Kadanoff-Baym Equations and Baryogenesis

Transcription

1 KBE, Kiel,

2 Nonequilibrium dynamics at high energy Heavy Ion Collisions LHC: ALICE RHIC Early universe Reheating after Inflation Dark matter freeze-out Baryogenesis...

3 Outline Matter-Antimatter asymmetry Baryogenesis and Leptogenesis Relativistic quantum kinetic theory for leptogenesis Results in Marcovian and Thermal-bath limits

4 Standard Model of Cosmology

5 No antimatter Ratio of cosmic ray fluxes observed at Earth Φ(anti-proton) Φ(proton) p/p BESS 2000 (Y. Asaoka et al.) 5 10 BESS 1999 (Y. Asaoka et al.) BESS polar 2004 (K. Abe et al.) CAPRICE 1994 (M. Boezio et al.) CAPRICE 1998 (M. Boezio et al.) HEAT pbar 2000 (A. S. Beach et al.) PAMELA kinetic energy [GeV] PAMELA Phys.Rev.Lett. 105 (2010) Antinuclei (e.g. anti-helium) < 10 6 A. Alcaraz et al., Phys. Lett. B 461, 387 (1999) Consistent with secondary production p + N p + p +...

6 No antimatter Galaxy: Antistars would accrete interstellar gas, leading to annihilation gamma-rays. Observations of discrete Galactic gamma-ray sources limit the fraction of antistars in the Galaxy to < 10 4 Galaxy clusters ( lyr): annihilations would produce high-energy gamma ray flux N + N π 0, π ± γ E γ 100MeV Non-observation yields a limit anti-matter matter G. Steigman JCAP 0810 (2008) 001

7 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?

8 Matter-Antimatter asymmetry Asymmetry parameter η = n b n b n γ n b = baryon density n b = anti-baryon density n γ = photon density Consistent value inferred from Big Bang Nucleosynthesis (T kev) and the cosmic microwave background (T ev) η = n b n b n γ = (6.21 ± 0.16) 10 10

9 Matter-Antimatter asymmetry

10 Matter-Antimatter asymmetry (1) Initial baryon asymmetry after Big Bang Problem: Diluted by inflation Washed out by B 0 processes at high energy

11 Matter-Antimatter asymmetry (1) Initial baryon asymmetry after Big Bang Problem: Diluted by inflation Washed out by B 0 processes at high energy (2) Dynamical creation: Baryogenesis

12 Baryogenesis Sakharov conditions Sakharov 1967 baryon number violation: B const. C,CP violation: γ(i f ) γ(ī f ) deviation from thermal equilibrium: γ(i f ) γ(f i)

13 Baryogenesis within the Standard Model of particles? SU(3) c SU(2) L U(1) Y

14 Baryogenesis within the Standard Model of particles? B-violation at quantum level t Hooft 76 µ j µ = g 2 32π 2 F µν µν F Exponentially suppressed for T < T EW 100GeV Γ proton e 16π2 /g 2 = Unsupressed for T > T EW (sphaleron) Shaposhnikov 85 B = L Klinkhamer, Manton 84; Kuzmin, Rubakov, CP-violation in quark mixing K 0 / K 0 decay, B d,s decay Electroweak phase-transition: first-order for m H < 60 80GeV LEP limit m H > 114GeV Third Sakharov condition is not fulfilled

15 Baryogenesis Baryogenesis in models beyond the Standard Model of particles Electroweak baryogenesis GUT baryogenesis Affleck-Dine baryogenesis Leptogenesis...

16 Motivation: Observation of massive neutrinos Neutrino oscillation (ν e ν µ ν τ ) mν 2 1 mν 2 2 = ev 2 mν 2 1 mν 2 3 = ev 2 Direct search ( 3 H 3 He + e + ν e ) m νe < 2.2eV 95%C.L. At least two massive neutrinos Tiny mass Superkamiokande Standard Model: neutrinos massless KATRIN (0.2eV)

17 SM + Right-handed neutrinos (ν R ) e,µ,τ Neutrino masses Majorana mass term L = L SM y ν R h l L M R ν R νr c See-saw mechanism m νl y 2 h 2 M R mev - ev

