Group Theoretic Approach to Theory of Fermion Production Minho Son - PowerPoint PPT Presentation

KEK-KIAS-NCTS Theory Workshop, Dec 4 - 7, 2018 Group Theoretic Approach to Theory of Fermion Production Minho Son Korea Advanced Institute of Science and Technology (KAIST) Based on Min, SON, Suh 1808.00939

Particle Production Relaxation with particle production + Leptogenesis SON, Ye, You 18’ Preheating via parametric resonance or • • Relaxation with particle production excitation in post-inflationary era Hook, Marques-Tavares 15’ SON, Ye, You 18’ Kofman, Linde, Starobinsky 97’ Fonseca, Morgante, Servant 18’ # ) or Axion-inflation via gauge boson ( 𝜚𝐺𝐺 … • % 𝜚 𝑘 %( ) production fermion ( 𝜖 Anbor, Sorbo 10’ Adshead, Pearce, Peloso, Roberts, Sorbo 18’ Gravitational waves from preheating • List goes on ….. Many literature (hard to list all here)

Particle Production Relaxation with particle production + Leptogenesis SON, Ye, You 18’ We are going to ‘ Reformulate ’ of theory of fermion production in a completely new manner

Traditional Approach To Theory of Fermion Production called technique of ‘Bogoliubov’ coefficient

The model 𝜔 + 1 < − 𝑊(𝜚) % 𝛿 5 𝐸 2 𝑗 𝑓 5 𝒯 = + 𝑒 - 𝑦 −𝑕 𝜔 % − 𝑛 + 𝑕 𝜚 2 𝜖 % 𝜚 On the metric: 𝑒𝑡 < = 𝑒𝑢 < − 𝑏 𝑢 < 𝑒𝐲 < = 𝑏 𝑢 < (𝑒𝜐 < − 𝑒𝐲 < ) Under rescaling 𝜔 → 𝑏 FG/< 𝜔 𝜔 + 1 2 𝑗 𝛿 % 𝜖 2 𝑏 < 𝜃 %K 𝜖 % 𝜚𝜖 K 𝜚 − 𝑏 - 𝑊(𝜚) ℒ = 𝜔 % − 𝑛𝑏 + 𝑕 𝜚 : Yukawa-type coupling ℎ𝜚 Common Interaction 1 L 𝑕 𝜚 = type in literature 𝑔 𝛿 % 𝛿 ( 𝜖 % 𝜚 : derivative coupling We will assume spatially homogenous scalar field : 𝜖 % 𝜚 = 𝜚̇ We will not distinguish 𝑢 and 𝜐 unless it is necessary

Fermion Production is formulated in Hamiltonian formalism % − 𝑛𝑏 − 1 𝑔 𝛿 P 𝛿 ( 𝜚̇ 𝜔 + 1 2 𝑗 𝛿 % 𝜖 2 𝑏 < 𝜃 %K 𝜖 % 𝜚𝜖 K 𝜚 − 𝑏 - 𝑊(𝜚) ℒ = 𝜔 A subtlety with derivative coupling Π R = 𝜀ℒ Π U = 𝜀ℒ 𝜀𝜚̇ = 𝑏 < 𝜚̇ − 1 2𝛿 P 𝛿 ( 𝜔 𝜀𝜔̇ = 𝑗𝜔 T 𝑔 𝜔 ℋ = Π R 𝜔̇ + Π U 𝜚̇ − ℒ 2𝛿 P 𝛿 ( 𝜔 < 2 −𝑗 𝛿 W 𝜖 W + 𝑛𝑏 + 1 𝑔 𝛿 P 𝛿 ( 𝜚̇ 𝜔 − 1 𝜔 + 1 < + 𝑏 ( 𝑊(𝜚) = 𝜔 2𝑏 < Π U 2𝑏 < 𝑔 < Adshead, Sfakianakis 15’ Definition of particle number is ambiguous Massless limit is not manifest

