phaleron decoupling and E hase transitio 2/45
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2 phaleron decoupling and E hase transitio 2/45
3 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. U(1) gauge-higgs model 2-dim. O(3) nonlinear sigma model 2-Higgs-Doublet Model MSSM with V eff (T ) Next-to-MSSM [Klinkhamer and Manton, Phys. Rev. D30 ('84)] [Bocharev and Shaposhnikov, Mod. Phys. Lett. A2 ('87)] [Mottola and Wipf, Phys. Rev. D39 ('89)] [Kastening, Peccei and Zhang, Phys. Lett. B266 ('91)] [Moreno, Oaknin and Quiros, Nucl. Phys. B483 ('97)] [KF, Kakuto, Tao and Toyoda, Prog. Theor. Phys. 114 ('05)] phaleron decoupling and E hase transitio 3/45
4 (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. D28 ('83)] phaleron decoupling and E hase transitio 4/45
5 B L µ j µ B+L = N f 16π 2[g2 2Tr(F µν F µν ) g 2 1B µν Bµν ] µ j µ B L = 0 N f = F µν 1 2 ϵµνρσ F ρσ B(t f ) B(t i ) = N f 32π 2 tf t i d 4 x [ g 2 2Tr(F µν F µν ) g 2 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 32π 2 d 3 x ϵ ijk [g 2Tr (F 2 ij A k 23 ) g 2A i A j A k g 21B ] ij B k t phaleron decoupling and E hase transitio 5/45
6 : E = 1 2 (E2 + B 2 ) = 0 F µν = B µν = 0 A µ = iu 1 µ U, B µ = µ v with U SU(2) U(x) : S 3 U SU(2) S 3 π 3 (S 3 ) Z = U(x) N CS ig2 3 d 3 x ϵ 48π 2 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 2π 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( ) = 2πN... N CS = N phaleron decoupling and E hase transitio 6/45
7 E N CS B 0 { e 2S instanton = e 8π2 /g 2 2 e e E sph/t... T > T C (B + L) 0 phaleron decoupling and E hase transitio 7/45
8 2.1 N CS [Atiyah and Singer, 1968] n R n L = ν = g2 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. B221 ('83)] p ǫ µν F µν 0 p vacuum particle production phaleron decoupling and E hase transitio 8/45
9 (1 + 1) Dirac eq. iγ µ ( µ iga µ (x))ψ(x) = 0 [γ 0 = σ 1, γ 1 = iσ 2 ; 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 = 2πn 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 L R L R L R phaleron decoupling and E hase transitio 9/45
10 2.2 fate of false vacuum T = 0: Callan-Coleman, Phys. Rev. D16 ('77); Coleman, The Uses of Instantons T 0: Affleck, Phys. Rev. Lett. 46 ('81) V(x) x metastable min. = false vacuum T = 0 : Γ 2 h ImE 0 ( ) 1/2 Scl e Scl/ h [1 + O( h)] 2π h E 0 = false vacuum localize S cl = bounce Euclid phaleron decoupling and E hase transitio 10/45