18 SM + Right-handed neutrinos (ν R ) e,µ,τ Neutrino masses Majorana mass term L = L SM y ν R h l L M R ν R ν c R See-saw mechanism m νl y 2 h 2 M R mev - ev Matter-Antimatter asymmetry B- via L-violation M R ν R ν c R 0νββ: (A, Z) (A, Z + 2) + 2e (Gerda, Nemo, Exo,... ) CP-violation in ν-mixing ν-oscillation (Double Chooz, Daya Bay,... ) Expanding universe H = ȧ/a

19 Leptogenesis Fukugita, Yanagida M νr,i l L,α h = y iα +... M νr,i l c L,α h = y iα +...

20 Leptogenesis Fukugita, Yanagida M νr,i l L,α h = y iα + y iβ y jβ y jα +... y jβ M νr,i l c L,α h = y iα + y iβ +... y jα Matter-antimatter (CP) asymmetry interference of tree and loop processes Γ(ν R,i l L,α h ) Γ(ν R,i l c L,α h) Im(y iαy iβ yjα y jβ ) Im

21 Leptogenesis B via L violation ν R,i lh ν R,i l c h CP violation ɛ i = Γ(ν R,i lh ) Γ(ν R,i l c h) Γ(ν R,i lh )+Γ(ν R,i l c h) ( Im + ) Deviation from equilibrium?

22 Leptogenesis thermal plasma l, h, q, W, Z, γ, gauge interactions fast g 4 T H T 2 /M pl right-handed neutrinos ν R,i N i slow Γ Ni = (y y) ii M Ni 8π g 4 T

23 Leptogenesis thermal plasma l, h, q, W, Z, γ, gauge interactions fast g 4 T H T 2 /M pl right-handed neutrinos ν R,i N i slow Γ Ni = (y y) ii M Ni 8π g 4 T... produced when T M Ni... equilibrate if Γ Ni e M N i /T > H... decay when T < M Ni and t > τ Ni

24 Leptogenesis thermal plasma l, h, q, W, Z, γ, gauge interactions fast g 4 T H T 2 /M pl right-handed neutrinos ν R,i N i slow Γ Ni = (y y) ii M Ni 8π g 4 T... produced when T M Ni... equilibrate if Γ Ni e M N /T i > H... decay when T < M Ni and t > τ Ni deviation from thermal equilibrium K (Γ Ni /H) T =MNi

25 Leptogenesis z eq eq N N K= eq N N K= N 5 N1 N N N 9 B L N B L z eq z=m 1 /T z=m 1 /T Effective rate equations Buchmüller, Di Bari, Plümacher,... dn Ni dt dn B L dt = (D i + S i )(N Ni N eq N i ) = i ɛ i D i (N Ni N eq N i ) } {{ } source term WN B L }{{} washout term

26 Leptogenesis final asymmetry η = N B L N γ = 3 4 ɛ 1κ f κ f K [N 1 -dominated, D+ID, unflavoured] Buchmüller, Di Bari, Plümacher

27 Leptogenesis M1 (GeV) m1 (ev) [N 1 -dominated, D+ID, unflavoured] Buchmüller, Di Bari, Plümacher lower bound on M 1 Davidson, Ibarra

28 Leptogenesis MEG M 1 (GeV) PRISM/PRIME m ~ 1 (ev) Probing supersymmetric leptogenesis with µ eγ Ibarra, Simonetto 09

29 Leptogenesis Baryogenesis via Leptogenesis CP violation in decay described by loop process deviation from thermal equilibrium Quantum nonequilibrium effects?

30 Kinetic theory: standard Boltzmann approach N L (t) = d 3 x g d 3 p (2π) 3 [f l (t, x, p) f l(t, x, p)] l

31 Kinetic theory: standard Boltzmann approach N L (t) = d 3 x g d 3 p (2π) 3 [f l (t, x, p) f l(t, x, p)] l p µ D µf l (t, x, p) = i dπ Ni dπ h (2π) 4 δ(p l + p h p Ni ) [ M 2 N i lh f N i (1 f l )(1 + f h ) +... ] M 2 lh N i f l f h (1 f Ni ) +... f ψ (t, x, p) : distribution function of on-shell particles M 2 : matrix elements computed in vacuum, off-shell effects

