Towards a quantum treatment of leptogenesis

Transcription

1 SEWM Swansea, July based on work in collaboration with Tibor Frossard, Andreas Hohenegger, Alexander Kartavtsev, Manfred Lindner [mainly ]

2 Physics beyond the Standard Model Collider exp. Baryon asymmetry.. Neutrino exp. +? Dark matter

3 Outline Leptogenesis Boltzmann approach Kadanoff-Baym approach Resonant enhancement on the closed time path

4 Leptogenesis Standard Model (SM) extended by three heavy singlet neutrino fields N i = N c i, i = 1, 2, 3 with Majorana masses ˆM = diag(m i ) in the mass eigenbasis L = L SM Ni / N 1 2 N ˆMN l φhp R N NP L h φ l Light neutrino masses via seesaw mechanism m ν = v 2 EW h ˆM 1 h T TeV M i M GUT for m e/v EW < h ij < 1

5 Leptogenesis Standard Model (SM) extended by three heavy singlet neutrino fields N i = N c i, i = 1, 2, 3 with Majorana masses ˆM = diag(m i ) in the mass eigenbasis L = L SM Ni / N 1 2 N ˆMN l φhp R N NP L h φ l Light neutrino masses via seesaw mechanism m ν = v 2 EW h ˆM 1 h T TeV M i M GUT for m e/v EW < h ij < 1 Baryogenesis via leptogenesis Fukugita, Yanagida 86 B-violation via L-violating Majorana masses M i CP-violation via Yukawa couplings Im[(h h) ij ] 0 Out-of-equilibrium (inverse) decay N i lφ and N i l c φ (Γ i /H) T =Mi m i /mev O(1) where m i = v 2 EW (h h) ii /M i (Γ SM /H) T =Mi g 4 M pl /M i 1 for M i GeV

6 Leptogenesis Vanilla leptogenesis for hierarchical spectrum M 1 M 2,3 requires large values of the reheating temperature T R O(M 1) 10 9 GeV Hamaguchi, Murayama, Yanagida; Davidson, Ibarra M1 (GeV) Gravitino production ( Ω th 3/2h 2 TR GeV m1 (ev) ) ( 10 GeV m 3/2 Buchmüller, Di Bari, Plümacher ) ( ) 2 m g 1 TeV Moroi, Murayama, Yamaguchi, 93; Bolz, Brandenburg, Buchmüller, 01;Pradler, Steffen, 06; Rychkov, Strumia, 07

7 Leptogenesis L-violating decay of heavy right-handed neutrino N i M Ni l αh = y iα +... M Ni l c α h = y iα +...

8 Leptogenesis L-violating decay of heavy right-handed neutrino N i M Ni l αh = y iα + y iβ y jβ y jα +... y jβ M Ni l c α h = y iα + y iβ +... y jα Matter-antimatter (CP) asymmetry interference of tree and loop processes ( Γ(N i l αh ) Γ(N i l c αh) Im(y iα y iβ yjαy jβ) Im + ) Sakharov: asymmetry from out-of-equilibrium N i decay/inverse decay

9 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

10 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

11 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, dispersion relation production rate from a thermal plasma for T M 1, T M 1 Covi, Rius, Roulet, Vissani 98; Giudice, Notari, Raidal, Riotto, Strumia 04; Besak, Bödeker 10 Kiessig, Thoma, Plümacher 10; Salvio Lodone Strumia 11, Anisimov, Besak, Bödeker 11; Laine, Schroder 11; Besak, Bodeker Flavour effects Nardi, Nir, Roulet, Racker 06; Abada, Davidson, Ibarra, Josse-Micheaux, Losada, Riotto 06; de Simone, Riotto 06; Blanchet, dibari 06;... Spectator processes, scatterings, N 2,...

12 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

13 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... derivation from first principles / generalization?

