1. Introduction overview of baryogenesis. Anomalous Baryon Number Nonconservation 3. Sphaleron Process 4. Requirements for EW Baryogenesis 5. Leptogen

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2 1. Introduction overview of baryogenesis. Anomalous Baryon Number Nonconservation 3. Sphaleron Process 4. Requirements for EW Baryogenesis 5. Leptogenesis 6. Discussions Sphaleron process and Leptogenesis /73

3 1. Introduction Baryon Asymmetry of the Universe n p n p M 10 1 M [Steigman, Ann. Rev. Astron. Astrophys. 14 ('76)] η n B n γ (95% CL) T < 1MeV Sphaleron process and Leptogenesis 3/73

4 n B s = ( ) s 7.04n γ at T < 1MeV B = 0 Sphaleron process and Leptogenesis 4/73

5 = (1) () C CP (3) B 0 ρ(t) ρ(t) = n p n ψ n (t) ψ n (t) O (t) = Tr [ ρ(t)o] ρ 0 n B 0 Tr [ρ 0 n B ] = 0 Sphaleron process and Leptogenesis 5/73

6 density operator ρ(t) i h ρ(t) t + [ρ(t), H] = 0 H ρ 0 H C CP = [ρ, C] = 0 or [ρ, CP] = 0 CBC 1 = B, CPB(CP) 1 = B B C B (ρ 0 ) H C CP n B n B = Tr[ρ n B ] = Tr[ρ Cn B C 1 ] = Tr[ρ n B ] = 0 or n B = Tr[ρ CPn B (CP) 1 ] = Tr[ρ n B ] = 0... n B = 0 C CP Sphaleron process and Leptogenesis 6/73

7 (1) tree-level global U(1) B -sym. SU(3) c SU() L U(1) Y q X µ l (squark) Affleck-Dine mechanism [Affleck & Dine, Nucl. Phys. B49 ('85); Dine, et al., Nucl. Phys. B458 ('96)] q Sphaleron process and Leptogenesis 7/73

8 (B + L) (chiral anomaly) [ = τ p > SU(5) (B + L) B 0 T = 0 ['t Hooft, Phys. Rev. D14 ('76)] Sphaleron process and Leptogenesis 8/73

9 () L g ( ψl γ µ A a µt a Lψ L + ψ R γ µ A a µt a Rψ R ) T a L T a R T a L = τ a and T a R = 0 and Y L Y R y AB q AL γ µ W µ q BL + h.c. (A B) strong CP phase ( θ QCD F µν F µν ) µ soft SUSY-breaking parameters µbφ d Φ u, Aϕ 3, Mχχ θ QCD 0 by nedm [Girardello & Grisaru, Nucl. Phys. B194 ('8)] Sphaleron process and Leptogenesis 9/73

10 (3) Γ B 0 H(T ) Sphaleron process and Leptogenesis 10/73

11 GUT Baryogenesis minimal SU(5) model: { 5 : ψ i matter: L d c R, l L 10 : χ [ij]l q L, u c R, ec R i = 1 5 (α = 1 3, a = 1, ) gauge: A µ = [Yoshimura, Phys. Rev. Lett. 41 ('78)] ( ) Gµ, B µ Xµ aα Xµ aα W µ, B µ L int g ψγ µ A µ ψ + gtr [ χγ µ {A µ, χ}] gxαµ a [ ε αβγū c Rγγ µ α q Lβa + ϵ ab ( q Lb γ µ e c R + l Lb γ µ d cα )] R X X B = 3 r 1 3 (1 r) 3 r+1 (1 r) = r r 3... C CP (r = r) = B = 0 B X qq r /3 X q l 1 r 1/3 X q q r /3 X q, l 1 r 1/3 Sphaleron process and Leptogenesis 11/73

12 B r r T m X X Γ D αm X (α 1/40) Γ D H(T = m X ) X X H(T ) 1.66 g T m Pl g : SU(5) model (B L) (B + L) (B + L) 0 B + L 0 (B + L) 0 B L 0 Leptogenesis: L 0 B = L Sphaleron process and Leptogenesis 1/73

13 . Anomalous Baryon Number Nonconservation B L µ j µ B+L = N f 16π [g Tr(F µν F µν ) g 1B µν Bµν ] µ j µ B L = 0 N f = F µν 1 ϵµνρσ F ρσ B(t f ) B(t i ) = N f 3π tf t i d 4 x [ g Tr(F µν F µν ) g 1B µν Bµν ] = N f [N CS (t f ) N CS (t i )] N CS Chern-Simons number: A 0 = 0-gauge N CS (t) = 1 3π d 3 x ϵ ijk [g Tr (F ij A k 3 ) g A i A j A k g 1B ] ij B k t Sphaleron process and Leptogenesis 13/73

14 : E = 1 (E + B ) = 0 F µν = B µν = 0 A µ = iu 1 µ U, B µ = µ v with U SU() U(x) : S 3 U SU() S 3 π 3 (S 3 ) Z = U(x) N CS ig 3 d 3 x ϵ 48π ijk Tr[U 1 i U U 1 j U U 1 k U] winding number U(1) axial U(1) anomaly axial fermion Q 5 = N CS (t) = g 4π g π tf dt dx ϵ µν F µν = N CS (t f ) N CS (t i ), t i dx A 1 (t, x). : A 1 (x) = 1 g xα(x) with α( ) α( ) = πn... N CS = N Sphaleron process and Leptogenesis 14/73