32 Corrections within Boltzmann framework Bose-enhancement, Pauli-Blocking; kinetic (non-)equilibrium quantum statistical factors 1 ± f k non-integrated Boltzmann equations Hannsestad, Basbøll 06; Garayoa, Pastor, Pinto, Rius, Vives 09; Hahn-Woernle, Plümacher, Wong 09 Thermal corrections via thermal QFT medium correction to CP-violating parameter ɛ = ɛ vac + δɛ th thermal masses, decay width Covi, Rius, Roulet, Vissani 98; Giudice, Notari, Raidal, Riotto, Stumia 04; Besak, Bödeker 10 Kiessig, Thoma, Plümacher 10;... Flavour effects Nardi, Nir, Roulet, Racker 06; Adaba, Davidson, Ibarra, Josse-Micheaux, Losada, Riotto 06; Blanchet, dibari 06;... Spectator processes, scatterings, N 2,...

33 Double Counting Problem Naive contribution from decay/inverse decay M 2 N i lh = M 0 2 (1 + ɛ i ) M 2 lh N i = M 0 2 (1 ɛ i ) M 2 N i l c h = M 0 2 (1 ɛ i ) M 2 l c h N i = M 0 2 (1 + ɛ i ) dn B L dt ( M 2 N i lh M 2 N i l c h)n Ni ( M 2 lh N i M 2 l c h N i )N eq N i

34 Double Counting Problem Naive contribution from decay/inverse decay M 2 N i lh = M 0 2 (1 + ɛ i ) M 2 lh N i = M 0 2 (1 ɛ i ) M 2 N i l c h = M 0 2 (1 ɛ i ) M 2 l c h N i = M 0 2 (1 + ɛ i ) dn B L dt ( M 2 N i lh M 2 N i l c h)n Ni ( M 2 lh N i M 2 l c h N i )N eq N i ɛ i (N Ni +N eq N i ) spurious generation of asymmetry even in equilibrium Origin: Double Counting Problem + need real intermediate state subtraction

35 Kinetic theory for leptogenesis Goal derivation of kinetic equations starting from first principles on-/off-shell treated in a unified way (avoid double-counting) thermal medium corrections,..., resonant leptogenesis, coherent flavor transitions

36 CTP/Kadanoff-Baym approach and leptogenesis Resolve double counting problems occuring in the Boltzmann approach associated to real intermediate states Buchmüller, Fredenhagen 00, De Simone, Riotto 05, MG, Kartavtsev, Hohenegger, Lindner 09,10, Beneke, Garbrecht, Herranen, Schwaller 10 Medium corrections to decay/inverse decay and scatterings MG, Kartavtsev, Hohenegger Lindner 09,10; Beneke, Garbrecht, Herranen, Schwaller 10; Garbrecht 10 see also Bödecker, Besak, Anisimov 10; Kiessig, Thoma, Plümacher 10; Salvio Lodone Strumia 11 Finite width of lepton, Higgs, and non-equilibrium Majorana neutrino evolution Anisimov, Buchmüller, Drewes, Mendizabal 08,10 Flavored leptogenesis: transition between flavored/unflavored regimes Beneke, 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 particles non-equilibrium propagators for unstable/off-shell particles inclusion of coherent N 1 N 2 flavor transitions (oscillations)

37 Kinetic theory for leptogenesis First Step: Bosonic toy model L = 1 2 µn i µ N i + µl µ l 1 2 M i N i N i y i N i ll y i N i l l λ 4 [l l] M Ni ll = M Ni l l = CP-violating parameter (vacuum, non-degenerate) x j = M 2 j /M 2 i ɛ vac i = Γ(N i ll) Γ(N i l l ) Γ(N i ll) + Γ(N i l l ) = j ( y j 2 yi y ) [ j 8πMj 2 Im yi x j ln y j ( 1 + xj x j ) ] 1 + 2(x j 1)

38 Kinetic theory for leptogenesis Schwinger-Keldysh propagator: x, y C D(x, y) = T C l(x)l (y) Schwinger-Dyson equation i( x + m 2 )D(x, y) = δ C (x y) + d 4 zσ(x, z)d(z, y) Integration over closed time path C d 4 z = d 4 z + d 4 z C t init z 0 max(x 0, y 0 ) Kadanoff-Baym equations x 0 ( x + m 2 )D (x, y) = i d 4 z (Σ > Σ <)(x, z)d (z, y) t init + i y 0 t init d 4 z Σ (x, z)(d > D <)(z, y)