14 Leptogenesis - 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

15 Leptogenesis - 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

16 Resonant leptogenesis Flanz Paschos Sarkar Weiss 96; effective Hamiltonian approach ɛ Ni = Im[(h h) 2 12 ] 16π(h h) ii M 1 (M 2 M 1 ) (M 2 M 1 ) 2 + M 2 1 (Re(h h) 12 /(16π)) 2 Covi Roulet 96; CP violating decay of mixing scalar fields described by effective mass matrix; formalism as in Liu Segre 93 Pilaftsis 97; Pilaftsis Underwood 03; Pole mass expansion of resummed propagators ɛ Ni = Im[(h h) 2 ij ] (h h) ii (h h) jj (M 2 i M 2 j )M i Γ j (M 2 i M 2 j )2 + M 2 i Γ2 j Buchmüller Plümacher 97; Diagonalization of resummed propagators ɛ Ni = Im[(h h) 2 12 ] M 1 M 2 (M2 2 M2 1 ) 8π(h h) ii (M2 2 M2 1 1 π Γ i M i ln M2 2 ) 2 + (M 1 Γ 1 M 2 Γ 2 ) 2 Rangarajan Mishra 99; comparison of different approaches Anisimov Broncano Plümacher 05; Reconciliation of diagonalization approach with the pole mass expansion approach Invariant quantity M 1 M 2 (M2 2 M2 1 )Im(h h) 2 12 related to CP violation appears in the enumerator M 2 1

17 Resonant leptogenesis The results can be summarized (neglecting log-corrections) as ɛ Ni = Im[(h h) 2 12] 8π(h h) ii R, R M1M2(M2 2 M 2 1 ) (M 2 2 M2 1 )2 + A 2 Different calculations correspond to different expressions for the regulator A 2 A 2 = ( ) 1 4 (M1 + Re(h 2 h)12 M2)4 Flanz Paschos Sarkar Weiss 96 16π M 2 i Γ 2 j Pilaftsis 97; Pilaftsis Underwood 03 (M 1Γ 1 M 2Γ 2) 2 Buchmüller Plümacher 97; Anisimov Broncano Plümacher 05; The regulator is relevant for determining the maximal possible resonant enhancement, which occurs for M 2 2 M 2 1 = ±A, and is given by R max = M1M2 2 A

18 Resonant leptogenesis The origin of the regulator is 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

19 Resonant leptogenesis The origin of the regulator is 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 In the maximal resonant case M 2 M 1 = O(Γ i ), the spectral functions overlap ρ(k 0 )/ρ max 1 m Γ Γ k 0 /m deviation from equilibrium is essential desirable to go beyond the quasi-particle approximation which underlies the conventional semi-classical Boltzmann approach.

20 CTP/Kadanoff-Baym approach to leptogenesis Goal(s) (1) derivation of kinetic equations starting from first principles on-/off-shell treated in a unified way (avoid double-counting) medium corrections,...

21 CTP/Kadanoff-Baym approach to leptogenesis Goal(s) (1) derivation of kinetic equations starting from first principles on-/off-shell treated in a unified way (avoid double-counting) medium corrections,... (2) check resonant enhancement w/o resorting to kinetic description (KB) coherent flavor transitions,...

22 CTP/Kadanoff-Baym approach to leptogenesis Goal(s) (1) derivation of kinetic equations starting from first principles on-/off-shell treated in a unified way (avoid double-counting) medium corrections,... (2) check resonant enhancement w/o resorting to kinetic description (KB) coherent flavor transitions,... (... incomplete list) Buchmüller, Fredenhagen hep-ph/004145; De Simone, Riotto hep-ph/ , Anisimov, Buchmüller, Drewes, Mendizabal , MG, Kartavtsev, Hohenegger, Lindner , Beneke, Fidler, Garbrecht, Herranen, Schwaller , Garbrecht ; Gagnon, Shaposhnikov (Poster) MG, Kartavtsev, Hohenegger , , Drewes, Mendizabal, Weniger Garbrecht, Herranen (next talk); Garbrecht ;Garbrecht, Drewes see also Fidler, Herranen, Kainulainen, Rahkila ; Canetti, Drewes, Shaposhnikov (Poster) Frossard, MG, Kartavtsev, Hohenegger (Poster); Huetig, Mendizabal, Philipsen (Poster)

23 CTP/Kadanoff-Baym approach to leptogenesis Lepton current j µ L (x) = α l α(x)γ µ l α(x) [ ] = tr γ µ S αβ l (x, x)

24 CTP/Kadanoff-Baym approach to leptogenesis Lepton current j µ L (x) = α l α(x)γ µ l α(x) [ ] = tr γ µ S αβ l (x, x) Lepton asymmetry n L (t) = 1 V V d 3 x j 0 L(t, x)