11 T 0 (harmonic osc. at x 0 ) 1 2 hω 0, T V 0 Γ(T ) 0 de e E/T Z 0 Γ(E), x 1 V(x) mω 02 =V (x 0 ) mω 2 = V (0) V 0 E x 0 x 2 O x 3 metastable min. = false vacuum x Z 0 = e hω0(n+1/2)/t = n=0 ( 2 sinh hω ) 1 0, Γ(E) = i h 2T 2m (ψ Eψ E ψeψ E ) T < hω 2π : T > hω 2π : Γ Im F Z 1 0 2π ht 2 h (E 0 ) 1/2 e S[x b]/ h Γ ω β π Im F Z 1 0 ω 4π sin(β hω /2) e βv 0 { xb (t) = bounce solution T (E) = period of the bounce phaleron decoupling and E hase transitio 11/45
12 SU(2) [Arnold and McLerran, Phys. Rev. D36 ('87)] broken phase WKB approx. of ImF (T ) Γ (b) sph (T ) k N tr N rot ω 2π ( αw (T )T 4π ) 3 e E sph/t zero modes: N tr = 26, N rot = for λ = g 2 negative mode: ω 2 ( )m 2 W for 10 2 λ/g 2 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 2 + Ae ΓV t κ = 1.09 ± 0.04 SU(2) pure gauge [Ambjørn and Krasnitz, P.L.B362('95)] phaleron decoupling and E hase transitio 12/45
13 2.3 H(t) < Γ(T ) = g k B = 1 f = F d 3 V = ±g T p [ ] (2π) log 1 e (ϵ p µ)/t 3 d 3 p ϵ p ϵ = g (2π) 3 e (ϵp µ)/t 1 d 3 p 1 n = g (2π) 3 e (ϵp µ)/t 1 s = f T ϵ p = p 2 + m 2, µ = phaleron decoupling and E hase transitio 13/45
14 f(= P ) ϵ n s (T m, µ) (T m) { } 1 π 2 g 7/8 90 T 4 { } ( ) 1 π 2 3/2 mt g 7/8 30 T 4 g m e (m µ)/t 2π { } ( ) 3/2 1 ζ(3) mt g 3/4 π T 3 g e (m µ)/t = ϵ 2 2π m { } 1 2π 2 g 7/8 45 T 3 m T m: n T 3 T m: kinetic equilibrium e m/t = phaleron decoupling and E hase transitio 14/45
15 t [T m] ( ) 1 = t λ : mean free path = σ = n(t ) g n ζ(3) π 2 T 3 σ λ = 1 n(t ) g n = B g B F g F σ λ m I T σ α2 s α2 T 2 (α = e2 4π ) s m I T... t = λ 10 gt 3 ( α 2 T 2 ) 1 = 10 gα 2 T phaleron decoupling and E hase transitio 15/45
16 H(T ) = 8πG 3 ρ(t ) 1.66 g T 2 m P ȷ g = m P = GeV H(T ) 1 t λ = 1 σn(t ) 1 α 2 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(2) 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... phaleron decoupling and E hase transitio 16/45
17 log t Hubble sphaleron electroweak GeV GeV T c log a ~ log(t 1 ) v(t C ) 200 T dec < T < T C = t (br) sph > H(T ) 1 T dec phaleron decoupling and E hase transitio 17/45
18 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 phaleron decoupling and E hase transitio 18/45
19 massless free-field approximation N = n n = m T µ T d 3 [ ] k 1 (2π) 3 e (ω k µ)/t 1 1 e (ω k+µ)/t 1 T 3 [ x 2 dx 2π 2 0 e x µ/t 1 x 2 ] e x+µ/t 1 T 3 3 µ T, (bosons) T 3 6 µ T, (fermions) cf. s = 2π2 45 g T 3 N s µ T [Dreiner & Ross, Nucl. Phys. B410 ('93)] phaleron decoupling and E hase transitio 19/45
20 Quantum number densities in terms of µ [Harvey & Turner, Phys. Rev. D42 ('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 + 2 Yukawa: µ 0 = µ ur µ ul = µ dl µ dr = µ il µ ir N N + 7 2(N + 2) = 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 + 2µ dl ) + i µ i = 0 phaleron decoupling and E hase transitio 20/45
