠk3
× (2π)4δ(p + k1 + k2)(2π)4δ(k2 − k3 + q)
× [D`ρ(X , k1)Dh
F (X , k2)DNj
h (X , k3) + {k1 ↔ k2}
+ D`h (X , k1)Dh
F (X , k2)DNjρ (X , k3) + {k1 ↔ k2}
+ D`ρ(X , k1)Dh
h (X , k2)DNj
F (X , k3)− {k1 ↔ k2}]Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Quantum corrections to the CP-violation parameter ε
SM+3νR MG, Kartavtsev, Hohenegger, Lindner 2010
ε(k,T )
εvac
M1/T
〈ǫ1〉 /ǫvac1
UR NR
ǫth1 /ǫvac1 , N1→φℓ˙
ǫth1¸
/ǫvac1 , N1→φℓ˙
ǫth,conv1
¸
/ǫvac1 , N1→φℓ
1
10
0.1 1 10
Bose enhancement of the vertex/self-energy one-loop diagram
larger than previously considered thermal QFT result (black dashed line)
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Kinetic theory for leptogenesis
How to solve the equations?
Solve two-time equations numerically
Marcovian limit (zero-width limit, one-time)
Buchmuller, Fredenhagen 00
MG, Kartavtsev, Hohenegger, Lindner 09,10
Beneke, Garbrecht, Herranen, Schwaller 10
Thermal bath limit (finite-width, two-time, neglect back-reaction)
Anisimov, Buchmuller, Drewes, Mendizabal 08,10
MG, Kartavtsev, Hohenegger, 11
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Motivation: resonant enhancement
Efficiency of leptogenesis depends on CP-violating parameter, which is one-loopsuppressed
εNi =Γ(Ni → `φ†)− Γ(Ni → `cφ)
Γ(Ni → `φ†) + Γ(Ni → `cφ)∝ Im
+
Self-energy (or ‘wave’) contribution to CP-violating parameter features aresonant enhancement for a quasi-degenerate spectrum M1 ' M2 � M3
εwaveNi
=Im[(h†h)2
12]
8π(h†h)ii× M1M2
M22 −M2
1
Flanz Paschos Sarkar 94/96; Covi Roulet Vissani 96;
On-shell initial N1: p2 = M21 Internal N2: i
p2−M22
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Motivation: resonant enhancement
Efficiency of leptogenesis depends on CP-violating parameter, which is one-loopsuppressed
εNi =Γ(Ni → `φ†)− Γ(Ni → `cφ)
Γ(Ni → `φ†) + Γ(Ni → `cφ)∝ Im
+
Self-energy (or ‘wave’) contribution to CP-violating parameter features aresonant enhancement for a quasi-degenerate spectrum M1 ' M2 � M3
εwaveNi
=Im[(h†h)2
12]
8π(h†h)ii× M1M2
M22 −M2
1
Flanz Paschos Sarkar 94/96; Covi Roulet Vissani 96;
On-shell initial N1: p2 = M21 Internal N2: i
p2−M22
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Motivation: resonant enhancement
The enhancement is limited by the finite width of N1 and N2
Off-shell initial N1: p2 = M21 + iM1Γ1 Internal N2: i
p2−M22−iM2Γ2
In the maximal resonant case the spectral functions overlap
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2
ρ(k 0
)/ρ m
ax
k0/m
Γ Γ
∆m
⇒ Need to go beyond the quasi-particle approximation which underlies theconventional semi-classical Boltzmann approach.
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Motivation: resonant enhancement
The enhancement is limited by the finite width of N1 and N2
Off-shell initial N1: p2 = M21 + iM1Γ1 Internal N2: i
p2−M22−iM2Γ2
In the maximal resonant case the spectral functions overlap
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2
ρ(k 0
)/ρ m
ax
k0/m
Γ Γ
∆m
⇒ Need to go beyond the quasi-particle approximation which underlies theconventional semi-classical Boltzmann approach.