25 CTP/Kadanoff-Baym approach to leptogenesis Lepton current j µ L (x) = α l α(x)γ µ l α(x) [ ] = tr γ µ S αβ l (x, x) Lepton asymmetry Equation of motion dn L dt = 1 V V n L (t) = 1 V d 3 x µj µ L (x) = 1 V V V d 3 x j 0 L(t, x) [ ] d 3 x tr γ µ( x µ + y µ )S αβ l (x, y) x=y

26 CTP/Kadanoff-Baym approach to leptogenesis Lepton current j µ L (x) = α l α(x)γ µ l α(x) [ ] = tr γ µ S αβ l (x, x) Lepton asymmetry Equation of motion dn L dt = 1 V V n L (t) = 1 V d 3 x µj µ L (x) = 1 V V V d 3 x j 0 L(t, x) Use KB equations for leptons on the right-hand side [ ] d 3 x tr γ µ( x µ + y µ )S αβ l (x, y) x=y dn L dt t [ = i dt d 3 p αγ tr Σ (2π) 3 l ρ p (t, t γβ )S l F p (t αγ, t) Σ l F p (t, t γβ )S l ρ p(t, t) 0 ] αγ S l ρ p (t, t γβ )Σ l F p (t αγ, t) + S l F p (t, t γβ )Σ l ρ p(t, t)

27 CTP/Kadanoff-Baym approach to leptogenesis N lφ N l c φ tree 2 tree vertex-corr. tree wave-corr. lφ l c φ s t s s, t t unified description of CP-violating decay, inverse decay, scattering

28 CTP/Kadanoff-Baym approach to leptogenesis N lφ N l c φ tree 2 tree vertex-corr. tree wave-corr. lφ l c φ s t s s, t t unified description of CP-violating decay, inverse decay, scattering dn L /dt vanishes in equilibrium due to KMS relations S eq F = 1 ( ) βk 0 2 tanh Sρ eq 2 Σ eq F = 1 ( ) βk 0 2 tanh Σ eq ρ 2 consistent equations free of double-counting problems MG, Kartavtsev, Hohenegger, Lindner 09; Beneke, Garbrecht, Herranen, Schwaller 10;

29 CTP/Kadanoff-Baym approach to leptogenesis Hierarchical case (η η vac )/η vac 10 2 η/ǫ vac η vac /ǫ vac 1 η η vac a b c d e f κ washout-parameter K = (Γ/H) T =M m 1/meV thermal initial abundance MG, Kartavtsev, Hohenegger, Lindner 10

30 Resonant enhancement Statistical propagator S F and spectral function S ρ are matrices in N 1, N 2, N 3 flavor space. We consider the sub-space N 1, N 2 of the quasi-degenerate states. ( ) S ij S 11 S (x, y) = T C N i (x) N 12 j (y) = S 21 S 22 coherent N 1 N 2 transitions out-of-equilibrium

31 Resonant enhancement Statistical propagator S F and spectral function S ρ are matrices in N 1, N 2, N 3 flavor space. We consider the sub-space N 1, N 2 of the quasi-degenerate states. ( ) S ij S 11 S (x, y) = T C N i (x) N 12 j (y) = S 21 S 22 coherent N 1 N 2 transitions out-of-equilibrium Self-energies for leptons and for Majorana neutrinos Γ 2 = φ N l N l l φ φ φ }{{}}{{} Σ αβ iγ l (x,y)= 2 S βα Σ ij N (x,y)= iγ 2 l (y,x) S ji (y,x) Solve KBEs treating lepton and Higgs as a thermal bath (no backreaction); include qualitative damping term for lepton/higgs (not essential, but practical; no consistent treatment of gauge-int. yet; see Poster by Janine Hütig) [ hierarchical case: Anisimov, Buchmüller, Drewes, Mendizabal 08,10 ] Important: lepton self-energy contains full Majorana propagator-matrix

32 Resonant enhancement Kadanoff-Baym Equations ((i / x M i )δ ik δσ N (x) ik )S kj F (x, y) = x 0 ((i / x M i )δ ik δσ N (x) ik )S kj ρ (x, y) = dz 0 0 y 0 0 x 0 y 0 dz 0 dz 0 d 3 ik z Σ N ρ (x, z)s kj F (z, y) d 3 z Σ N ik F (x, z)s kj ρ (z, y) d 3 z Σ N ik ρ (x, z)s kj ρ (z, y)