15 E N CS { B 0 e S instanton = e 8π /g e e E sph/t E sph = Sphaleron T > T C (B + L) 0 Sphaleron process and Leptogenesis 15/73

16 N CS [Atiyah and Singer, 1968] n R n L = ν = g 16π d 4 xtr(f µν F µν ) (chiral fermions) = Pontrjagin index = instanton number ψ L E ψ R E [Ambjørn, et al. Nucl. Phys. B1 ('83)] p ǫ µν F µν 0 p vacuum particle production Sphaleron process and Leptogenesis 16/73

17 (1 + 1) Dirac eq. iγ µ ( µ iga µ (x))ψ(x) = 0 [γ 0 = σ 1, γ 1 = iσ ; γ 3 = γ 0 γ 1 = σ 3 ] { A 0 =0 i( x iga 1 (x))ψ L (x) i t ψ(x) = Hψ(x) iσ 3 ( x iga 1 (x)) ψ(x) = i( x iga 1 (x))ψ R (x) ψ(x + L) = ψ(x) x ) t-indep. gauge trf. ψ(x) = exp (ig dx A 1 (x) ψ(x) H ψ(x) = iσ 3 x ψ(x) with ψ(x) = e ipx with p = πn L A 1 (x) = 0 A 1 (x) = 0 0 ψ(x + L) = e ig R L 0 dx A1(x) ψ(x + L) = e iαl ψ(x) { H ψl (x) = +p + α (n Z) ψ(x) H ψ R (x) = p ψ(x) E=0 α π L : L R L R L R Sphaleron process and Leptogenesis 17/73

18 3. Sphaleron Process Sphaleron σφαλϵρos = ready-to-fall, deceitful [cf. a sphalt] [Klinkhamer and Manton, Phys. Rev. D30 ('84)] 4-dim. SU() gauge + 1-doublet Higgs -dim. U(1) gauge-higgs model -dim. O(3) nonlinear sigma model -Higgs-Doublet Model MSSM with V eff (T ) Next-to-MSSM [Klinkhamer and Manton, Phys. Rev. D30 ('84)] [Bocharev and Shaposhnikov, Mod. Phys. Lett. A ('87)] [Mottola and Wipf, Phys. Rev. D39 ('89)] [Kastening, Peccei and Zhang, Phys. Lett. B66 ('91)] [Moreno, Oaknin and Quiros, Nucl. Phys. B483 ('97)] [KF and Senaha, hep-ph/0905.0] [KF, Kakuto, Tao and Toyoda, Prog. Theor. Phys. 114 ('05)] Sphaleron process and Leptogenesis 18/73

19 (saddle point) = least-energy path maximum-energy configuration Energy vacuum configuration space N CS =1 vacuum N CS =0 least-energy path/gauge trf. = noncontractible loop highest symmetry config. [Manton, Phys. Rev. D8 ('83)] Sphaleron process and Leptogenesis 19/73

20 [1/volume/time] SU() broken phase WKB approx. of ImF (T ) ( Γ (b) sph (T ) k N ω αw (T )T tr N rot π 4π [Arnold and McLerran, Phys. Rev. D36 ('87)] ) 3 e E sph/t zero modes: N tr = 6, N rot = for λ = g negative mode: ω ( )m W for 10 λ/g 10 symmetric phase Γ (s) sph (T ) κ(α W T ) 4 k O(1) MC simulation N CS (t)n CS (0) N CS + Ae ΓV t κ = 1.09 ± 0.04 SU() pure gauge [Ambjørn and Krasnitz, P.L.B36('95)] Sphaleron process and Leptogenesis 0/73

21 H(t) < Γ(T ) = g k B = 1 f = F d 3 V = ±g T p [ ] (π) log 1 e (ϵ p µ)/t 3 d 3 p ϵ p ϵ = g (π) 3 e (ϵp µ)/t 1 d 3 p 1 n = g (π) 3 e (ϵp µ)/t 1 s = f T ϵ p = p + m, µ = Sphaleron process and Leptogenesis 1/73

22 f(= P ) ϵ n s (T m, µ) (T m) { } 1 π g 7/8 90 T 4 { } ( ) 1 π 3/ mt g 7/8 30 T 4 g m e (m µ)/t π { } ( ) 3/ 1 ζ(3) mt g 3/4 π T 3 g e (m µ)/t = ϵ π m { } 1 π g 7/8 45 T 3 m T m: n T 3 T m: kinetic equilibrium e m/t = Sphaleron process and Leptogenesis /73

23 t [T m] ( ) 1 = t λ : mean free path = σ = n(t ) g n ζ(3) π T 3 σ λ = 1 n(t ) g n = B g B F g F σ λ m I T σ α s α T (α = e 4π ) s m I T... t = λ 10 gt 3 ( α T ) 1 = 10 gα T Sphaleron process and Leptogenesis 3/73

