Probing the Early Universe with Baryogenesis & Inflation - PowerPoint PPT Presentation

Probing the Early Universe with Baryogenesis & Inflation Wilfried Buchmüller DESY, Hamburg ICTP Summer School,Trieste, June 2015

[adapted from Schmitz ’12] • When and how was the baryon asymmetry generated? • What caused inflation, and at which energy scale? • How are inflation, baryogenesis and dark matter related?

Outline • BARYOGENESIS 1. Electroweak baryogenesis 2. Leptogenesis 3. Other models • INFLATION 1. The basic picture 2. Recent developments

BARYOGENESIS What is the origin of matter, i.e., the baryon-to-photon ratio η B = n B n γ = (6 . 1 ± 0 . 1) × 10 − 10 ? ? ? ? Key references A. D. Sakharov, JETP Lett. 5 (1967) 24 G. ‘t Hooft, Phys. Rev. Lett. 37 (1976) 8 V. A. Kuzmin, V. A. Rubakov and M. E. Shaposhnikov, Phys. Lett. B 155 (1985) 36 J. A. Harvey and M. S. Turner, Phys. Rev. D 42 (1990) 3344

Sakharov’s conditions Necessary conditions for generating a matter-antimatter asymmetry: • baryon number violation • C and CP violation • deviation from thermal equilibrium Check of 3rd condition: h B i = Tr( e − β H B ) = Tr( ΘΘ − 1 e − β H B ) = � Tr( e − β H B ) , Θ = CPT Alternative mechanisms: dynamics of scalar fields, e.g. Affleck-Dine baryogenesis, heavy moduli decay, ...

Sphaleron processes Baryon and lepton number not conserved in Standard Model, X � � µ = 1 J B q L γ µ q L + u R γ µ u R + d R γ µ d R , 3 generations X � � J L µ = l L γ µ l L + e R γ µ e R , generations divergence given by triangle anomaly, ∂ µ J B µ = ∂ µ J L µ ⇣ B µ ν ⌘ = N f W Iµ ν + g 0 2 B µ ν e µ ν f − g 2 W I ; 32 π 2 N f : number of generations; W I µ , B µ : SU (2) and U (1) gauge fields, gauge couplings g and g 0 .

Change in baryon and lepton number related to the change in topological charge the gauge field, Z t f Z d 3 x @ µ J B B ( t f ) − B ( t i ) = dt µ t i = N f [ N cs ( t f ) − N cs ( t i )] , with g 3 Z d 3 x ✏ ijk ✏ IJK W Ii W Jj W Kk N cs ( t ) = 96 ⇡ 2 E T=0, sphaleron / E sph − 1 0 1 W T=0, instanton Non-abelian gauge theory: ∆ N cs = ± 1 , ± 2 , . . . . Jumps in Chern-Simons asso- ciated with changes of baryon and lepton number, ∆ B = ∆ L = N f ∆ N cs .

s L s t L L c L b L Sphaleron d b L L d ν τ L u ν µ L ν e In SM, e ff ective 12-fermion interaction Y O B + L = ( q Li q Li q Li l Li ) , ∆ B = ∆ L = 3 i u c + d c + c c → d + 2 s + 2 b + t + ν e + ν µ + ν τ

Sphaleron rate crucially depends on temperature, only relevant in early universe: Γ ⇠ e − S inst = e − 4 π α = O 10 − 165 � � zero temperature: = κ M 7 Γ B + L W ( α T ) 3 exp ( � β E sph ( T )) , EW phase transition: V E sph ( T ) ' 8 π with g v ( T ) Γ B + L = κ s α 5 T 4 ⇠ 10 − 6 high temperature phase: V “consensus” among theorists: B+L violating processes in thermal equilibrium in temperature range: T EW ∼ 100 GeV < T < T sph ∼ 10 12 GeV ... but no direct experimental evidence for instanton or sphaleron processes ...

Chemical potentials In SM, with Higgs doublet H and N f generations, there are 5 N f + 1 chemical potentials ( q i , ` i , u i , d i , e i ); for non-interacting gas of massless particles µ i give asymmetries in particle and antiparticle number densities, 8 ⇣ ( � µ i ) 3 ⌘ � µ i + O fermions , n i − n i = gT 3 < ⇣ ( � µ i ) 3 ⌘ 6 2 � µ i + O bosons , : Relations between the various chemical potentials: SU (2) instantons: X (3 µ qi + µ li ) = 0 i QCD instantons: X (2 µ qi − µ ui − µ di ) = 0 i

vanishing hypercharge of plasma: ✓ ◆ µ qi + 2 µ ui − µ di − µ li − µ ei + 2 X µ H = 0 N f i Yukawa interactions: µ qi − µ H − µ dj = 0 , µ qi + µ H − µ uj = 0 , µ li − µ H − µ ej = 0 Relations determine B and L in terms of B-L : X X B = (2 µ qi + µ ui + µ di ) , L = (2 µ li + µ ei ) i i 8 N f + 4 B = c s ( B − L ) , L = ( c s − 1)( B − L ) , c s = 22 N f + 13 Electroweak baryogenesis and leptogenesis fundamentally different! In EWBG B-L conserved, generation of B in strong phase transition; in LG generation of B from initial generation of B-L (then unaffected by sphaleron processes)

I. Electroweak Baryogenesis Key references D. A. Kirzhnits and A. D. Linde, Phys. Lett. B 42 (1972) 471 A. G. Cohen, D. B. Kaplan and A. E. Nelson, Phys. Lett. B 336 (1994) 41 Reviews W. Bernreuther, Lect. Notes Phys. 591 (2002) 237 D. E. Morrissey and M. J. Ramsey-Musolf, New J. Phys. 14 (2012) 125003 T. Konstandin, Physics - Uspekhi 56 (8) 747 (2013)

V V T = T > T e ff e ff 2 C T 2 T C T = T C T = T < T T 1 C 1 0 0 φ φ φ c r i t 2nd order vs 1st order (electroweak) phase transition, as universe cools down. What determines the shape of the effective potential? How does the phasetransition proceed in an expanding universe? How can the complicated nonequilibrium process be calculated, where all masses are generated?