A way out: field redefinition % − 𝑛𝑏 − 1 𝑔 𝛿 P 𝛿 ( 𝜚̇ 𝜔 + 1 2 𝑗 𝛿 % 𝜖 2 𝑏 < 𝜃 %K 𝜖 % 𝜚𝜖 K 𝜚 − 𝑏 - 𝑊(𝜚) ℒ = 𝜔 Adshead, Sfakianakis 15’ 𝜔 → 𝑓 FWZ [ U/\ 𝜔 % − 𝑛𝑏 cos2𝜚 𝑔 + 𝑗 𝑛𝑏 sin 2𝜚 𝑔 𝛿 ( 𝜔 + 1 2 𝑗 𝛿 % 𝜖 2 𝑏 < 𝜃 %K 𝜖 % 𝜚𝜖 K 𝜚 − 𝑏 - 𝑊(𝜚) ℒ = 𝜔 = 𝑛 X = 𝑛 Y Adshead, Pearce, Peloso, Roberts, Sorbo 18’ Hamiltonian formalism Π R = 𝜀ℒ Π U = 𝜀ℒ 𝜀𝜔̇ = 𝑗𝜔 T 𝜀𝜚̇ = 𝑏 < 𝜚̇ 2 −𝑗 𝛿 W 𝜖 W + 𝑛 X − 𝑗 𝑛 Y 𝛿 ( 𝜔 + 1 < + 𝑏 - 𝑊(𝜚) ℋ = 𝜔 2𝑏 < Π U No 𝜔 - dependence in conjugate momentum Π U Entire fermion sector is quadratic in 𝜔 : particle number is unambiguously defined Massless limit is manifest

Fermion production 2 −𝑗 𝛿 W 𝜖 W + 𝑛 X − 𝑗 𝑛 Y 𝛿 ( 𝜔 + 1 < + 𝑏 - 𝑊(𝜚) ℋ = 𝜔 2𝑏 < Π U Garbrecht, Prokopec, Schmidt 02’ for generic complex mass To estimate Fermion Production, we quantize 𝜔 while keeping pseudo-scalar as a classical field Quantum field 𝜔 We follow notation and convention in Adshead, Pearce, Peloso, Roberts, Sorbo 18’ 𝑒 G 𝑙 2𝜌 G/< 𝑓 W𝐥⋅𝐲 f 𝑉 h 𝐥,𝑢 𝑏 h 𝐥 + 𝑊 T −𝐥 𝜔 = + h −𝐥,𝑢 𝑐 h hk± 𝑣 h 𝐥,𝑢 𝜓 h (𝐥) 𝑉 h = 𝑠 𝑤 h 𝐥,𝑢 𝜓 h (𝐥) 𝑙 + 𝑠 𝜏 ⃗ ⋅ 𝐥 1 𝜓̅ F = 0 𝜓 h 𝐥 = 𝜓̅ h where 𝜓̅ T = 0 , 2𝑙 𝑙 + 𝑙 G 1 ** helicity basis for an arbitrary 𝐥

∗ 𝑏 h (𝐥) 𝐵 h 𝐶 h T 𝐥 , 𝑐 h −𝐥 ℋ R = f + 𝑒𝑙 G 𝑏 h T (−𝐥) 𝐶 h −𝐵 h 𝑐 h hk± 𝐵 h = 1 2 − 𝑛 X < − 𝑙 h − 𝑠𝑛 Y < − 𝑤 h ∗ 𝑤 ∗ 𝑤 𝑣 h 2𝜕 𝑆𝑓 𝑣 h 2𝜕 𝐽𝑛(𝑣 h h ) 4𝜕 𝐶 h = 𝑠 𝑓 Wh• € < − 𝑤 h < − 𝑗𝑠𝑛 Y (𝑣 h < + 𝑤 h < ) 2 𝑛 X 𝑣 h 𝑤 h − 𝑙 𝑣 h 2 Fermion number density for a particle with helicity 𝑠 T 𝐥;𝑢 𝑏 h (𝐥; 𝑢) 0 𝑜 h,y = 0 𝑏 h T (𝐥;𝑢) are diagonalized 𝑏 h (𝐥) , 𝑏 h T (𝐥) at 𝑢 ≠ 0 w/ 𝑏 h (𝐥; 𝑢) , 𝑏 h At 𝑢 ≠ 0 At 𝑢 = 0 𝑏 h (𝐥) 0 = 0 𝑏 h 𝐥; 𝑢 0 ≠ 0 T 𝐥 T 𝐥 𝑏 h 𝐥 , 𝑏 h 𝑏 h 𝐥 , 𝑏 h ↔ one-particle state ↮ one-particle state due to 𝐶 h = 0 anymore due to 𝐶 h ≠ 0

