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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


Outline

Matter-Antimatter asymmetry

Baryogenesis and Leptogenesis

Relativistic quantum kinetic theory for leptogenesis

Results in Marcovian and Thermal-bath limits


Standard Model of Cosmology


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 + . . .


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


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?



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



(1) Initial baryon asymmetry after Big Bang

Problem:

Diluted by inflationWashed out by ∆B 6= 0 processes at high energy

(2) Dynamical creation: 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)


Baryogenesis within the Standard Model of particles ?

SU(3)c × SU(2)L × U(1)Y


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


Baryogenesis

Baryogenesis in models beyond the Standard Model of particles

Electroweak baryogenesis

GUT baryogenesis

Affleck-Dine baryogenesis

Leptogenesis

. . .


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)


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


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


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


Leptogenesis Fukugita, Yanagida

MνR,i→`L,αh† =yiα + . . .

MνR,i→`cL,αh =

y∗iα + . . .


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


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 ?


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


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


Leptogenesis

final asymmetry

η =NB−L

Nγ= −3

4ε1κf

K

κf

[N1-dominated, D+ID, unflavoured] Buchmuller, Di Bari, Plumacher


Leptogenesis

m1 (eV)

M1(G

eV)

[N1-dominated, D+ID, unflavoured] Buchmuller, Di Bari, Plumacher

lower bound on M1 Davidson, Ibarra


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


Leptogenesis

Baryogenesis via Leptogenesis

CP violation in decay described by loop process

deviation from thermal equilibrium

Quantum nonequilibrium effects ?


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


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, . . .


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


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


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)



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)

]



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)



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



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) = + + . . .



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



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


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

)


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


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}]


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)



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


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


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.


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


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


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.


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.


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)


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)


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?


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


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


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))


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


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


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)



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


Thermal initial correlations: 2PIExample: 2PI three-loop truncation (〈Φ〉 = 0)

= +

Iterative Solution:

= +

+ + +

+ . . .



= +

Iterative Solution:

= +

+

+ +

+ . . .



= +

Iterative Solution:

= +

+ + +

+ . . .



= +

= ︸︷︷︸ΠGauss

+

︸︷︷︸ΠNon−Gauss

Non-Gaussian self-energy contains αthn (x1, . . . , xn) for all n ≥ 4

Πnon−Gauss (x , z) = + +

+ + + . . .

MG, Muller (09)



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


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



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



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



-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



-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’



-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’



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


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