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Thermal bath limit
Self-energies for leptons and for Majorana neutrinos
�N
φ�ℓ
φ�ℓ
φ︸ ︷︷ ︸Σ`
αβ(x , y) =∂iΓ2
∂S`βα(y , x)
︸ ︷︷ ︸ΣN
ij (x , y) =∂iΓ2
∂S ji (y , x)
Idea: treat the medium of Standard Model particles as a thermal bath to whichthe right-handed neutrinos are weakly coupled
neglecting back-reaction ⇒ linearized KBEs ⇒ analytical solution
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Thermal bath limit
Analytical solution in the hierarchical case M1 � M2
Anisimov, Buchmuller, Drewes, Mendizabal 08,10 Phys.Rev.Lett. 104 (2010) 121102
SN,A(∆t) =
(iγ0 cos(ω∆t) +
M − ~p~γω
sin(ω∆t)
)e−Γ|∆t|/2
SN,F (t,∆t) = −(iγ0 cos(ω∆t)− M − ~p~γ
ωsin(ω∆t)
)×[ tanh(βω
2)
2e−Γ|∆t|/2︸ ︷︷ ︸
EquilibriumDamped w.r.t ∆t
− δfN (ω)e−Γt︸ ︷︷ ︸Non-equilibrium
Undamped w.r.t ∆t
]
∆t = x0 − y 0, t = (x0 + y 0)/2
SA = i(S> − S<)/2, SF = (S> + S<)/2
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Thermal bath limit
Cross-check with a full numerical solution of the two-time KBEs (for φ4-theory)Garbrecht, MG 2011
0.01
0.1
-30 -20 -10 0 10 20 30
relative time ∆t [1/mth]
|k| = 2.96mth
t = 17.5/mth
∆Fϕ (t,∆t,k)
∆Aϕ (t,∆t,k)
Statistical propagator ∆Fϕ = (∆>
ϕ + ∆<ϕ )/2 (red) and spectral function
∆Aϕ = i2(∆>
ϕ −∆<ϕ ) (blue) obtained from a numerical solution of the KBEs .
Dependence on the relative time ∆t for fixed central time t.
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Thermal bath limit
Garbrecht, MG 2011
0.01
0.1
-30 -20 -10 0 10 20 30
relative time ∆t [1/mth]
|k| = 3.0mth
t = 17.5/mth
∆Fϕ (t,∆t,k)
∆Aϕ (t,∆t,k)
Excited momentum mode |k| = 3.0mth. The statistical propagator
∆Fϕ = (∆>
ϕ + ∆<ϕ )/2 can be described by the sum of an exponentially damped
equilibrium contribution and an undamped non-equilibrium contribution.
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Thermal bath limit
Evolution of damped and un-damped components with the central time tGarbrecht, MG 2011
0.001
0.01
0.1
1
10
0 5 10 15 20 25 30 35 40 45
a(t)[1/m
th]
central time t [1/mth]
aundamped
adamped
δf(k) e−Γkt/ωk
(1 + 2fBE)/(2ωk)
∆Fϕ,fit ≡ (adamped (t)e−Γk |∆t|/2 + aundamped (t)) cos(ωk ∆t)
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Thermal bath limit
Where does it come from?
∆<,>ϕ (x0, y 0) = −
∞∫t0
dw 0
∞∫t0
dz0 ∆Rϕ(x0,w 0)Π<,>ϕ (w 0, z0)∆A
ϕ(z0, y 0)
+∆Rϕ(x0, t0)[∂x0∂y0 ∆<,>
ϕ (t0, t0)]∆Aϕ(t0, y
0)
+∆Rϕ(x0, t0)[∂y0 ∆<,>
ϕ (t0, t0)]∆Aϕ(t0, y
0)
+ ∆Rϕ(x0, t0)[∂x0 ∆<,>
ϕ (t0, t0)]∆Aϕ(t0, y
0)
+∆Rϕ(x0, t0)[∆<,>
ϕ (t0, t0)]∆Aϕ(t0, y
0)
For a thermal bath and in Breit-Wigner approximation
∆R,A(x0, y 0) ' ±Θ(±(x0 − y 0))e−Γϕ|x0−y0|/2 sin(ωϕ(x0 − y 0))
ωϕ
Then, one finds (finite t0):
i∆<,>ϕ (x0, y 0) = i∆<,>
ϕ,th (x0 − y 0) +1
ωϕcos[ωϕ(x0 − y 0)]e−Γϕ(x0+y0)/2δf (k)
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Thermal bath limit
Formulated in Wigner space:
i∆<,>ϕ (k, t) =i∆st<,>
ϕ (k) + δfϕ(k)2πδ(k2 −m2ϕ − ΠH
ϕ)e−Γϕ(k)t
Equilibrium contribution
i∆st<ϕ (k) = f eq(k)i∆Aϕ (k)
i∆st>ϕ (k) = (1 + f eq(k))i∆Aϕ (k)
with finite width
i∆Aϕ (k) =2ΠAϕ
(k2 −m2ϕ − ΠH
ϕ)2 + ΠAϕ2
Non-equilibrium contribution with zero-width?