24 H(T ) = 8πG 3 ρ(t ) 1.66 g T m P ȷ g = m P = GeV H(T ) 1 t λ = 1 σn(t ) 1 α T t (sym) sph 1 αw 4 T t (br) sph 1 αw 4 T ee sph/t m P 1.66 g T GeV 1 at T = 100GeV 1 10GeV 1 (strong EW int.) 10 3 GeV 1 T = T C 100GeV = SU() L U(1) Y Φ T 0 T > T C = t QCD < t EW < t (sym) sph H(T ) 1... T < T C = t QCD < t EW H(T ) 1... Sphaleron process and Leptogenesis 4/73

25 log t Hubble sphaleron electroweak GeV 10 1 GeV T c log a ~ log(t 1 ) v(t C ) 00 T dec < T < T C = t (br) sph > H(T ) 1 T dec Sphaleron process and Leptogenesis 5/73

26 B (B L) [ ] Q i [H, Q i ] = 0 Z(T, µ) Tr e (H P i µ iq i )/T Q i (T, µ) = T µ i log Z(T, µ) Q i µ i LFV : Q i = B/N f L i, unbroken gauge charge Z(T, µ) (... ) [Khebnikov & Shaposhnikov, Phys. Lett. B387 ('96); Laine & Shaposhnikov, Phys. Rev. D61 ('00) ] µ Q i µ µ µ A + µ B = µ C Sphaleron process and Leptogenesis 6/73

27 massless free-field approximation N = n n = m T µ T d 3 [ ] k 1 (π) 3 e (ω k µ)/t 1 1 e (ω k+µ)/t 1 T 3 [ x dx π 0 e x µ/t 1 x ] e x+µ/t 1 T 3 3 µ T, (bosons) T 3 6 µ T, (fermions) cf. s = π 45 g T 3 N s µ T [Dreiner & Ross, Nucl. Phys. B410 ('93)] Sphaleron process and Leptogenesis 7/73

28 µ [Harvey & Turner, Phys. Rev. D4 ('90)] N N H Higgs doublets (ϕ 0 ϕ ) W u L(R) d L(R) e il(r) ν il ϕ 0 ϕ µ W µ ul(r) µ dl(r) µ il(r) µ i µ 0 µ (3N + 7) µ s W ϕ 0, color, charge neutrality µ gluon = µ Z,γ = 0 quark mixing, LFV { gauge: µw = µ dl µ ul = µ il µ i = µ + µ 0 N + Yukawa: µ 0 = µ ur µ ul = µ dl µ dr = µ il µ ir N N + 7 (N + ) = N + 3 µ : (µ W, µ 0, µ ul, µ i ) sphaleron process : 0 i (u L d L d L ν L ) i N(µ ul + µ dl ) + i µ i = 0 Sphaleron process and Leptogenesis 8/73

29 [T /6 ] B = N(µ ul + µ ur + µ dl + µ dr ) = 4Nµ ul + Nµ W, L = i (µ i + µ il + µ ir ) = 3µ + Nµ W Nµ 0 Q = 3 N(µ u L + µ ur ) N(µ d L + µ dr ) 3 i (µ il + µ ir ) µ W N H µ = Nµ ul µ (4N N H )µ W + (4N + N H )µ 0 I 3 = 1 N(µ u L µ dl ) i (µ i µ il ) µ W 1 N H(µ 0 + µ ) = (N + N H + 4)µ W µ i µ i Sphaleron process and Leptogenesis 9/73

30 T > T C (symmetric phase) Q = I 3 = 0 (µ W = 0) B = 8N + 4N H N + 13N H (B L), L = 14N + 9N H N + 13N H (B L) T < T C (broken phase) Q = 0 and µ 0 = 0 (... ϕ 0 condensates) B = 8N + 4(N H + ) 4N + 13(N H + ) (B L), L = 16N + 9(N H + ) 4N + 13(N H + ) (B L) (B L) primordial = 0 B = L = 0... (i) (ii) B L 0 B + L Sphaleron process and Leptogenesis 30/73

31 4. Requirements for EW Baryogenesis (1) = ( (B + L) 0) () = SUSY-SM, extra Higgs, (3) t EW = 10GeV < t (sym) sph = 103 GeV H(T ) 1 = GeV at T 100GeV = Sphaleron process and Leptogenesis 31/73

32 B 0 B = 0 v v C s T R L B = 0, broken phase s T R R v w s R R L symmetric phase B 0 ψ R v co z bubble wall (= Higgs + gauge config.) B-conserving ψ L ψ R chiral charge (Q L Q R ) flux (Q L Q R ) (RR L s RL R) s µ B 0 ṅ B = µ B T Γ(s) sph = Sphaleron process and Leptogenesis 3/73

33 m h > 114GeV (LEPII) = q W U m j U q m i m j O(α W ) [Farrar and Shaposhnikov, Phys. Rev. D50 ('94)] Weak int. QCD correction (short range) decoherence bubble wall n B < 10 6 s [Gavela, et al., Nucl. Phys. B430 ('94)] [Huet and Sather, Phys. Rev. D51 ('95)] 1st-order EWPT & CP violtion Sphaleron process and Leptogenesis 33/73