21 [T 2 /6 ] B = N(µ ul + µ ur + µ dl + µ dr ) = 4Nµ ul + 2Nµ W, L = i (µ i + µ il + µ ir ) = 3µ + 2Nµ W Nµ 0 Q = 2 3 N(µ u L + µ ur ) N(µ d L + µ dr ) 3 i (µ il + µ ir ) 2 2µ W 2N H µ = 2Nµ ul 2µ (4N N H )µ W + (4N + 2N H )µ 0 I 3 = 1 2 N(µ u L µ dl ) i (µ i µ il ) 2 2µ W N H(µ 0 + µ ) = (2N + N H + 4)µ W µ i µ i phaleron decoupling and E hase transitio 21/45
22 T > T C (symmetric phase) Q = I 3 = 0 (µ W = 0) B = 8N + 4N H 22N + 13N H (B L), L = 14N + 9N H 22N + 13N H (B L) T < T C (broken phase) Q = 0 and µ 0 = 0 (... ϕ 0 condensates) B = 8N + 4(N H + 2) 24N + 13(N H + 2) (B L), L = 16N + 9(N H + 2) 24N + 13(N H + 2) (B L) (B L) primordial = 0 B = L = 0... (i) (ii) B L 0 B + L phaleron decoupling and E hase transitio 22/45
23 T = 100GeV, t EW 10GeV 1 H(T ) GeV 1 (M) Φ(x) = v F (M; T ) = a(t )M 2 + b(t )M 4 Effective potential V eff (v; T ) T = 0 Φ = v V eff (v) v [Coleman, Secret Symmetry] V eff (v) = Γ[φ(x) = v]/ d 4 x Γ[φ] = generating functional of 1PI vertex functions phaleron decoupling and E hase transitio 23/45
24 Effective potential = V eff (v; T ) = 1 V T log Z = 1 V T log Tr [ e H/T ] Φ =v v at T path integral: Tr(e H/T ) = N(T ) pbc ( [dϕ] exp [0, 1/T ] Tr 1/T 0 ) d 4 x E L E (ϕ) { ϕ(0, x) = ϕ(1/t, x) boson ψ(0, x) = ψ(1/t, x) fermion Fourier mode k 0 = iω n with ω n = { 2nπT (boson) (2n + 1)πT (fermion) phaleron decoupling and E hase transitio 24/45
25 V eff (v;t) V eff (0;T) T > T C > 0 T > T C > 0 v C v 0 v v 0 v order parameter: Φ = 1 ( ) 0 2 v v(t) v(t)... 1st order EWPT v 0 T C T v 0 v C T C T v C lim T T C v(t ) 0 2nd order PT 1st order PT phaleron decoupling and E hase transitio 25/45
26 T = 0 Matsubara frequency ω n d 4 k V eff (v) V eff (v; T ) by (4π) i d 3 k 4 β (2π) 3( ) n= 1-loop effective potentail i β n d 3 k (2π) 3 log(k2 m 2 ) = = d 4 k (4π) 4 log(k2 m 2 )± 2i β d 4 k (4π) 4 log(k2 m 2 )± it 4 π 2 0 k 0 =iω n d 3 ) k k (1 (2π) log e β 2 +m 2 3 x dx x 2 log (1 e 2 +(m/t ) 2) T = 0-contribution T = 0 counterterm T = 0 counterterm phaleron decoupling and E hase transitio 26/45
27 1-loop W Z V eff (φ; T ) = 1 2 µ2 φ 2 + λ ( ) φ 4 φ4 + 2Bv0φ Bφ [log 4 2 v ] + V (φ; T ) where B = 3 64π 2 v0 4 (2m 4 W + m 4 Z 4m 4 t), V (φ; T ) = T 4 I B,F (a) 2π [6I B(a 2 W ) + 3I B (a Z ) 6I F (a t )], (a A = m A (φ)/t ) ) x dx x 2 log (1 e 2 +a 2. 0 high-temperature expansion [m/t << 1] γ E = I B (a) = π4 45 +π2 12 a2 π ( 6 (a2 ) 3/2 a4 a 16 log 2 4π a4 γ E 3 ) + O(a 6 ) 16 4 ( I F (a) = 7π4 360 π2 24 a2 a4 a 16 log 2 π a4 γ E 3 ) + O(a 6 ) 16 4 phaleron decoupling and E hase transitio 27/45