39 Kinetic theory for leptogenesis B-L current j µ(x) = 2i [D µl(x)] l (x) l(x) D µl (x) = (n B L, j B L )

40 Kinetic theory for leptogenesis B-L current j µ(x) = 2i [D µl(x)] l (x) l(x) D µl (x) = (n B L, j B L ) dn B L dt = d 3 x g D µ j µ

41 Kinetic theory for leptogenesis B-L current j µ(x) = 2i [D µl(x)] l (x) l(x) D µl (x) = (n B L, j B L ) dn B L dt = d 3 x g D µ j µ = 2i d 3 x g [D µ D µl(x)] l (x) l(x) D µ D µl (x) = 2i d 3 x g x [D >(x, y) D >(y, x)] x=y

42 Kinetic theory for leptogenesis dn B L dt = d 3 x t g d 4 z [ g Σ <(x, z)d >(z, x) Σ >(x, z)d <(z, x) t init +Σ c < (x, z)dc > (z, x) Σc > (x, z)dc < (z, x) ] x 0 =t Σ (x, y) =

43 Kinetic theory for leptogenesis N ll N l l tree 2 tree vertex-corr. tree wave-corr. ll l l s t, s u, t u s s, t t, u u unified description of CP-violating decay, inverse decay, scattering

44 Kinetic theory for leptogenesis How to solve the equations? Solve two-time equations numerically Marcovian limit (zero-width limit, one-time) Buchmüller, Fredenhagen 00 MG, Kartavtsev, Hohenegger, Lindner 09,10 Beneke, Garbrecht, Herranen, Schwaller 10 Thermal bath limit (finite-width, two-time, neglect back-reaction) Anisimov, Buchmüller, Drewes, Mendizabal 08,10 MG, Kartavtsev, Hohenegger, 11

45 Kinetic theory for leptogenesis How to solve the equations? Solve two-time equations numerically Marcovian limit (zero-width limit, one-time) Buchmüller, Fredenhagen 00 MG, Kartavtsev, Hohenegger, Lindner 09,10 Beneke, Garbrecht, Herranen, Schwaller 10 Thermal bath limit (finite-width, two-time, neglect back-reaction) Anisimov, Buchmüller, Drewes, Mendizabal 08,10 MG, Kartavtsev, Hohenegger, 11

46 Marcovian limit Separation of fast/short and slow/large scales x interaction, λ de Broglie λ mfp, l horizon 1/M, 1/T 1/Γ, 1/y 2 T, 1/H

47 Marcovian limit Separation of fast/short and slow/large scales x interaction, λ de Broglie λ mfp, l horizon 1/M, 1/T 1/Γ, 1/y 2 T, 1/H Wigner transformation k s = x y, X = (x + y)/2 D(X, k) = d 4 s e iks D(X + s/2, X s/2) y X x s

48 Marcovian limit Separation of fast/short and slow/large scales x interaction, λ de Broglie λ mfp, l horizon 1/M, 1/T 1/Γ, 1/y 2 T, 1/H Wigner transformation k s = x y, X = (x + y)/2 D(X, k) = d 4 s e iks D(X + s/2, X s/2) y X x s Gradient expansion X k slow fast Γ,H,y 2 T M,T d 4 z Σ(x, z)d(z, y) Σ(X, k)d(x, k) + i 2 ( Σ D X Σ D k k X )

49 Marcovian limit Marcovian evolution equation for N B L dn B L dt = d 3 X g d 4 k [ (2π) 4 Σ <(X, k)d >(X, k) Σ >(X, k)d <(X, k) ] +Σ c <(X, k)d>(x c, k) Σ c >(X, k)d<(x c, k)

50 Marcovian limit Marcovian evolution equation for N B L dn B L dt = d 3 X g d 4 k [ (2π) 4 Σ <(X, k)d >(X, k) Σ >(X, k)d <(X, k) ] +Σ c <(X, k)d>(x c, k) Σ c >(X, k)d<(x c, k) In equilibrium: Kubo-Martin-Schwinger relations (β = 1/T ) D eq > = e βk0 D eq < Σ eq > = e βk0 Σ eq < vanishes automatically in equilibrium consistent equations free of double-counting problems