34 boson loop V eff (v; T ) T ( m(v) ) 3/ Higgs boson boson m(v) g v (for v 0) HDM, SUSY-SM sfermion [light stop m t 1 < m t ] m(v) = m 0 + g v (m 0 g v0) NMSSM [KF, Tao and Toyoda, PTP 114 ('05)] sphaleron decoupling condition scalar self-interaction complex parameters λ 6,7 in HDM, µb, A in SUSY-SM complex Majorama mass gaugino, Higgsino soft masses in SUSY-SM complex T C Sphaleron process and Leptogenesis 34/73

35 { I. chiaral charge flux II. diffusion equation I. chiral charge flux diffusion eq. source term (i / m(x)) ψ(x) = 0 m(x) = CP-violating wave equation { y ϕ(x) = y v(x)e iθ(x) ϕ(x) mass matrix [SUSY] [Nelson, et al, Nucl. Phys. B373 ('9); KF, et al, Prog. Theor. Phys. 95 ('96)] expansion w.r.t. Im m(x) [KF, et al, Phys. Rev. D50 ('94)] expansion w.r.t. m(x) [Huet and Nelson, Phys. Lett. B355 ('95); Phys. Rev. D53 ('96)] derivative expansion [Carena, et al., Nucl. Phys. B503 ('97)] Nonequilibrium field theory Closed-Time-Path formalism [Riotto, Phys. Rev. D53 ('96): Kainulainen, et al., JHEP 06 ('01); Phys. Rev. D66 ('0)] Sphaleron process and Leptogenesis 35/73

36 II. diffusion equation symmetric phase Q i Q i (t, x) = D Qi Q i j Γ ij c j Q j + [source term] D Q : Q diffusion const. mean-free path Γ ij : Q i c i [Cohen, Kaplan, Nelson, Phys. Lett. B336 ('94); Joyce, Prokopec, Turok, Phys. Rev. D53 ('96)] ( Q(t, x) = 0) (Γ 0) D Q Sphaleron process and Leptogenesis 36/73

37 {u L, d L, u R, d R, ν L, e L, e R, ϕ 0, ϕ, W, Z} B = L = 0 Y µ B Q a = {B L, Y, I 3, B} µ B L, µ Y, µ I3, µ B : n i = T 6 k iµ i = T 6 k { i qi a ki = 1(Weyl fermion) µ Q a k i = (complex boson) a q a i = i Qa n ul (d L ) = T ( µ B µ B L µ Y + ( ) 1 ) µ I 3, n ur = T ( µ B µ B L + ) 3 µ Y, n dr = T ( µ B µ B L 1 ) 3 µ Y, n νl (e L ) = T ( µ B L 1 6 µ Y + ( ) 1 ) µ I3, n er = T 6 ( µ B L µ Y ), n ϕ 0 (ϕ ) = T ( 1 3 µ Y + ( ) 1 ) µ I3, n W = T 3 ( µ I 3 ). Sphaleron process and Leptogenesis 37/73

38 Q a : Q a = i q a i n i = T 6 k i qi a qi b µ Qb i,b µ Y = µ I3 B = ( ) T color nul + n dl + n ur + n dr = 6 ( 4 3 (µ B + µ B L ) + ) 3 µ Y, B L = B T 6 ( 3µ B L µ Y ), [ 1 Y = 3 6 (n u L + n dl ) + 3 n u R 1 ] 3 n d R 1 (n ν L + n el ) n er m (n ϕ 0 + n ϕ ) = T [ 6 3 µ B + 8 ( ) ] 10 3 µ B L m µ Y (m = ) B = L = 0 µ B L = 3 µ Y, µ B = 1 6 µ Y Y = T 6 ( m + 5 ) µ Y 3... µ B = Y (m + 5/3)T Sphaleron process and Leptogenesis 38/73

39 n B = 3 Γ sph T dt µ B = 3 Γ sph (m + 5/3)T 3 z/vw dt ρ Y (z v w t) v w ρ Y (z): z Y = z Y ρ Y (z) v w O z z v w t z z/vw dt ρ Y (z v w t) = 1 v w 0 dzρ Y (z) F Y τ v w (F Y = injected flux) τ : diffusion length Sphaleron process and Leptogenesis 39/73

40 BAU: n B s 3N 100 π g κα 4 W F Y v w T 3 τt N O(1), τ m.f.p. τt { 1 for quarks for leptons MC simulation = forward scattering enhanced : for top quark τt max. at v w 1/ 3 for this optimal case n B s 10 3 F Y v w T 3 F Y v w T 3 O(10 7 ) = Sphaleron process and Leptogenesis 40/73

41 toy model m(z) = m tanh(az) exp ( iπ 1 tanh(az) ) no CP violation in the broken phase [z ] R R s R L R s R L [KF et al., Phys. Rev. D50 ('94); Prog. Theor. Phys. 95 ('96)] wall width wave length of the carrier R O(1) R a=1 a= 0.5 m =1 0 a= p L /m 0 Sphaleron process and Leptogenesis 41/73

42 chiral charge flux F Q T 3 (Q L Q R ) [dimensionless] at T = 100GeV log (m /T) v = 0.1 w log (a/t) log (m /T) v = 0.58 w log (a/t) log (m /T) v = 0.9 w log (a/t) n B s 3 N 100 π g κα 4 W F Y optimal τt 10 3 v w T 3 F Y v w T 3 O(1) Sphaleron process and Leptogenesis 4/73