28 T > m W, m Z, m t where V eff (φ; T ) D(T 2 T 2 0 )φ 2 E T φ 3 + λ T 4 φ4 D = 1 8v0 2 (2m 2 W + m 2 Z + 2m 2 t), E = 1 λ T = λ 3 16π 2 v 4 0 T 2 0 = 1 2D (µ2 4Bv 2 0), 4πv0 3 (2m 3 W + m 3 Z) 10 2 ( 2m 4 W log m2 W α B T + 2 m4 Z log m2 Z α B T 2 4m4 t log log α F (B) = 2 log (4)π 2γ E ) m2 t α F T 2 T C φ = 0 φ C : φ C = 2E T C λ TC Γ (br) sph < H(T C) φ C T C > 1 = λ [m H = 2λv 0 ] m h < 46GeV = m SM h > 114GeV by LEPII phaleron decoupling and E hase transitio 28/45
29 Monte Carlo simulations effective fermion mass : m f (T ) O(T ) ω n = (2n + 1)πT πt... { scalar fields: ϕ(x) (site) gauge fields: U µ (x) Z = [dϕ du µ ] exp { S E [ϕ, U µ ]} 3-dim. SU(2) system with a Higgs doublet and a triplet time-component of U µ [Laine & Rummukainen, hep-lat/ ] 4-dim. SU(2) system with a Higgs doublet Em h < 66.5 ± 1.4GeV [Csikor, hep-lat/ ] end-point m h = { 72.3 ± 0.7 GeV 72.1 ± 1.4 GeV no PT (cross-over) in the Standard Model phaleron decoupling and E hase transitio 29/45
30 Sphaleron decoupling condition broken phase beyond one-loop order [Arnold and McLerran, Phys. Rev. D36 ('87)] 1 db B dt 13N f ω 128π 2 αw 3 κn tr N rot e E sph/t H(T ) = 1.66 g T 2 /m P E sph 4πv g 2 E v T > E [ ( ) ω log(κn tr N rot ) + log m W 2 log ( T )] 100GeV E = 2.00, N tr N rot = 80.13, ω 2 = 2.3m 2 W ( at λ/g2 2 = 1), κ = 1, T = 100GeV, v T > ( ) = 1.20 main, zero modes 10% phaleron decoupling and E hase transitio 30/45
31 MSSM (Minimal Supersymmetric Standard Model) [ + 1 Higgs doublet] + superpartners { Supersymmetric Yukawa int. ( superpotential) gauge anomaly cancellation 2HDM (two-higgs-doublet Model) [ + 1 Higgs doublet], Φ 1, Φ 2 Yukawa int. or Higgs doublet Higgs potential variation NMSSM (Next-to-Minimal Supersymmetric Standard Model) MSSM + 1 Singlet Superfield [KF, Tao and Toyoda, Prog. Theor. Phys. 114 ('05)] phaleron decoupling and E hase transitio 31/45
32 Higgs potential (tree level) MSSM: superpotential W = f (e) AB H dl A E B + f (d) AB H dq A D B f (u) AB H uq A U B µh d H u V 0 = m 2 1Φ d Φ d + m 2 2Φ uφ u ( m 2 3Φ d Φ u + h.c. ) + g2 2 + g soft-susy-br. terms ( Φ d Φ d Φ uφ u ) 2 + g D-term potential Φ d Φ u 2 2HDM: gauge-inv. renormalizable potential V 0 = m 2 1Φ 1 Φ 1 + m 2 2Φ 2 Φ 2 + (m 2 3Φ 1 Φ 2 + h.c.) λ 1(Φ 1 Φ 1) λ 2(Φ 2 Φ 2) 2 + λ 3 (Φ 1 Φ 1)(Φ 2 Φ 2) λ 4 (Φ 1 Φ 2)(Φ 2 Φ 1) { 1 2 λ 5(Φ 1 Φ 2) 2 + [ λ 6 (Φ 1 Φ 1) + λ 7 (Φ 2 Φ 2) ] } (Φ 1 Φ 2) + h.c. 2HDM MSSM by Φ 1 Φ d, Φ 2 Φ u, λ 1 = λ 2 = λ 3 = g2 2 + g1 2, λ 4 = g , λ 5,6,7 = 0 phaleron decoupling and E hase transitio 32/45