51 Quantum corrections dn B L dt D µ j µ 2 y 1 2 dπ pdπ k dπ q Θ(p 0 )(2π) 4 δ(k p q) ( ) D N 1 > (X, k)dl < (X, p)dh < (X, q) DN 1 < (X, k)dl > (X, p)dh > (X, q) ɛ 1 (X, k, p, q)

52 Quantum corrections dn B L dt D µ j µ 2 y 1 2 dπ pdπ k dπ q Θ(p 0 )(2π) 4 δ(k p q) ( ) D N 1 > (X, k)dl < (X, p)dh < (X, q) DN 1 < (X, k)dl > (X, p)dh > (X, q) ɛ 1 (X, k, p, q) ɛ vertex i (X, k, p, q) = j y j 2 Im ( yi yj ) yi dπ y dπ k1 k2 dπ k3 j (2π) 4 δ(p + k 1 + k 2 )(2π) 4 δ(k 2 k 3 + q) [D l ρ(x, k 1 )D h F (X, k 2)D N j h (X, k 3) + {k 1 k 2 } + D l h (X, k 1)D h F (X, k 2)D N j ρ (X, k 3 ) + {k 1 k 2 } + D l ρ (X, k 1)D h h (X, k 2)D N j F (X, k 3) {k 1 k 2 }]

53 Quantum corrections to the CP-violation parameter ɛ SM+3ν R MG, Kartavtsev, Hohenegger, Lindner ǫ1 /ǫ vac 1 ǫ th 1 /ǫ vac 1, N1 φl ǫth vac 1 /ǫ 1, N1 φl ǫth,conv vac 1 /ǫ 1, N1 φl ɛ(k, T ) ɛ vac UR M1/T NR Bose enhancement of the vertex/self-energy one-loop diagram larger than previously considered thermal QFT result (black dashed line)

54 Kinetic theory for leptogenesis How to solve the equations? Solve two-time equations numerically Marcovian limit (zero-width limit, one-time) Buchmüller, Fredenhagen 00 MG, Kartavtsev, Hohenegger, Lindner 09,10 Beneke, Garbrecht, Herranen, Schwaller 10 Thermal bath limit (finite-width, two-time, neglect back-reaction) Anisimov, Buchmüller, Drewes, Mendizabal 08,10 MG, Kartavtsev, Hohenegger, 11

55 Motivation: resonant enhancement Efficiency of leptogenesis depends on CP-violating parameter, which is one-loop suppressed ɛ Ni = Γ(N i lφ ) Γ(N i l c φ) Γ(N i lφ ) + Γ(N i l c φ) Im + Self-energy (or wave ) contribution to CP-violating parameter features a resonant enhancement for a quasi-degenerate spectrum M 1 M 2 M 3 Flanz Paschos Sarkar 94/96; Covi Roulet Vissani 96; ɛ wave N i = Im[(h h) 2 12] 8π(h h) ii M1M2 M 2 2 M2 1

56 Motivation: resonant enhancement Efficiency of leptogenesis depends on CP-violating parameter, which is one-loop suppressed ɛ Ni = Γ(N i lφ ) Γ(N i l c φ) Γ(N i lφ ) + Γ(N i l c φ) Im + Self-energy (or wave ) contribution to CP-violating parameter features a resonant enhancement for a quasi-degenerate spectrum M 1 M 2 M 3 Flanz Paschos Sarkar 94/96; Covi Roulet Vissani 96; ɛ wave N i = Im[(h h) 2 12] 8π(h h) ii M1M2 M 2 2 M2 1 On-shell initial N 1: p 2 = M 2 1 Internal N 2: i p 2 M2 2

57 Motivation: resonant enhancement The enhancement is limited by the finite width of N 1 and N 2 Off-shell initial N 1: p 2 = M im 1Γ 1 Internal N 2: i p 2 M 2 2 im 2Γ 2

58 Motivation: resonant enhancement The enhancement is limited by the finite width of N 1 and N 2 Off-shell initial N 1: p 2 = M im 1Γ 1 Internal N 2: In the maximal resonant case the spectral functions overlap i p 2 M 2 2 im 2Γ 2 ρ(k 0 )/ρ max 1 m Γ Γ k 0 /m Need to go beyond the quasi-particle approximation which underlies the conventional semi-classical Boltzmann approach.