43 5. Leptogenesis 5.1 Seesaw mechanism 5. CP violation in N R -decay 5.3 Calculation of generated lepton number review articles Buchmüller, Di Bari and Plümacher, Ann. Phys. 315 ( 05) Buchmüller, Peccei and Yanagida, Ann. Rev. Nucl. Part. Sci. 55 ( 05) [hep-ph/050169] Buchmüller, hep-ph/ Sphaleron process and Leptogenesis 43/73

44 5.1 Seesaw mechanism Minimal SM + singlet N R (-spinor notation) L Y = y ij ϵ ab l ail e c jrφ b h ij ϵ ab l ail N c jr Φ b 1 M ijn c irn c jr + h.c. SU() doublets: l L = ( νl e L ), Φ = ( ) ϕ 0, Φ = iτ Φ = ϕ ( ) ϕ + ϕ 0 y h M N f N f h 0 M 0 = L-violation T = 0 vacuum: Φ = ( ) 0 v 0 / L Y y ijv 0 e il e c jr h ijv 0 ν il N c ir 1 M ijn c irn c jr + h.c. = e T Lm e e c R ν T Lm ν N c R 1 N c T R MN c R + h.c. Sphaleron process and Leptogenesis 44/73

45 m e m ν U (e) L m eu (e) R = diag(m e, m µ, m τ ), S L m ν S R = Λ D = diagonal e c R = U (e) R e c R, M = S T R MS R e L = U (e)t L e L, N c R = S R N c R, ν L = S T Lν L V = L m = m i e e T il e c ir 1 ( ν T L N ct R ) ( 0 Λ D Λ D M ) ( ν L N c R ) + h.c. ( ) 1 1 Λ D M M 1 V V = 1 + O(Λ D Λ D 1 M ) ( ) ( ) V T 0 ΛD ΛD M V 1 Λ D 0 M 0 M Λ D T T L (Λ D M 1 Λ D )T L = Λ l, T T R MT R = Λ h L ν m = 1 ( ν T L N ct R ( ) ( ) ( ) V T L 0 Λl 0 T ) 0 TR L 0 0 Λ h 0 T R V ( ν L N c R ) + h.c. Sphaleron process and Leptogenesis 45/73

46 mass eigenstates : { ηll = T [ L ν L Λ D ( M ] 1 ) N R c η hl = T [ R N c R + ( M ] 1 ) Λ D ν L L CC g [ē L σ µ ν L + ν L σ µ ē L ] Wµ + h.c. g [ ] ē L σ µ (U (e) L ST LT L )η ll + η ll σ µ (TL T S L U (e) L )ē L Wµ + h.c. (U MNS ) fi = ( ) U (e) L ST L T L fi f =lepton flavor, i =mass eigenstate 3 CP phases = { ΛD U MNS Sphaleron process and Leptogenesis 46/73

47 5. CP violation in N R -decay T > O(100)GeV symmetric phase gauge boson, lepton massless Higgs boson (ϕ 0, ϕ ) heavy neutrino N decay asymmetry { Γ(Ni e j ϕ+ ) = Γ(N i ν j ϕ 0 ) Γ(N i l j ϕ) SU() symmetry Γ(N i e + j ϕ ) = Γ(N i ν j ϕ 0 ) Γ(N i l j ϕ) partial decay asym. ε i j Γ(N i l j ϕ) Γ(N i l j ϕ) Γ(N i l j ϕ) + Γ(N i l j ϕ) total decay asym. ε i j Γ(N i l j ϕ) j Γ(N i l j ϕ) j Γ(N i l j ϕ) + j Γ(N i l j ϕ) Leptogenesis U MNS Sphaleron process and Leptogenesis 47/73

48 5.3 Calculation of generated lepton number [in comoving frame] dn ψ (t) dt + 3H(t)n ψ (t) = [γ(ψ i + j + ) γ(i + j + ψ)] i,j, [γ(ψ + a i + j + ) γ(i + j + ψ + a)] a,i,j, γ γ(ψ + a + b + i + j + ) = d p ψ d p a d p j (π) 4 δ 4 (p ψ + p a + p i p j ) M(ψ + a + b + i + j + ) f ψ f a f b (1 ± f i )(1 ± f j ) d p d3 p (π) 3 E p, f ψ (p, t) = ψ, ± = { boson fermion Sphaleron process and Leptogenesis 48/73

49 n ψ (t) = d 3 p (π) 3 f ψ(p, t) = 0 f eq 1 ± f eq ψ eq eq (1 ± fi )(1 ± fj ) = 1 e βe i e βej e βe ψ 1e βe i 1 e βe j 1 = e βe ψ γ(ψ i + j + ) γ(i + j + ψ) = 1 1 e βe ψ 1e βe i 1 e βe j 1 = f eq d p ψ d p i (π) 4 δ 4 (p ψ p i p j )f eq ψ [ M(ψ i + j + ) M(i + j + ψ) ] 1 e βe 1 = i f eq j eβe e βe 1 eq (1 ± fψ ) eq eq (1 ± fi )(1 ± fj ) Sphaleron process and Leptogenesis 49/73