33 Decay of nearly-degenerate Majorana neutrinos Goal: obtain analytical result by taking essential features into account (width, coherent flavor-mixing, memory integrals) [later: compare with numerical treatment]

34 Decay of nearly-degenerate Majorana neutrinos Goal: obtain analytical result by taking essential features into account (width, coherent flavor-mixing, memory integrals) [later: compare with numerical treatment] First step: Non-equilibrium Majorana propagator in BW appr. S ij ij th F p (t, t ) = SF p (t t ) + S ij F p (t, t )

35 Decay of nearly-degenerate Majorana neutrinos Goal: obtain analytical result by taking essential features into account (width, coherent flavor-mixing, memory integrals) [later: compare with numerical treatment] First step: Non-equilibrium Majorana propagator in BW appr. S ij ij th F p (t, t ) = SF p (t t ) + S ij F p (t, t ) Second step: Lepton asymmetry t t n L (t) = i(h h) ji dt dt 0 0 d 3 p (2π) 3 ( ) tr [P R S ij F p (t, t ) S ji F p(t, t ) }{{} Deviation from equilibrium, CP-violation S ji F p(t, t ) = CP S ij F p (t, t ) T (CP) 1 lepton-higgs loop S lφ = S l φ ] P L S lφρ p (t t )

36 Majorana neutrino propagator Retarded and advanced propagators S R (x, y) = Θ(x 0 y 0 )S ρ(x, y) S A (x, y) = Θ(y 0 x 0 )S ρ(x, y) The Kadanoff-Baym equation for the statistical propagator can be written as 0 [( (i d 4 ) z / x M i δ ik δσ ik N (x)) ] ik δ(x z) Σ N R (x, z) = Special solution of the inhomogeneous equation S ij F (x, y) inhom = 0 d 4 us ik R (x, u) 0 0 S kj F (z, y) d 4 ik z Σ N F (x, z)s kj A (z, y) d 4 kl z Σ N F (u, z)s lj (z, y) General solution of the homogeneous equation S ij F (x, y) hom = d 3 u SR ik (x, (0, u)) d 3 v A kl (u, v)s lj A ((0, v), y) where A kl (u, v) is a free function A

37 Majorana neutrino propagator 10 0 pole mass difference and effective width pole pole pole M pole 2 M pole 1, Γ pole I [M1] Re(h h)12/(8π) M pole M pole 1 Γ pole 1 Γ pole (M 2 M 1 )/M 1 Effective masses M pole I ω pi p=0 and widths Γ pole I Γ pi p=0 of the sterile Majorana neutrinos extracted from complex poles of resummed ret/adv prop. for (h h) 11 = 0.03, (h h) 22 = 0.045, (h h) 12 = 0.03 e iπ/4 and T = 0.25M 1.

38 Majorana neutrino propagator Regime (M 2 M 1)/M 1 Re(h h) 12/(8π) M pole i M i, Γ pole i Γ i (h h) ii 8π M i Regime (M 2 M 1)/M 1 Re(h h) 12/(8π) M pole i Γ pole i M i 16π ( ) e M i /T 1 M1 + M2 (M 2 M 1)((h h) 22 (h h) 11) ± 2 2 ((h h) 22 (h h) 11) 2 + 4(Re(h h), 12) 2 ( ) 2 ( 1 + (h h) e M i /T 11 + (h h) 22 1 ± ) ((h h) 22 (h h) 11) 2 + 4(Re(h h) 12) 2 The relation between the mass- and Yukawa coupling matrices at zero and finite temperature is M(T ) = (P L Z(T ) T + P R Z(T ) )M(T = 0)(P L Z(T ) + P R Z(T ) ), (h h)(t ) = Z(T ) T (h h)(t = 0)Z(T ), where Z ij (T ) V ik (T )(δ kj + (h h) kj T 2 /(6µ 2 )), V (T ) V (T ) = 1

39 Result for the lepton asymmetry n L (t) = d 3 p d 3 q d 3 k (2π) 9 2q 2k (2π)3 δ(p k q) I,J=1,2 ɛ n=±1 F ɛn JI Lɛn IJ (t)