Finite-temperature effective potential Massive scalar field: ✓ 1 2( ∂ τ φ ) 2 + 1 ◆ Z 2( ∂ i φ ) 2 + V ( φ ) S β = β Z β V ( φ ) = 1 β = 1 2 µ 2 + λ Z Z 4 φ 4 , d 3 x , = d τ T β 0 Euclidean field theory with finite time range β ; add source term, calculate free energy (constant source, volume Ω ): Z Z Z β [ j ] = D φ exp ( � S β [ φ ] � j φ ) = exp ( � β Ω W β ( j )) , β β 1 ∂ W β Z = β Ω h φ ( x ) i ⌘ ϕ ∂ j β

Legendre transformation yields effective potential: j = − ∂ V β V β ( ϕ ) = W β ( j ) − ϕ j , ∂φ explicit calculation, high temperature expansion: V β ( ϕ ) = V T =0 ( ϕ ) � π 2 90 T 4 + 1 1 24 m 2 ( ϕ ) T 2 � 12 π m 3 ( ϕ ) T + . . . m ( ϕ ) m 2 ( ϕ ) = m 2 + 3 λϕ 2 , ⌧ 1 T = 1 2( m 2 + λ 4 T 2 ) ϕ 2 + λ 4 ϕ 2 + . . . 2nd term is free energy of massless boson; thermal bath generates “thermal mass” of boson; usefull concept to understand some effects in thermal field theory qualitatively, but different from kinematic mass; in gauge theories problem of gauge invariance ...

Higgs model & symmetry breaking In (Abelian) Higgs symmetry “broken” in ground state: Z ( D µ φ ) ∗ D µ φ + µ 2 | φ | 2 + λ | φ | 4 � � S β = , β √ µ 2 < 0 , � 1 / 2 ≡ ϕ 0 / − µ 2 / λ � D µ = ∂ µ + igA µ , Re φ 0 = 2 , √ m A = g ϕ 0 , m H = 2 λ ϕ 0 finite-temperature potential (with “barrier temperature”, where the barrier dissappears): 3 T ϕ 3 + λ V β ( ϕ ) = a b ) ϕ 2 − b 4 ϕ 4 + . . . 2( T 2 − T 2 ∂ 2 V β b = − µ 2 a = 3 g 2 16 + λ � ϕ =0 = 0 : T 2 a , � ∂ 2 ϕ 2 �

Cooling down, at critical temperature, Higgs vev jumps to critical vev: V β c ( ϕ c ) = V β c (0) : T 2 c � T 2 ' b 2 b a λ > 0 , T 2 c = 2 b ϕ c 3 λ T c phase transition weak for large Higgs mass (small coupling λ ); Standard model and extensions: “a” and “b” in effective potential more complicated functions of gauge and Yukawa couplings. Much work on effective potential (mostly mid-nineties): loop corrections, gauge dependence, infrared divergencies, treatment of Goldstone bosons, resummations, nonperturbative effects (gap equations, rigorous lattice studies!), beautiful work ... Baryogenesis needs strong phase transition: ϕ c > 1 T c not possible in SM, but possible in extensions ...

Phase diagram [Jansen ’96] [Csikor, Fodor, Heitger ’98] Nonperturbative effects change 1st order transition to crossover at critical Higgs mass: critical endpoint, lattice: R HW = m H m c , H = 72 . 1 ± 1 . 4 GeV m W ◆ 1 / 2 ✓ 3 gap equations, magnetic mass: m c H = ' 74 GeV , 4 π C m SM = Cg 2 T , C ' 0 . 35

Bubble nucleation & growth _ ~ T T C b u bb l e s fo r m li q u i d a n d e x p a n d T < T C no 1st-order phase transition in t = t 0 SM, but in extensions (singlet model, 2HDM,...) nucleation rate per volume: Γ V = A exp ( − Γ eff [ Φ ]) , t > t 0 Φ : saddle point of e ff ective action, interpolating between the two phases , Langer’s theory, ...

v W a ll unbroken phase broken phase q q φ = 0 q q becomes our world C P C P S p h _ ~ Γ B + L 0 _ _ q q _ _ q q S p h Γ B + L >> H CP violating scatterings at bubble wall (one-dimensional approximation): ✓ ◆ φ ( z ) = ρ ( z ) ρ ( z ) = v c 1 − tanh z 2 e i θ ( z ) , y ψ ¯ X L f = − ψ L ψ R φ , √ 2 L w ψ

Calculating the baryon asymmetry very difficult, series of approximations: Schwinger-Keldysh → Boltzmann equations → diffusion equations ... ; CP violating interactions with bubble wall generate in front of wall excess of left-handed “tops”, converted to baryon asymmetry by sphaleron processes; in frame of wall chemical potentials only depend on distance from wall: µ q L ( z ) = 3( µ q 1 ( z ) + µ q 2 ( z ) + µ q 3 ( z )) chemical potentials: = 3 ∂ n B n B Γ sph ⇣ ⌘ baryon number density: µ q L − κ cs T 2 2 ∂ t T X Γ ij µ j + · · · = S i di ff usion equations: v w ∂ z µ i − j Z ∞ n B = 3 k B = 3 κ cs Γ sph Γ sph µ q L ( z ) e − k B z , final result: 2 2 v w T 3 v w T 0 important parameters: critical Higgs vev, bubble wall velocity, bubble wall width, diffusion parameters, ...