50 unitarity: [Kolb and Wolfram, Appendix of Nucl. Phys. B17] i,j, M(ψ i + j + ) (1 ± f eq eq i )(1 ± fj ) = i,j, M(i + j + ψ) (1 ± f eq eq i )(1 ± fj )... i,j, [γ(ψ i + j + ) γ(i + j + ψ)] = 0 Sphaleron process and Leptogenesis 50/73

51 CP-symmetry n ψ n ψ f ψ (t) = f ψ(t), M(α β) = M(ᾱ β) n ψ n ψ γ(ψ i + j + ) γ(i + j + ψ) [ γ( ψ ī + j + ) γ(ī + j + ψ) ] = d p ψ (π) 4 δ 4 (p ψ p i p j ) = 0 { [ M(ψ i + j + ) M( ψ ī + j + ) ] f ψ (1 ± f i )(1 ± f j ) [ M(i + j + ψ) M(ī + j + ψ) } ] f i f j (1 ± f ψ ) Sphaleron process and Leptogenesis 51/73

52 f(t, p) n(t) Boltzmann eq. f(p, t) = n(t) n eq f eq (p) [#(elastic scatt.) #(inelastic scatt)] ṅ ψ (t) + 3H(t)n(t) [ n ψ = i,j, a,i, [ n eq ψ n ψ n a n eq ψ neq a γ eq (ψ i + j + ) n in j n eq i neq γ eq (ψ + a i + j + ) n in j γ eq ( ) = f eq (p) γ( ) j γeq (i + j + ψ) n eq i neq j γeq (i + j + ψ + a) ] ] Sphaleron process and Leptogenesis 5/73

53 n ψ (t) s entropy density: s = π 45 g T 3 g = B Hubble parameter in flat RD universe: H(t) g B F g F ( ) 1/ 8πG 3 ρ r = ( 8πG 3 π ) 1/ 30 g T 4 a(t) t 1/ T 1, H(t) = ȧ(t) a(t) = 1 t ṡ = 3 T sdt dt Y ψ n ψ s = 3sd log T dt = 3s 1 t = 3sH(t) ṅ ψ = sẏψ + ṡy ψ = sẏψ 3sH(t)Y ψ = sẏψ 3H(t)n ψ ṅ ψ (t) + 3H(t)n ψ (t) = sẏψ(t) Sphaleron process and Leptogenesis 53/73

54 t z = M T M =heavy-ν mass t z d dt = M dt T dt d dz = zd log T dt d dz = H(t)z d dz = ( 4π 3 45 g ) 1/ T m Pl z d dz = ( 4π 3 45 g ) 1/ M 1 d m Pl z dz s dy ψ dt = ( 4π 3 45 g ) 1/ π C M 4 1 z 4 dy ψ dz 45 g T 3 M 1 m Pl z dy ψ dz = ( π 45 g ) 3/ M 5 1 dy ψ π m Pl z 4 dz C = π ( ) π 3/ M 45 g m Pl [M m Pl ; cf. CDM] Sphaleron process and Leptogenesis 54/73

55 Boltzmann eq. C M 4 dy ψ z 4 dz = i,j, a,i, [ ] Y ψ Y eq γ eq (ψ i + j + ) Y iy j ψ Y eq i Y eq j γeq (i + j + ψ) [ ] Y ψ Y a Y eq ψ Y a eq γ eq (ψ + a i + j + ) Y iy j Y eq i Y eq j γeq (i + j + ψ + a) (ψ, a, i, j) = N i, l, l, ϕ, ϕ Sphaleron process and Leptogenesis 55/73

56 System of (N i, l, l, ϕ, ϕ) L = ±1 : N i lϕ, N i l ϕ, lϕ N i, l ϕ N i L = ± : lϕ l ϕ, l ϕ lϕ T m ϕ n eq l = n eq l = ζ(3) ( ) 3 π 4 3 gen isospin T 3, n eq ϕ = neq ϕ = ζ(3) π T 3 ν f eq (p) e E p/t n eq N = d 3 p (π) 3 e p +M /T = T 3 π = T 3 π z K (z) 0 dx x e x +z (z = M/T ) (K n (z) : modified Bessel function) Sphaleron process and Leptogenesis 56/73

57 C M i 4 dy Ni z 4 dz Boltzmann equations = Y N i Y eq N i [ γ eq (N i lϕ) + γ eq (N C = π l ϕ) ] ( π 45 g ) 3/ Mi m Pl + Y ly ϕ Y eq l Y eq ϕ γ eq (lϕ N i ) + Y ly ϕ Y eq l Y eq ϕ γ eq ( l ϕ N i ) C M i 4 dy l z 4 dz = Y N i Y eq N i γ eq (N i lϕ) Y ly ϕ Y eq l Y eq ϕ γ eq (lϕ N i ) + Y ly ϕ Y l eq Y ϕ eq γ eq ( l ϕ lϕ) Y ly ϕ Y eq l Y eq ϕ γ eq (lϕ l ϕ) C M i 4 dy l z 4 dz = Y N i Y eq N i γ eq (N i l ϕ) Y ly ϕ Y eq l Y eq ϕ γ eq ( l ϕ N i ) Y ly ϕ Y l eq Y ϕ eq γ eq ( l ϕ lϕ) + Y ly ϕ Y eq l Y eq ϕ γ eq (lϕ l ϕ) Y ϕ Y ϕ Sphaleron process and Leptogenesis 57/73