40 Result for the lepton asymmetry n L (t) = d 3 p d 3 q d 3 k (2π) 9 2q 2k (2π)3 δ(p k q) I,J=1,2 ɛ n=±1 F ɛn JI Lɛn IJ (t) Coefficients F depend on Yukawa couplings, thermal distributions of lepton and Higgs, resummed ret/adv propagators and initial conditions F ɛn JI = ( ( (h h) 1 ji + f 2 φ(q) ) ( + ɛ 1 2ɛ 3 f 2 l(k) )) [ tr P L ( k γ 0 + ɛ 2kγ) ijkl=1,2 ( S ikɛ 4 RI γ 0 S kl F p(0, 0)γ 0S ljɛ 1 AJ S jkɛ 4 RI )] γ 0 S kl liɛ F p(0, 0)γ 0 S 1 AJ

41 Result for the lepton asymmetry n L (t) = d 3 p d 3 q d 3 k (2π) 9 2q 2k (2π)3 δ(p k q) I,J=1,2 ɛ n=±1 F ɛn JI Lɛn IJ (t) Coefficients F depend on Yukawa couplings, thermal distributions of lepton and Higgs, resummed ret/adv propagators and initial conditions F ɛn JI = ( ( (h h) 1 ji + f 2 φ(q) ) ( + ɛ 1 2ɛ 3 f 2 l(k) )) [ tr P L ( k γ 0 + ɛ 2kγ) ijkl=1,2 ( S ikɛ 4 RI γ 0 S kl F p(0, 0)γ 0S ljɛ 1 AJ S jkɛ 4 RI )] γ 0 S kl liɛ F p(0, 0)γ 0 S 1 AJ Time-dependence: flavor diagonal and off-diagonal contributions: ( ) L ± II (t) = 1 e Γ pi t Γ lφ Re Γ pi (ω pi k q + iγ pi /2) 2 + Γ 2 lφ /4 ( L ± 21(t) = 1 e i(ω p1 ω p2 )t e (Γ p1+γ p2 )t/2 Γ lφ Γ p1 + Γ p2 ± 2i(ω p1 ω p2) (ω p1 k q ± iγ p1/2) 2 + Γ 2 lφ /4 ) Γ lφ + (ω p2 k q iγ p2/2) 2 + Γ 2 lφ /4 = L ± 12(t)

42 Resonant enhancement Comparison KB Boltzmann: hierarchical limit n Boltzmann n L (t) = Im[(h h) 2 12] M 1 8π M 2 L (t) = Im[(h h) 2 12] 8π (2π) 3 δ(p k q) Re d 3 p d 3 q d 3 k 4k (2π) 9 p1 ω p1 2q 2k (1 + f φ (q) f l (k)) f FD (ω p1) 1 e Γ p1t M 1 M 2 Γ lφ (ω p1 k q + iγ p1/2) 2 + Γ 2 lφ /4 Γ p1 d 3 p d 3 q d 3 k 4k (2π) 9 p1 ω p1 2q 2k (2π) 3 δ(p k q) 2πδ(ω p1 k q) (1 + f φ (q) f l (k)) f FD (ω p1) 1 e Γ p1t Γ p1. The thermal width of lepton and Higgs Γ lφ = Γ l + Γ φ leads to a replacement of the on-shell delta function in the Boltzmann equations by a Breit-Wigner curve, in accordance with Anisimov, Buchmüller, Drewes, Mendizabal 10 The coherent contributions are suppressed with Γ p1/ω p2

43 Resonant enhancement Comparison KB Boltzmann: degenerate case n L (t) = Im[(h h) 2 12] 8π M 1M 2(M2 2 M1 2 ) (M2 2 M2 1 )2 + (M 1Γ 1 M 2Γ 2) 2 Γ lφ d 3 p d 3 q d 3 k (2π) 9 ω p 2q 2k 4k p (2π) 3 δ(p k q) (ω p k q) 2 + Γ 2 lφ /4 (1 + f φ f l )f FD (ω p) [ ( 1 e Γ pi t 1 e i(ω p1 ω p2 )t e (Γ p1+γ p2 ) ] )t/2 4 Re Γ p1 + Γ p2 + 2i(ω p1 ω p2) I =1,2 n Boltzmann L (t) = Im[(h h) 2 12] 8π Γ pi M 1M 2(M2 2 M1 2 ) (M2 2 M2 1 )2 + (M 1Γ 1 M 2Γ 2) 2 d 3 p d 3 q d 3 k (2π) 9 ω p 2q 2k 4k p (2π) 3 δ(p k q) 2πδ(ω p k q) (1 + f φ f l )f FD (ω p) [ ] 1 e Γ pi t I =1,2 Γ pi