59 Thermal bath limit Self-energies for leptons and for Majorana neutrinos N l l φ }{{} Σ αβ iγ 2 l (x, y) = S βα l (y, x) φ φ }{{} Σ N ij (x, y) = iγ2 S ji (y, x) Idea: treat the medium of Standard Model particles as a thermal bath to which the right-handed neutrinos are weakly coupled neglecting back-reaction linearized KBEs analytical solution

60 Thermal bath limit Analytical solution in the hierarchical case M 1 M 2 Anisimov, Buchmüller, Drewes, Mendizabal 08,10 Phys.Rev.Lett. 104 (2010) S N,A ( t) = ( iγ 0 cos(ω t) + M p γ ω ( S N,F (t, t) = iγ 0 cos(ω t) M p γ ω [ tanh( βω 2 ) e Γ t /2 } 2 {{} Equilibrium Damped w.r.t t ) sin(ω t) e Γ t /2 ) sin(ω t) δf N (ω)e Γt }{{} Non-equilibrium Undamped w.r.t t ] t = x 0 y 0, t = (x 0 + y 0 )/2 S A = i(s > S <)/2, S F = (S > + S <)/2

61 Thermal bath limit Cross-check with a full numerical solution of the two-time KBEs (for φ 4 -theory) Garbrecht, MG 2011 F ϕ (t, t, k) A ϕ (t, t, k) k = 2.96m th t = 17.5/m th relative time t [1/m th ] Statistical propagator F ϕ = ( > ϕ + < ϕ )/2 (red) and spectral function A ϕ = i 2 ( > ϕ < ϕ ) (blue) obtained from a numerical solution of the KBEs. Dependence on the relative time t for fixed central time t.

62 Thermal bath limit Garbrecht, MG 2011 F ϕ (t, t, k) A ϕ (t, t, k) k = 3.0m th t = 17.5/m th relative time t [1/m th ] Excited momentum mode k = 3.0m th. The statistical propagator F ϕ = ( > ϕ + < ϕ )/2 can be described by the sum of an exponentially damped equilibrium contribution and an undamped non-equilibrium contribution.

63 Thermal bath limit Evolution of damped and un-damped components with the central time t Garbrecht, MG a undamped a damped 1 δf(k) e Γkt /ω k (1 + 2f BE)/(2ω k ) a(t) [1/mth] central time t [1/m th ] F ϕ,fit (a damped (t)e Γ k t /2 + a undamped (t)) cos(ω k t)

64 Thermal bath limit Where does it come from? <,> ϕ (x 0, y 0 ) = dw 0 dz 0 R ϕ(x 0, w 0 )Π <,> ϕ (w 0, z 0 ) A ϕ(z 0, y 0 ) t 0 t 0 + R ϕ(x 0, t 0)[ x 0 y 0 <,> ϕ (t 0, t 0)] A ϕ(t 0, y 0 ) + R ϕ(x 0, t 0)[ y 0 <,> ϕ (t 0, t 0)] A ϕ(t 0, y 0 ) + R ϕ(x 0, t 0)[ x 0 <,> ϕ (t 0, t 0)] A ϕ(t 0, y 0 ) + R ϕ(x 0, t 0)[ <,> ϕ (t 0, t 0)] A ϕ(t 0, y 0 ) For a thermal bath and in Breit-Wigner approximation R,A (x 0, y 0 ) ±Θ(±(x 0 y 0 ))e Γϕ x0 y 0 /2 sin(ω ϕ(x 0 y 0 )) ω ϕ Then, one finds (finite t 0): i <,> ϕ (x 0, y 0 ) = i <,> ϕ,th (x 0 y 0 ) + 1 cos[ω ϕ(x 0 y 0 )]e Γϕ(x 0 +y 0 )/2 δf (k) ω ϕ

65 Thermal bath limit Formulated in Wigner space: i <,> ϕ Equilibrium contribution (k, t) =i st<,> (k) + δf ϕ(k)2πδ(k 2 mϕ 2 Π H ϕ)e Γϕ(k)t ϕ i st< ϕ (k) = f eq (k)i A ϕ (k) i st> ϕ (k) = (1 + f eq (k))i A ϕ (k) with finite width i A ϕ (k) = 2Π A ϕ (k 2 m 2 ϕ Π H ϕ) 2 + Π A ϕ 2 Non-equilibrium contribution with zero-width?