58 γ eq [f eq e E/T, 1 ± f eq 1] γ eq (N lϕ) = = = p 1 d 3 p 1 e E 1/T (π) 3 E 1 d p 1 f eq N (p 1)(π) 4 δ 4 (p 1 p p 3 ) M(N lϕ) d 3 p 1 (π) 3 E 1 e E 1/T M Γ rs (N lϕ) d 3 p 1 (π) 3 M E 1 e p 1 +M /T = M π d 3 p (π) 3 E d 3 p 3 (π) 3 E 3 (π) 4 δ 4 (p 1 p p 3 ) M(N lϕ) 0 decay width in the rest frame of N p dp p + M e p +M /T = T 3 π z K 1 (z) CPT-inv. γ eq (N lϕ) = γ eq ( l ϕ N) = T 3 π z K 1 (z) Γ rs (N lϕ) γ eq (N l ϕ) = γ eq (lϕ N) = T 3 π z K 1 (z) Γ rs (N l ϕ) Sphaleron process and Leptogenesis 58/73

59 Γ rs (N lϕ) Γ rs (N l ϕ) L Y = h ij ( N i 1 γ 5 ) ν j ϕ 0 N 1 γ 5 i e j ϕ + h ij ( ν j 1 + γ 5 ) N i ϕ γ 5 ē j N i ϕ amplitudes: p l j im(n i l j ϕ) = N i p 1 p 3 φ l j l j l j l j = N i N i N k N i l m l m N k N i l m N k φ φ φ φ im 0 + im A + im B + im C + Sphaleron process and Leptogenesis 59/73

60 Feynman rule for Majorana fermions Majorana 4-fermion N(x): N c (x) = C( N(x)) T = N(x), or ( N(x)) T = C 1 N(x) propagator: N a Nb = ( ) i, p/ M ab N a N b = C bc N a Nc = end point of an external line: ( ) i p/ M CT, N a Nb = ab ( C 1 ) i p/ M ab N a p, s = U s a(p), p, s N a = Ū s a(p), Na p, s = ( C 1) ab U s b (p), p, s N a = Ū s b (p) ( C T ) ba Dirac fermion: ψ a p, s = u s a(p), p, s ψ a = ū s a(p) : fermion in p, s ψ a p, s = v s a(p), p, s ψ a = v s a(p) : anti-fermion in p, s Sphaleron process and Leptogenesis 60/73

61 tree level : im 0 = ih ij ū s j (p ) 1 + γ 5 Ui s (p 1 ) one-loop : d im A = h im h kmh 4 k kj (π) 4 ūs j (p ) 1 + γ 5 M k k Mk ) ū s Ui s (p 1 ) = i(hh ) ik h kj C( M k M i j (p ) 1 + γ 5 k/ p/ 3 (k p 3 ) U s i (p 1 ) 1 (k + p ) im B = h im h kmh kj d 4 k (π) 4 ūs j (p ) 1 + γ 5 M k p 1 M k k/ k U s i (p 1 ) 1 (k + p 1 ) = i(hh ) ik h kj A(Mi M i M k ) Mi M k ū s j (p ) 1 + γ 5 Ui s (p 1 ) im C = h imh km h kj d 4 k (π) 4 ūs j (p ) 1 + γ 5 p/ 1 p 1 M k k/ k U s i (p 1 ) 1 (k p 1 ) M i = i(hh ) ki h kj A(Mi ) Mi M k A(p ) C(ξ) ū s j (p ) 1 + γ 5 Ui s (p 1 ) Sphaleron process and Leptogenesis 61/73

62 A(p ) = C(ξ) = 1 16π ξ 16π dx x [ log(x x ) + log( p iϵ) ] dx 1 x 0 dy 1 x (1 x y)ξ xy iϵ A(p ) is MS-regulated, while the Im-part is finite. total decay width tree-level contribution [ Γ(Ni l j ϕ) + Γ(N i l ] j ϕ) j = M i = 1 8π (hh ) ii M i d 3 p (π) 3 E d 3 p 3 (π) 3 E 3 (hh ) ii (p 1 p )(π) 4 δ 4 (p 1 p p 3 ) Sphaleron process and Leptogenesis 6/73

63 [ Γ(Ni l j ϕ) Γ(N i l ] j ϕ) j = 1 4π = [ ((hh M i Im ) ] [ M i M k ) ki M k i i M k [ ((hh Im ) ] ) ki [f(ξ k ) + g(ξ k )] M i (8π) k i ξ k M k/m i, f(ξ) = ξ ImA(M ) = 1 16π ξ ImC(ξ) = 16π ξ = 16π ξ = 16π )] M ImA(Mi ) + ImC( k Mi [ 1 (1 + ξ) log 1 + ξ ], g(ξ) = ξ dx x Im log( M iϵ) = 1 16π ( π) = 1 16π dx dx 1 x 0 1 x 0 dx 1 x x + ξ 1 x dy Im (1 x y)ξ xy iϵ dy π(1 x)δ ((1 x)ξ (x + ξ)y) 1 x 0 dy δ(y 1 x x + ξ ) = ξ 16π 1 0 dx ξ 1 ξ ( ) 1 + ξ x + ξ 1 Sphaleron process and Leptogenesis 63/73