∗ 𝑏 h (𝐥) 𝐵 h 𝐶 h T 𝐥 , 𝑐 h −𝐥 ℋ R = f + 𝑒𝑙 G 𝑏 h T (−𝐥) 𝐶 h −𝐵 h 𝑐 h hk± 𝐵 h = 1 2 − 𝑛 X < − 𝑙 h − 𝑠𝑛 Y < − 𝑤 h ∗ 𝑤 ∗ 𝑤 𝑣 h 2𝜕 𝑆𝑓 𝑣 h 2𝜕 𝐽𝑛(𝑣 h h ) 4𝜕 𝐶 h = 𝑠 𝑓 Wh• € < − 𝑤 h < − 𝑗𝑠𝑛 Y (𝑣 h < + 𝑤 h < ) 2 𝑛 X 𝑣 h 𝑤 h − 𝑙 𝑣 h 2 Fermion number density for a particle with helicity 𝑠 T 𝐥; 𝑢 𝑏 h (𝐥; 𝑢) 0 = 𝛾 h < 𝑜 h,y = 0 𝑏 h T (𝐥;𝑢) are diagonalized 𝑏 h (𝐥) , 𝑏 h T (𝐥) at 𝑢 ≠ 0 w/ 𝑏 h (𝐥; 𝑢) , 𝑏 h = 1 2 − 𝑛 X < − 𝑙 ∗ 𝑤 h − 𝑠𝑛 Y < − 𝑤 h ∗ 𝑤 h ) 𝑣 h 2𝜕 𝑆𝑓 𝑣 h 2𝜕 𝐽𝑛(𝑣 h 4𝜕 Bogoliubov coeff. ∗ 𝑐 h T (𝐥) 𝑏 h (𝐥; 𝑢) = 𝛽 h 𝑏 h (𝐥) − 𝛾 h T (𝐥;𝑢) = 𝛾 h 𝑏 h (𝐥) + 𝛽 h ∗ 𝑐 h T (𝐥) 𝑐 h Diag. ops In terms of diag. at 𝑢 ≠ 0 ops at 𝑢 = 0

looks too technical … Any simplication? T 𝐥;𝑢 𝑏 h (𝐥; 𝑢) 0 𝑜 h,y = 0 𝑏 h = 1 2 − 𝑛 X < − 𝑙 ∗ 𝑤 h − 𝑠 𝑛 Y < − 𝑤 h ∗ 𝑤 h ) 𝑣 h 2𝜕 𝑆𝑓 𝑣 h 2𝜕 𝐽𝑛(𝑣 h 4𝜕 Solving EOM of 𝑣 h , 𝑤 h with correct initial condition is another source of confusion

looks too technical … Any simplication? T 𝐥;𝑢 𝑏 h (𝐥; 𝑢) 0 𝑜 h,y = 0 𝑏 h = 1 2 − 𝑛 X < − 𝑙 ∗ 𝑤 h − 𝑠 𝑛 Y < − 𝑤 h ∗ 𝑤 h ) 𝑣 h 2𝜕 𝑆𝑓 𝑣 h 2𝜕 𝐽𝑛(𝑣 h 4𝜕 Solving EOM of 𝑣 h , 𝑤 h with correct initial condition is another source of confusion Recall a Fourier mode in ‘helicity’ basis T (−𝐥) 𝜔 ∼ 𝑉 h 𝐥, 𝑢 𝑏 h 𝐥 + 𝑊 h −𝐥,𝑢 𝑐 h 𝑣 h 𝑣 h 𝐥, 𝑢 𝜓 h (𝐥) 𝑉 h = 𝑠 𝑤 h 𝐥, 𝑢 𝜓 h (𝐥) = 𝑠𝑤 h ⊗ 𝜓 h ≡ 𝜊 h ⊗ 𝜓 h