33 Higgs particles in the MSSM (tree level) Φ d, Φ u : NG modes = 5 = 3 (neutral) + 2 (H ± ) V 0 V 0 = min at Φ d = 1 ( ) v0 cos β, Φ 2 0 u = 1 ( 0 2 v 0 sin β ) m 2 1 = m 2 3 tan β 1 2 m2 Z cos(2β), m 2 2 = m 2 3 cot β m2 Z cos(2β) Φ d,u m 2 3 V 0 Higgs fields mass matrix of the Higgs particles m 2 m 2 3 A = m 2 h,h = 1 2 sin β cos β [ m 2 Z + m 2 A ] (m 2 Z + m2 A )2 4m 2 Z m2 A cos2 (2β) CP-odd CP-even m 2 H ± = m 2 W + m 2 A phaleron decoupling and E hase transitio 33/45
34 ϕ 4 g2, 1 g1 2 m h min { m 2 Z, ma} 2, mh max { } m 2 Z, m 2 A new upper bound on m h : m 2 h m 2 Z cos 2 (2β) + 3 2π 2 m 4 t v 2 0 [Okada, Yamaguchi and Yanagida, Prog. Theor. Phys. 85 ('91)] log ( m 2 t + m2 t m 2 t ) h-h mixing ZZh-coupling e Z Z e + h phaleron decoupling and E hase transitio 34/45
35 allowed region for the lightest neutral Higgs boson allowed region for the pseudoscalar Higgs boson m t = GeV tan 10 (b) m h -max 99.7%CL Excluded by LEP 95%CL tanβ µ<0 no mixing m h - max D CDF Tevatron Preliminary ττ 95% CL Exclusion D (1.0 fb -1 ) CDF (1.8 fb -1 ) 1 Theoretically Inaccessible m h (GeV/c 2 ) 0 LEP 2 no mixing m A (GeV/c 2 ) m h -max benchmark scenario [PDG: C. Amsler et al., Phys. Lett. B667, 1 (2008)] phaleron decoupling and E hase transitio 35/45
36 MSSM EWPT order parameter: Φ d = 1 2 ( v1 0 ), Φ u = 1 2 ( 0 v 2 + iv 3 V eff (v; T ) v = (v 1, v 2, v 3 ) (at each T ) symmetric under Φ 1 Φ 2 ) ( ) = eiθ 0 2 v u m 1 = m 2 = m 3 = 0 (tan 4 β = λ 1 /λ 2 ) [KF, Kakuto, Takenaga, Prog. Theor. Phys. 91] Higgs mass 2 matrix: M 2 ij 2 V eff (T = 0) ϕ i ϕ j = Γ (2) 1PI (p = 0) { mhi g ZZHi phaleron decoupling and E hase transitio 36/45
37 V eff (v; T = 0) = V 0 (v) π 2 m 2 a: field (Φ d, Φ u )-dependent mass 2, : a = t, t, b, b, W, Z, [Carena, Ellis, Pilaftsis and Wagner, Nucl. Phys. B586 ('00)] a c a ( m 2 a) 2 ( log m2 a M c a : ) field parametrization VEV + fluctuation ( 1 ) ( 2 (v d + h d + ia d ) Φ d =, Φ u = e iθ ϕ d 1 ϕ + u 2 (v u + h u + ia u ) ) = V eff v d = v 0 cos β, v u = v 0 sin β, θ soft mass 0 = 1 Veff = m 2 1 Re(m 2 v d h 3e iθ )tan β + 1 d 2 m2 Zcos(2β) +, 0 = 1 Veff = m 2 2 Re(m 2 v u h 3e iθ )cot β 1 u 2 m2 Zcos(2β) +, 0 = 1 Veff = 1 Veff = Im(m 2 v u a d v d a 3e iθ ) +. u phaleron decoupling and E hase transitio 37/45
38 0 neutral Higgs boson charged Higgs bosonmass: M 2 = m 2 H ± = 2 Veff h 2 d 2 Veff h d h u 2 Veff h d a u 1 cos β 1 sin β cos β 2 Veff 2 V eff ϕ + d ϕ u h d h u 2 Veff h 2 u 1 2 Veff sin β h u a d input m 2 H ± Re(m 2 3e iθ ) = 1 cos β 1 sin β 1 sin β cos β [NG mode ] 2 Veff h d a u 2 Veff h u a d 2 Veff a d a u 1 sin β cos β Re(m2 3e iθ ) + m 2 W + mass eigenstates H i h d h u = O a H 1 H 2 H 3, O T M 2 O = diag(m 2 H 1, m 2 H 2, m 2 H 3 ) phaleron decoupling and E hase transitio 38/45