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Thermal bath limit
KB Equation in Wigner space[k2 − 1
4∂2
t + ik0∂t −m2ϕ
]∆<,>ϕ − e−i�{ΠH
ϕ}{∆<,>ϕ } − e−i�{Π<,>ϕ }{∆H
ϕ}(1)
=1
2e−i� ({Π>ϕ }{∆<
ϕ } − {Π<ϕ }{∆>ϕ })
�{A}{B} =1
2(∂xA) (∂kB)− 1
2(∂kA) (∂xB) , (2)
Problem: k-derivatives hitting the on-shell delta function in ∆<,>ϕ are
unsuppressed ⇒ gradient expansion breaks down
Idea: resummation of gradients
e−i�{Π>ϕ }{∆<ϕ } = Π>ϕ∆<
ϕ,eff
Result
i∆<ϕ,eff (k) = (f eq(k) + δf (k))i∆Aϕ (k)
i∆>ϕ,eff (k) = (1 + f eq(k) + δf (k))i∆Aϕ (k)
⇒ recover KB Ansatz effectively
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Comparison KB – Boltzmann in resonant leptogenesis
0
0.2
0.4
0.6
0.8
1
0.1 1 10
t [1/Γ1]
nL(t,p)/nBoltzmannL (t = ∞,p)
Boltzmann
KB M2 = 1.5M1
KB M2 = 1.1M1
KB M2 = 1.025M1
KB M2 = 1.005M1
MG, Hohenegger, Kartavtsev; work in progress
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Comparison KB – Boltzmann in resonant leptogenesis
1
10
100
0.001 0.01 0.1 1
R
(M22 −M2
1 )/M21
RKBmax
RBoltzmannmaxRBoltzmannmax
M1Γ1+
M2Γ2
|M1Γ1−
M2Γ2|
RBoltzmann
RKB(t = ∞)
RKB(t = 1/Γ1)
RKB(t = 0.25/Γ1)
Γ2/Γ1 = 1.5
RBoltzmannmax = M1M2/(2|Γ1M1 − Γ2M2|), RKB
max = M1M2/(2(Γ1M1 + Γ2M2))
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Conclusions
↔ ↔
Neutrino oscillationsmν ∼ meV− eVseesaw-mechanism
SM + 3νR
Matter-antimatter asym.(nb − nb)/nγ ∼ 10−10
leptogenesis
Baryogenesis via leptogenesis is a non-equilibrium, quantum process
Closed-time-path approach can resolve double-counting issues of standardBoltzmann approach
Resonant leptogenesis: finite width + non-equilibrium⇒ Kadanoff-Baym in thermal bath limit
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Conclusions
↔ ↔
Neutrino oscillationsmν ∼ meV− eVseesaw-mechanism
SM + 3νR
Matter-antimatter asym.(nb − nb)/nγ ∼ 10−10
leptogenesis
Baryogenesis via leptogenesis is a non-equilibrium, quantum process
Closed-time-path approach can resolve double-counting issues of standardBoltzmann approach
Resonant leptogenesis: finite width + non-equilibrium⇒ Kadanoff-Baym in thermal bath limit
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Thermal initial correlations: 2PI
Closed time path C with αn
Feynman rules
−iλ∫C α3 α4 α5 α6 . . .