44 Resonant enhancement Comparison KB Boltzmann: degenerate case n L (t) = Im[(h h) 2 12] 8π M 1M 2(M2 2 M1 2 ) (M2 2 M2 1 )2 + (M 1Γ 1 M 2Γ 2) 2 Γ lφ d 3 p d 3 q d 3 k (2π) 9 ω p 2q 2k 4k p (2π) 3 δ(p k q) (ω p k q) 2 + Γ 2 lφ /4 (1 + f φ f l )f FD (ω p) [ ( 1 e Γ pi t 1 e i(ω p1 ω p2 )t e (Γ p1+γ p2 ) ] )t/2 4 Re Γ p1 + Γ p2 + 2i(ω p1 ω p2) I =1,2 n Boltzmann L (t) = Im[(h h) 2 12] 8π Γ pi M 1M 2(M2 2 M1 2 ) (M2 2 M2 1 )2 + (M 1Γ 1 M 2Γ 2) 2 d 3 p d 3 q d 3 k (2π) 9 ω p 2q 2k 4k p (2π) 3 δ(p k q) 2πδ(ω p k q) (1 + f φ f l )f FD (ω p) [ ] 1 e Γ pi t I =1,2 Γ pi Regulator M 1Γ 1 M 2Γ 2 is confirmed

45 Resonant enhancement Comparison KB Boltzmann: degenerate case n L (t) = Im[(h h) 2 12] 8π M 1M 2(M2 2 M1 2 ) (M2 2 M2 1 )2 + (M 1Γ 1 M 2Γ 2) 2 Γ lφ d 3 p d 3 q d 3 k (2π) 9 ω p 2q 2k 4k p (2π) 3 δ(p k q) (ω p k q) 2 + Γ 2 lφ /4 (1 + f φ f l )f FD (ω p) [ ( 1 e Γ pi t 1 e i(ω p1 ω p2 )t e (Γ p1+γ p2 ) ] )t/2 4 Re Γ p1 + Γ p2 + 2i(ω p1 ω p2) I =1,2 n Boltzmann L (t) = Im[(h h) 2 12] 8π Γ pi M 1M 2(M2 2 M1 2 ) (M2 2 M2 1 )2 + (M 1Γ 1 M 2Γ 2) 2 d 3 p d 3 q d 3 k (2π) 9 ω p 2q 2k 4k p (2π) 3 δ(p k q) 2πδ(ω p k q) (1 + f φ f l )f FD (ω p) [ ] 1 e Γ pi t I =1,2 Γ pi Regulator M 1Γ 1 M 2Γ 2 is confirmed Additional oscillating contribution due to coherent N 1 N 2 transitions

46 Resonant enhancement MG, Kartavtsev, Hohenegger 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 ] n L (t) = d 3 p (2π) 3 n L (t, p), Γ 1 = 0.01M 1, Γ 2 = 0.015M 1, Γ lφ 0

47 Resonant enhancement Resonant enhancement within the Boltzmann approach R Boltzmann M1M2(M2 2 M 2 1 ) (M 2 2 M2 1 )2 + A 2

48 Resonant enhancement Resonant enhancement within the Boltzmann approach R Boltzmann M1M2(M2 2 M 2 1 ) (M 2 2 M2 1 )2 + A 2 Resonant enhancement within the Kadanoff-Baym approach, including coherent contributions ( (h h) 12 (h h) ii ) R KB (t) = M 1M 2(M 2 2 M 2 1 ) (M 2 2 M2 1 )2 + (M 1Γ 1 M 2Γ 2) 2 (1 f coherent(t))