39 gauge and Yukawa interactions ( L gauge g 2 m W g V V Hi W µ + W µ + Z µz µ ) H 2 cos 2 i + θ W g ( 2 g ZHi H 2 cos θ j Z µ ) H i µh j W L Y g 2m b 2m W b(g S bbhi + iγ 5 g P bbh i )b H i corrections to the couplings [ : g V V H = 1, g ZHH = 0, g bbh = 1] g V V Hi = O 1i cos β + O 2i sin β g ZHi H j = 1 2 [(O 3iO 1j O 3j O 1i ) sin β + (O 3i O 2j O 3j O 2i ) cos β] g S bbh i = O 1i 1 cos β, gp bbh i = O 3i tan β, g 2 bbh i = ( g S bbh i ) 2 + ( g P bbhi ) 2 v 0 = 246GeV, tan β, m H ± µ, A q, scalar soft mass, m 2 1, m 2 2, m 2 3 Higgs mass vs EWPT phaleron decoupling and E hase transitio 39/45
40 light stop scenario [de Carlos, Espinosa, Nucl. Phys. B503 ('97)] stop mass 2 matrix: M 2 t = m2 t L + ( ) g g2 2 8 (v 2 u v 2 d ) + y2 t 2 v 2 u y t 2 (µv d + A(v 2 iv 3 )) m 2 t R g2 1 6 (v 2 u v 2 d )+ y2 t 2 v 2 u m 2 t L = 0 or m 2 t R = 0 = smaller eigenvalue: m 2 t 1 O(v 2 )... V t(v; T ) T 6π (m2 t 1 ) 3/2 T v 3 stronger 1st order PT effective for larger y t smaller tan β m t = 1 2 y t v 0 sin β phaleron decoupling and E hase transitio 40/45
41 (CP-conserving case) tan β = 6, m h = 82.3GeV, m A = 118GeV, m t 1 = 168GeV T C = 93.4GeV, v C = 129GeV [KF, Prog. Theor. Phys. 101 ('99)] v v 1 2.0x x x x x x x x x x x v C / TC tanβ GeV m~ tr (GeV) m~ tr (GeV) m ~ t1 m A m h V eff (v 1, v 2, v 3 = 0; T C ) m tr -dependence (tan β = 6) phaleron decoupling and E hase transitio 41/45
42 Lattice MC studies 3d reduced model strong 1st order for m t 1 < m t and m h 110GeV [Laine et al. hep-lat/ ] 4d model with SU(3), SU(2) gauge bosons, 2 Higgs doublets, stops, sbottoms A t,b = 0, tan β 6 [Csikor, et al. hep-lat/ ] m h v /T =1 C C m A = 500 GeV v C /T C > 1 below the steeper lines max. m h = 103 ± 4 GeV for m t L 560 GeV phaleron decoupling and E hase transitio 42/45
43 (i) B L (ii) B + L (i) Leptogenesis, (B L)-nonconserving GUTs, Affleck-Dine (ii) Higgs Standard Model X MSSM with m t 1 m t, m h 105GeV 2HDM NMSSM phaleron decoupling and E hase transitio 43/45
44 Sphaleron decoupling condition revisited (preliminary) collaboration with E. Senaha T C T N ( ) 3/2 Γ N (T ) T 4 Ecb (T ) e E cb(t )/T 2πT [Linde, Nucl. Phys. B216 ('83)] E cb (T ) : critical bubble V eff (T ) V eff (T N ) v N v ϕ (r = 0) = 0 ϕ(r = ) = 0 v N 0 r phaleron decoupling and E hase transitio 44/45