Example
Thermal time path C + I
Feynman rules
−iλ∫C −iλ
∫I
Example
Challenge
Encapsulate integrations over I into initial correlations αthn
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Thermal initial correlations: 2PI
Closed time path C with αn
Feynman rules
−iλ∫C α3 α4 α5 α6 . . .
Example
Thermal time path C + I
Feynman rules
−iλ∫C −iλ
∫I
Example
Challenge
Encapsulate integrations over I into initial correlations αthn
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Thermal initial correlations: 2PI
Closed time path C with αn
Feynman rules
−iλ∫C α3 α4 α5 α6 . . .
Example
Thermal time path C + I
Feynman rules
−iλ∫C −iλ
∫I
Example
Challenge
Encapsulate integrations over I into initial correlations αthn
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Thermal initial correlations
Thermal time path C + ISelf-consistent Schwinger-Dyson equation
G−1th (x , y) = G−1
0,th(x , y)− Πth(x , y) ⇔
(�x + m2)Gth(x , y) = −iδC+I(x − y)− i
∫C+I
d4zΠth(x , z)Gth(z, y)︸ ︷︷ ︸
Closed time path C with initial correlations αKadanoff-Baym equation for a Non-Gaussian initial state
(�x + m2)G(x , y) = −iδC(x − y)
− i
∫C
d4z [ΠGauss (x , z) + Πnon−Gauss (x , z)] G(z, y)
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Thermal initial correlations: 2PI
Thermal 2PI propagator connecting real and imaginary times
Gth(−iτ , t) = =
“Connection” MG, Muller (2009)
=
︸ ︷︷ ︸∝ δC(t − 0±)
+
Important: Same 2PI truncation for Matsubara and real-time propagators
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Thermal initial correlations: 2PIExample: 2PI three-loop truncation (〈Φ〉 = 0)
= +
Iterative Solution:
= +
+ + +
+ . . .
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Thermal initial correlations: 2PIExample: 2PI three-loop truncation (〈Φ〉 = 0)
= +
Iterative Solution:
= +
+ + +
+ . . .
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Thermal initial correlations: 2PIExample: 2PI three-loop truncation (〈Φ〉 = 0)
= +
Iterative Solution:
= +
+
+ +
+ . . .
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Thermal initial correlations: 2PIExample: 2PI three-loop truncation (〈Φ〉 = 0)
= +
Iterative Solution:
= +
+ + +
+ . . .
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Thermal initial correlations: 2PIExample: 2PI three-loop truncation (〈Φ〉 = 0)
= +
= ︸ ︷︷ ︸ΠGauss
+
︸ ︷︷ ︸ΠNon−Gauss
Non-Gaussian self-energy contains αthn (x1, . . . , xn) for all n ≥ 4
Πnon−Gauss (x , z) = + +
+ + + . . .
MG, Muller (09)
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Thermal initial correlations: 2PIExample: 2PI three-loop truncation (〈Φ〉 = 0)
= +
= ︸ ︷︷ ︸ΠGauss
+
︸ ︷︷ ︸ΠNon−Gauss
Non-Gaussian self-energy contains αthn (x1, . . . , xn) for all n ≥ 4
Πnon−Gauss (x , z) = + +
+ + + . . .