49 Resonant enhancement Resonant enhancement within the Boltzmann approach R Boltzmann M1M2(M2 2 M 2 1 ) (M 2 2 M2 1 )2 + A 2 Resonant enhancement within the Kadanoff-Baym approach, including coherent contributions ( (h h) 12 (h h) ii ) R KB (t) = M 1M 2(M 2 2 M 2 1 ) (M 2 2 M2 1 )2 + (M 1Γ 1 M 2Γ 2) 2 (1 f coherent(t)) Partial cancellation of Boltzmann- and coherent contribution cuts off the enhancement in the doubly degenerate limit M 1 M 2 and Γ 1 Γ 2 R KB (t) t 1/ΓpI M 1M 2(M 2 2 M 2 1 ) (M 2 2 M2 1 )2 + (M 1Γ 1 + M 2Γ 2) 2

50 Resonant enhancement MG, Kartavtsev, Hohenegger 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))

51 Resonant enhancement MG, Kartavtsev, Hohenegger 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 R KB, h h12 h h11 = 0.5 R KB, h h12 h h11 = 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))

52 Conclusions First-principles methods like Kadanoff-Baym equations are helpful to scrutinize classical approximations Resonant enhancement on the closed time path valid for M Γ, effective width/mass, flavor coherence smaller enhancement than Boltzmann Rmax Boltzmann = M 1 M 2 /(2 Γ 1 M 1 Γ 2 M 2 ) Rmax KB M 1 M 2 /(2(Γ 1 M 1 + Γ 2 M 2 )) numerical solution for h h 12 h h ii (check BW appr.) relevant e.g. for flavor symmetries ( M/M, Γ/Γ 1) effective density matrix approach in zero-width limit

53 Conclusions First-principles methods like Kadanoff-Baym equations are helpful to scrutinize classical approximations Resonant enhancement on the closed time path valid for M Γ, effective width/mass, flavor coherence smaller enhancement than Boltzmann Rmax Boltzmann = M 1 M 2 /(2 Γ 1 M 1 Γ 2 M 2 ) Rmax KB M 1 M 2 /(2(Γ 1 M 1 + Γ 2 M 2 )) numerical solution for h h 12 h h ii (check BW appr.) relevant e.g. for flavor symmetries ( M/M, Γ/Γ 1) effective density matrix approach in zero-width limit thank you!

54 Majorana neutrino propagator Treating lepton/higgs as a thermal bath, the Majorana neutrino propagator is given by S ij ij th F p (t, t ) = SF p (t t ) + SR ik p(t)γ 0 SF kl p(0, 0)γ 0S lj A p ( t ) }{{} S ij F p (t, t ) The resummed retarded and advanced functions can be obtained by solving a SD equation (we use MS at T = 0 and µ = (M 1 + M 2)/2) [ (/ ) ] p M i δ ik δσ ik ik N (p) Σ N R(A)(p) S kj R(A) (p) = δij Σ N ij R(A) (p) = 2 [ (h h) ij P L + (h h) ji P R ] S lφr(a) (p) vac S lφ R(A) (p) = 1 ( 1 32π 2 ɛ γ E + ln(4π) + 2 ln ( p 2 µ 2 ) ) ± iπθ(p 2 )sign(p 0) /p S lφ th R(A) (p) T 2 /p/(12p 2 ) ± 2i /p/(e p 0 /T 1)Θ(p 2 )sign(p 0)/(32π)

55 Majorana neutrino propagator Solve SD equation for S R(A) (p) in Breit-Wigner approximation: S ij R(A) (p) Z ij 1R(A) + Z ij 2R(A) p 2 x 1 p 2 x 2 with residua Z ij IR(A) and complex poles x I (basis independent) where x 1,2 = (V ± W )2 4Q 2 ( ω pi i Γ ) 2 pi p 2 2 V = (η 1M 1(1 + iγ 22) η 2M 2(1 + iγ 11)) 2 4η 1η 2M 1M 2(Reγ 12) 2 W = (η 1M 1(1 + iγ 22) + η 2M 2(1 + iγ 11)) 2 + 4η 1η 2M 1M 2(Imγ 12) 2 Q = det Ω LR = det Ω RL = (1 + iγ 11)(1 + iγ 22) + γ 12 2 γ ij (h h) ij Θ(p2 )sign(p 0) 16π ( ) e p 0 /T 1 + i 2 p ln µ 2 16π + T 2 2 6p T 2 2 6µ 2