45 T N (nucleation temp.) Γ N (T N )H(T N ) 3 = H(T N ) E cb(t N ) T N 3 2 log E cb(t N ) T N = log g (T N ) 4 log T N 100GeV T C = GeV, v C = GeV T N = 116.9GeV, v C = 117.9GeV v C v N = 0.86 = % T C T N sphaleron decoupling condition E sph (T ) V eff (T ) zero-mode factor in progress phaleron decoupling and E hase transitio 45/45
1. Introduction overview of baryogenesis. Anomalous Baryon Number Nonconservation 3. Sphaleron Process 4. Requirements for EW Baryogenesis 5. Leptogen
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1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N m 1.
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[ ] (Ising model) 2 i S i S i = 1 (up spin : ) = 1 (down spin : ) (4.38) s z = ±1 4 H 0 = J zn/2 i,j S i S j (4.39) i, j z 5 2 z = 4 z = 6 3
![[ ] (Ising model) 2 i S i S i = 1 (up spin : ) = 1 (down spin : ) (4.38) s z = ±1 4 H 0 = J zn/2 i,j S i S j (4.39) i, j z 5 2 z = 4 z = 6 3](/thumbs/103/161261933.jpg "[ ] (Ising model) 2 i S i S i = 1 (up spin : ) = 1 (down spin : ) (4.38) s z = ±1 4 H 0 = J zn/2 i,j S i S j (4.39) i, j z 5 2 z = 4 z = 6 3")
4.2 4.2.1 [ ] (Ising model) 2 i S i S i = 1 (up spin : ) = 1 (down spin : ) (4.38) s z = ±1 4 H 0 = J zn/2 S i S j (4.39) i, j z 5 2 z = 4 z = 6 3 z = 6 z = 8 zn/2 1 2 N i z nearest neighbors of i j=1
遍歴電子磁性とスピン揺らぎ理論 - 京都大学大学院理学研究科 集中講義
email: takahash@sci.u-hyogo.ac.jp August 3, 2009 Title of Lecture: SCR Spin Fluctuation Theory 2 / 179 Part I Introduction Introduction Stoner-Wohlfarth Theory Stoner-Wohlfarth Theory Hatree Fock Approximation
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1 Chapter 1 Why supper symmetry? 2 Chapter 2 Representaions of the supersymmetry algebra SUSY Q a d 3 xj 0 α J x µjµ = 0 µ SUSY ( {Q A α,q βb } = 2σ µ α β P µδ A B (2.1 {Q A α,q βb } = {Q αa,q βb } = 0
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No.7, No.8, No.9 email: takahash@sci.u-hyogo.ac.jp Spring semester, 2012 Introduction (Critical Behavior) SCR ( b > 0) Arrott 2 Total Amplitude Conservation (TAC) Global Consistency (GC) TAC 2 / 25 Experimental
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0406_total.pdf
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QMI_09.dvi
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QMI_10.dvi
25 3 19 Erwin Schrödinger 1925 3.1 3.1.1 σ τ x u u x t ux, t) u 3.1 t x P ux, t) Q θ P Q Δx x + Δx Q P ux + Δx, t) Q θ P u+δu x u x σ τ P x) Q x+δx) P Q x 3.1: θ P θ Q P Q equation of motion P τ Q τ σδx
IA hara@math.kyushu-u.ac.jp Last updated: January,......................................................................................................................................................................................