66 Thermal bath limit KB Equation in Wigner space [ k 2 1 ] 4 2 t + ik 0 t mϕ 2 <,> ϕ e i {Π H ϕ}{ <,> ϕ } e i {Π <,> ϕ }{ H ϕ} = 1 2 e i ( {Π > ϕ }{ < ϕ } {Π < ϕ }{ > ϕ } ) (1) {A}{B} = 1 2 ( xa) ( kb) 1 2 ( ka) ( xb), (2) Problem: k-derivatives hitting the on-shell delta function in <,> ϕ unsuppressed gradient expansion breaks down Idea: resummation of gradients are Result e i {Π > ϕ }{ < ϕ } = Π > ϕ < ϕ,eff i < ϕ,eff (k) = (f eq (k) + δf (k))i A ϕ (k) i > ϕ,eff (k) = (1 + f eq (k) + δf (k))i A ϕ (k) recover KB Ansatz effectively

67 Comparison KB Boltzmann in resonant leptogenesis n L (t, p)/n Boltzmann L (t =, p) Boltzmann KB M 2 = 1.5M 1 KB M 2 = 1.1M 1 KB M 2 = 1.025M 1 KB M 2 = 1.005M t [1/Γ 1 ] MG, Hohenegger, Kartavtsev; work in progress

68 Comparison KB Boltzmann in resonant leptogenesis 100 R max Boltzmann R Boltzmann R KB (t = ) R KB (t = 1/Γ 1) R KB (t = 0.25/Γ 1) R 10 R KB max 1 M1Γ1 M2Γ2 M1Γ1 + M2Γ (M 2 2 M 2 1 )/M 2 1 Γ 2 /Γ 1 = 1.5 R Boltzmann max = M 1M 2/(2 Γ 1M 1 Γ 2M 2 ), R KB max = M 1M 2/(2(Γ 1M 1 + Γ 2M 2))

69 Conclusions Neutrino oscillations m ν mev ev seesaw-mechanism SM + 3ν R Matter-antimatter asym. (n b n b)/n γ leptogenesis

70 Conclusions Neutrino oscillations m ν mev ev seesaw-mechanism SM + 3ν R Matter-antimatter asym. (n b n b)/n γ leptogenesis Baryogenesis via leptogenesis is a non-equilibrium, quantum process Closed-time-path approach can resolve double-counting issues of standard Boltzmann approach Resonant leptogenesis: finite width + non-equilibrium Kadanoff-Baym in thermal bath limit

71 Thermal initial correlations: 2PI Closed time path C with α n Feynman rules iλ C α 3 α 4 α 5 α 6... Example

72 Thermal initial correlations: 2PI Closed time path C with α n Thermal time path C + I Feynman rules Feynman rules iλ C α 3 α 4 α 5 α 6... Example iλ C Example iλ I

73 Thermal initial correlations: 2PI Closed time path C with α n Thermal time path C + I Feynman rules Feynman rules iλ C α 3 α 4 α 5 α 6... Example iλ C Example iλ I Challenge Encapsulate integrations over I into initial correlations α th n

74 Thermal initial correlations Thermal time path C + I Self-consistent Schwinger-Dyson equation G 1 1 th (x, y) = G0,th (x, y) Π th(x, y) ( x + m 2 )G th (x, y) = iδ C+I (x y) i d 4 zπ th (x, z)g th (z, y) C+I }{{} Closed time path C with initial correlations α Kadanoff-Baym equation for a Non-Gaussian initial state ( x + m 2 )G(x, y) = iδ C (x y) i d 4 z [Π Gauss (x, z) + Π non Gauss (x, z)] G(z, y) C

75 Thermal initial correlations: 2PI Thermal 2PI propagator connecting real and imaginary times G th ( iτ, t) = = Connection MG, Müller (2009) = + }{{} δ C (t 0 ± ) Important: Same 2PI truncation for Matsubara and real-time propagators