MG, Muller (09)
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Thermal initial correlations: 2PI
Thermal correlation energy (computed at initial time):
E eqcorr (t = 0) =
i
4
∫C
d4z [ΠGauss (x , z) + Πnon−Gauss (x , z)] G(z, x)
∣∣∣∣x=0
=
∣∣∣∣x=0
+
∣∣∣∣x=0
+
∣∣∣∣x=0︸ ︷︷ ︸
x0∫0
dz0 → 0
︸ ︷︷ ︸x0∫0
dz0 → 0
=
∣∣∣∣x=0
= E eq4−p. corr (t = 0)
...only the thermal 4-point correlation contributes
⇒ truncate initial correlations with n > 4
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Leading non-Gaussian correction for λΦ4 2PI 3-loop
Gaussian IC
G(x , y)|x0,y0=0 = Gth(x , y)|x0,y0=0
α4(x1, . . . , x4) = 0
αn(x1, . . . , xn) = 0 for n > 4
Non-Gaussian IC with αth4
G(x , y)|x0,y0=0 = Gth(x , y)|x0,y0=0
α4(x1, . . . , x4) = αth4 (x1, . . . , x4)
αn(x1, . . . , xn) = 0 for n > 4
Truncate thermal initial correlations
⇒ nonequilibrium initial states
The upper states are ‘as thermal as possible’
Expectation: Non-Gaussian state closer to equilibrium
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Leading non-Gaussian correction for λΦ4 2PI 3-loop
0.3
0.4
0.5
0.6
0.7
0.8
0.01 0.1 1 10 100
GF(t
,t,k)
t mR
k = 0
k = mR
k = 2mR
500 1000 1500 2000t mR
KB, Gauss (A)
KB, Non-Gauss (B)
ThQFT
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Leading non-Gaussian correction for λΦ4 2PI 3-loop
0.95
1
1.05
0 1 2 3 4 5 6 7GF(t
,t,k)
/GF(0
,0,k
)
k/mR
t mR = 2000
0.95
1
1.05
GF(t
,t,k)
/GF(0
,0,k
)
t mR = 10
0.95
1
1.05
GF(t
,t,k)
/GF(0
,0,k
)
t mR = 0.5
0.95
1
1.05
GF(t
,t,k)
/GF(0
,0,k
)
t mR = 0.0 KB, Gauss
0.95
1
1.05
GF(t
,t,k)
/Gth
(k)
t mR = 2.0
0 1 2 3 4 5 6 7
0.95
1
1.05
k/mR
t mR = 2000
0.95
1
1.05t mR = 10
0.95
1
1.05t mR = 0.5
0.95
1
1.05t mR = 0.0 KB, Non-Gauss
0.95
1
1.05t mR = 2.0
all modes remain close to equilibrium
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Leading non-Gaussian correction for λΦ4 2PI 3-loop
-0.3-0.2-0.1
0 0.1 0.2 0.3 0.4
0.01 0.1 1 10 100
µ/m
R
t mR
2
2.1
2.2
2.3T
/mR
500 1000 1500 2000t mR
KB, Gauss (A)
KB, Non-Gauss (B)
Thermal Eq.
︸ ︷︷ ︸build-up
︸ ︷︷ ︸kinetic -
︸ ︷︷ ︸chemical equilibration
correlated system ∆T ' 0
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Leading non-Gaussian correction for λΦ4 2PI 3-loop
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0 1 2 3 4 5
Eco
rr(t
,k)
t [m-1R]
kmax=7mR
Correlation energy (Gaussian part)
Contribution from initial 4-point correlation
Sum
⇒ particles in initial state are ‘well-dressed’
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Leading non-Gaussian correction for λΦ4 2PI 3-loop
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0 1 2 3 4 5
Eco
rr(t
,k)
t [m-1R]
kmax=7mR
kmax=10mR
kmax=7mR
kmax=5mR
Correlation energy (Gaussian part)
Contribution from initial 4-point correlation
Sum
⇒ particles in initial state are ‘well-dressed’
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis
Leading non-Gaussian correction for λΦ4 2PI 3-loop
0.9 1
1.1 1.2
0 1 2 3 4 5 6 7
GF(t
,t,k)
/Gth
(k)
k/mR
t mR = 2000
0.9 1
1.1 1.2
GF(t
,t,k)
/Gth
(k)
t mR = 10
0.9 1
1.1 1.2
GF(t
,t,k)
/Gth
(k)
t mR = 0.5
0.9 1
1.1 1.2
GF(t
,t,k)
/Gth
(k)
t mR = 0.0 KB, Gauss
0.9 1
1.1 1.2
GF(t
,t,k)
/Gth
(k)
t mR = 2.0
0 1 2 3 4 5 6 7
0.9 1 1.1 1.2
k/mR
t mR = 2000
0.9 1 1.1 1.2
t mR = 10
0.9 1 1.1 1.2
t mR = 0.5
0.9 1 1.1 1.2
t mR = 0.0 KB, Non-Gauss
0.9 1 1.1 1.2
t mR = 2.0
UV modes are not excited
⇒ suppresses unwanted back-reaction on IR modes
Mathias Garny (DESY Hamburg) Kadanoff-Baym Equations and Baryogenesis