64 N i ε i = 1 8π(hh ) ii k i Γ(N i lϕ) = 1 + ε i Γ(N i l ϕ) = 1 ε i [ ((hh Im ) ] ) ki [f(ξ k ) + g(ξ k )] Γ = (hh ) ii 16π (1 + ε i)m i Γ = (hh ) ii 16π (1 ε i)m i γ eq (N i lϕ) = γ eq ( l ϕ N i ) = (hh ) ii 3π 3 T 4 z 3 K 1 (z) (1 + ε i ) γ eq (N i l ϕ) = γ eq (lϕ N i ) = (hh ) ii 3π 3 T 4 z 3 K 1 (z) (1 ε i ) z = M i T Sphaleron process and Leptogenesis 64/73

65 L = ±-scattering terms: N k N k M os (lϕ lϕ) = M(lϕ N i ) πδ(s M i ) M i Γ i M(Ni l ϕ) 1 ϵ i 4 πδ(s Mi ) Γ i M i Γ i l i l j l i l j γ eq (lϕ l ϕ) = N k N k on-shell φ = T [ ] 4 (hh ) kk 3π 16π γ eq ( l ϕ lϕ) = T [ ] 4 (h h T ) kk 3π 16π φ 0 0 φ dt t K 1 (t) f(t /z ) dt t K 1 (t) f(t /z ) φ [See, Appendix of Buchmüller, et al., Ann. Phys. 315; Kolb and Wolfram, Nucl. Phys. B17 ( 80), for detail] Sphaleron process and Leptogenesis 65/73

66 toy model with flavors M 1 = 10 6 m Pl, M /M 1 = 10, ε 1 = ε = 10 8 initial conditions: Y N = Y eq N, Y l = Y l = Y eq l, Y ϕ = Y ϕ = Y eq ϕ at z = M 1 /T = x10-1x10-3 1x10-4 1x10-5 1x10-6 1x10-7 1x10-8 1x10-9 1x x x10-1 1x10-13 (hh ) 11 =(hh ) =10 3 Y L Y eq N Y N 1x z = M 1 /T Y eq N 1 Y N1 1x10-1x10-3 1x10-4 1x10-5 1x10-6 1x10-7 1x10-8 1x10-9 1x x x10-1 1x10-13 (hh ) 11 =(hh ) =10 6 Y eq N Y L 1x z = M 1 /T Y eq N 1 Y N Y N1 Sphaleron process and Leptogenesis 66/73

67 1x10-9 Y l Y l Y ϕ Y ϕ constant (hh ) = 100(hh ) 11 (hh ) =(hh ) 11 L Y L 1x10-10 (hh ) = 10(hh ) 11 (hh ) =10 (hh ) 11 (hh ) 11 (hh ) (hh ) =10 1 (hh ) 11 1x10-1x10-3 (hh ) 11 =(hh ) =10 1x x10-6 1x10-5 1x10-4 1x10-3 1x10-1x10-1 (hh ) 11 1x10-4 1x10-5 Y N Y N1 1x10-6 1x10-7 1x10-8 1x10-9 1x x10-11 Y L Y N = 0 N L 1x10-1 1x x z = M 1 /T Sphaleron process and Leptogenesis 67/73

68 { hij ν L N R M i U MNS y ij (l L e R ), h ij, M ij GUTs lepton sector: (y ij, h ij ) quark sector: (y (d), y (u) ) (,3)-model N i ν i [Endo, et al., Phys. Rev. Lett. 89 ('0)] oscillation data : h O(1) M GeV m ν (hv 0 ) /M h O(0.01) M GeV h M < T R... e.g. weak scale M EW leptogenesis [Hill, Murayama, Perez, hep-ph/050448] Sphaleron process and Leptogenesis 68/73

69 6. Discussions Baryogenesis, Leptogenesis Sphaleron process and Leptogenesis 69/73

70 Leptogenesis, GUTs baryogenesis, CDM abundance spatially uniform system Boltzmann equation f(p, t) n(t) n eq f eq (p) Full Boltzmann equations Garayoa, Pastor, Pino, Rius and Vives, hep-ph/ Basbøll and Hannested, JCAP 0701 ('07) [hep-ph/060905] full Boltzmann K = Γ(N lϕ) H(T = M R ) integrated Boltzmann large K large Yukawa coupling Sphaleron process and Leptogenesis 70/73

71 For small K ( 0.1): η L lϕ l ϕ effective for thermalization For K 1: For K > 1: η L N R (K = 1) η L = n L n γ K η full by full Boltzmann η int by integrated Boltzmann Sphaleron process and Leptogenesis 71/73

72 GUTs & EW baryogenesis, Leptogenesis, CDM abundance, etc formulations integrated Boltzmann equation full Boltzmann equation Kadanoff-Baym equation Sphaleron process and Leptogenesis 7/73

73 Sphaleron process and Leptogenesis 73/73

phaleron decoupling and E hase transitio 2/45

1/45 phaleron decoupling and E hase transitio 2/45 Sphaleron σφαλϵρos = ready-to-fall, deceitful [cf. a sphalt] [Klinkhamer and Manton, Phys. Rev. D30 ('84)] 4-dim. SU(2) gauge + 1-doublet Higgs 2-dim.