76 Thermal initial correlations: 2PI Example: 2PI three-loop truncation ( Φ = 0) = +

77 Thermal initial correlations: 2PI Example: 2PI three-loop truncation ( Φ = 0) = + Iterative Solution: = +

78 Thermal initial correlations: 2PI Example: 2PI three-loop truncation ( Φ = 0) = + Iterative Solution: = + +

79 Thermal initial correlations: 2PI Example: 2PI three-loop truncation ( Φ = 0) = + Iterative Solution: =

80 Thermal initial correlations: 2PI Example: 2PI three-loop truncation ( Φ = 0) = + = } {{ } Π Gauss + } {{ } Π Non Gauss

81 Thermal initial correlations: 2PI Example: 2PI three-loop truncation ( Φ = 0) = + = } {{ } Π Gauss + } {{ } Π Non Gauss Non-Gaussian self-energy contains α th n (x 1,..., x n) for all n 4 Π non Gauss (x, z) = MG, Müller (09)

82 Thermal initial correlations: 2PI Thermal correlation energy (computed at initial time): Ecorr eq (t = 0) = i 4 C = = d 4 z [Π Gauss (x, z) + Π non Gauss (x, z)] G(z, x) x=0 + + x=0 x=0 }{{} x 0 dz x=0 = E eq 4 p. corr (t = 0) }{{} x 0 dz x=0...only the thermal 4-point correlation contributes truncate initial correlations with n > 4

83 Leading non-gaussian correction for λφ 4 2PI 3-loop Gaussian IC Non-Gaussian IC with α th 4 G(x, y) x 0,y 0 =0 = G th (x, y) x 0,y 0 =0 α 4 (x 1,..., x 4 ) = 0 α n(x 1,..., x n) = 0 for n > 4 G(x, y) x 0,y 0 =0 = G th (x, y) x 0,y 0 =0 α 4 (x 1,..., x 4 ) = α th 4 (x 1,..., x 4 ) α n(x 1,..., x n) = 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

84 Leading non-gaussian correction for λφ 4 2PI 3-loop 0.8 k = G F (t,t,k) k = m R k = 2m R KB, Gauss (A) KB, Non-Gauss (B) ThQFT t m R t m R

85 Leading non-gaussian correction for λφ 4 2PI 3-loop G F (t,t,k)/g F (0,0,k) G F (t,t,k)/g F (0,0,k) G F (t,t,k)/g th G (k) F (t,t,k)/g F (0,0,k) G F (t,t,k)/g F (0,0,k) 1.05 t m R = 0.0 KB, Gauss t m R = 0.0 KB, Non-Gauss t m R = 0.5 t m R = t m R = 2.0 t m R = t m R = 10 t m R = t m R = 2000 t m R = k/m R k/m R all modes remain close to equilibrium

86 Leading non-gaussian correction for λφ 4 2PI 3-loop T/m R 2.3 KB, Gauss (A) KB, Non-Gauss (B) 2.2 Thermal Eq µ/m R t m R t m R } {{ } build-up } {{ } kinetic - } {{ } chemical equilibration correlated system T 0

87 Leading non-gaussian correction for λφ 4 2PI 3-loop 0.4 Correlation energy (Gaussian part) 0.2 Contribution from initial 4-point correlation Sum 0 E corr (t,k) k max =7m R t [m -1 R ] particles in initial state are well-dressed

88 Leading non-gaussian correction for λφ 4 2PI 3-loop E corr (t,k) 0.4 Correlation energy (Gaussian part) 0.2 Contribution from initial 4-point correlation Sum 0 k max =5m R -0.2 k max =7m R -0.4 k max =10m R t [m -1 R ] particles in initial state are well-dressed

89 Leading non-gaussian correction for λφ 4 2PI 3-loop G F (t,t,k)/g th (k) G F (t,t,k)/g th (k) G F (t,t,k)/g th (k) G F (t,t,k)/g th (k) G F (t,t,k)/g th (k) t m R = 0.0 t m R = 0.5 t m R = 2.0 t m R = 10 t m R = 2000 KB, Gauss t m R = 0.0 t m R = 0.5 t m R = 2.0 t m R = 10 t m R = 2000 KB, Non-Gauss k/m R k/m R UV modes are not excited suppresses unwanted back-reaction on IR modes