looks too technical … Any simplication? T 𝐥;𝑢 𝑏 h (𝐥; 𝑢) 0 𝑜 h,y = 0 𝑏 h = 1 2 − 𝑛 X < − 𝑙 ∗ 𝑤 h − 𝑠 𝑛 Y < − 𝑤 h ∗ 𝑤 h ) 𝑣 h 2𝜕 𝑆𝑓 𝑣 h 2𝜕 𝐽𝑛(𝑣 h 4𝜕 Solving EOM of 𝑣 h , 𝑤 h with correct initial condition is another source of confusion Recall a Fourier mode in ‘helicity’ basis T (−𝐥) 𝜔 ∼ 𝑉 h 𝐥, 𝑢 𝑏 h 𝐥 + 𝑊 h −𝐥,𝑢 𝑐 h 𝑣 h 𝑣 h 𝐥, 𝑢 𝜓 h (𝐥) 𝑉 h = 𝑠 𝑤 h 𝐥, 𝑢 𝜓 h (𝐥) = 𝑠𝑤 h ⊗ 𝜓 h ≡ 𝜊 h ⊗ 𝜓 h Then we realize that 𝜂 h ‹ = 1 ∗ 𝑤 h + 𝑣 h 𝑤 h ∗ ) = 𝑠 𝑆𝑓(𝑣 h ∗ 𝑤 h ) 2 𝑠(𝑣 h ⃗ T 𝜏 𝜂 h = 𝜊 h ⃗ 𝜊 h 𝜂 h < = − 𝑗 ∗ 𝑤 h − 𝑣 h 𝑤 h ∗ ) = 𝑠 𝐽𝑛(𝑣 h ∗ 𝑤 h ) 2𝑠(𝑣 h 𝜂 h G = 1 collapses into one vector < − 𝑤 h < 𝑣 h 2

T 𝐥;𝑢 𝑏 h (𝐥; 𝑢) 0 𝑜 h,y = 0 𝑏 h = 1 2 − 𝑛 X < − 𝑙 ∗ 𝑤 h − 𝑠 𝑛 Y < − 𝑤 h ∗ 𝑤 h ) 𝑣 h 2𝜕 𝑆𝑓 𝑣 h 2𝜕 𝐽𝑛(𝑣 h 4𝜕 𝑣 h ⃗ T 𝜏 𝐫 = 𝑠𝑙 𝑦 • ‹ + 𝑛 Y 𝑦 • < + 𝑛 X 𝑦 • G 𝜂 h = 𝜊 h ⃗ 𝜊 h w/ 𝜊 h ≡ 𝑠𝑤 h 𝜂 h ‹ = 1 * We will see the origin ∗ 𝑤 h + 𝑣 h 𝑤 h ∗ ) = 𝑠 𝑆𝑓(𝑣 h ∗ 𝑤 h ) 2 𝑠(𝑣 h of this vector later 𝜂 h < = − 𝑗 ∗ 𝑤 h − 𝑣 h 𝑤 h ∗ ) = 𝑠 𝐽𝑛(𝑣 h ∗ 𝑤 h ) 2𝑠(𝑣 h 𝜂 h G = 1 < − 𝑤 h < 𝑣 h 2

T 𝐥;𝑢 𝑏 h (𝐥; 𝑢) 0 𝑜 h,y = 0 𝑏 h = 1 2 − 𝑛 X < − 𝑙 ∗ 𝑤 h − 𝑠 𝑛 Y < − 𝑤 h ∗ 𝑤 h ) 𝑣 h 2𝜕 𝑆𝑓 𝑣 h 2𝜕 𝐽𝑛(𝑣 h 4𝜕 𝑣 h ⃗ T 𝜏 𝐫 = 𝑠𝑙 𝑦 • ‹ + 𝑛 Y 𝑦 • < + 𝑛 X 𝑦 • G 𝜂 h = 𝜊 h ⃗ 𝜊 h w/ 𝜊 h ≡ 𝑠𝑤 h 𝜂 h ‹ = 1 * We will see the origin ∗ 𝑤 h + 𝑣 h 𝑤 h ∗ ) = 𝑠 𝑆𝑓(𝑣 h ∗ 𝑤 h ) 2 𝑠(𝑣 h of this vector later 𝜂 h < = − 𝑗 ∗ 𝑤 h − 𝑣 h 𝑤 h ∗ ) = 𝑠 𝐽𝑛(𝑣 h ∗ 𝑤 h ) 2𝑠(𝑣 h 𝜂 h G = 1 < − 𝑤 h < 𝑣 h 2 𝜂 𝑠 ,𝐫 behave like vector reps of SO(3) What is this mysterious SO(3)?

Group Theoretic Approach