Real-time Evolution of U ( 1 ) Chiral Charge Daniel Figueroa Adrien Florio Mikhail Shaposhnikov Lattice 2018, 23 th of July 2018
Overview Lattice Model Results/Outlooks
Overview Lattice Model Results/Outlooks
Anomalous Processes Gauge bosons Axion Physics Anomaly Chiral Magnetic Effect Baryogenesis Fermionic charge A. Florio, Lattice2018, 23/07/18 1
Anomalous Processes Gauge bosons Axion Physics Anomaly Chiral Magnetic Effect Baryogenesis Fermionic charge U ( 1 ) A. Florio, Lattice2018, 23/07/18 1
Anomalous Processes Model Gauge bosons 4 F µν F µν + ¯ L = − 1 Ψ / D Ψ +( D µ φ ) ∗ D µ φ − V ( φ ) Axion Physics Anomaly Chiral Magnetic Effect 5 = ¯ Ψ γ µ γ 5 Ψ j µ with: Baryogenesis V ( φ ) = m 2 | φ | 2 + λ | φ | 4 Fermionic charge U ( 1 ) A. Florio, Lattice2018, 23/07/18 1 - 2
Anomalous Processes Model Gauge bosons 4 F µν F µν + ¯ L = − 1 Ψ / D Ψ +( D µ φ ) ∗ D µ φ − V ( φ ) Axion Physics Anomaly Chiral Magnetic Effect 5 = ¯ Ψ γ µ γ 5 Ψ j µ with: Baryogenesis V ( φ ) = m 2 | φ | 2 + λ | φ | 4 e 2 8 π 2 F µν ˜ Fermionic charge Anomaly: ∂ µ j µ 5 = F µν U ( 1 ) A. Florio, Lattice2018, 23/07/18 1 - 2
Model Abelian Instabilities 4 F µν F µν + ¯ L = − 1 Ψ / D Ψ +( D µ φ ) ∗ D µ φ − V ( φ ) Integrate out Ψ F CS = µ N CS 5 = ¯ Ψ γ µ γ 5 Ψ j µ with: V ( φ ) = m 2 | φ | 2 + λ | φ | 4 e 2 8 π 2 F µν ˜ Anomaly: ∂ µ j µ 5 = F µν A. Florio, Lattice2018, 23/07/18 2 - 3
Model Abelian Instabilities F µν = ∂ µ K µ 16 π 2 F µν ˜ e 2 4 F µν F µν + ¯ L = − 1 Ψ / D Ψ x 3 K 0 = N CS = x 3 � A · � � d � � d � α B 2 π +( D µ φ ) ∗ D µ φ − V ( φ ) Integrate out Ψ F CS = µ N CS 5 = ¯ Ψ γ µ γ 5 Ψ j µ with: V ( φ ) = m 2 | φ | 2 + λ | φ | 4 e 2 8 π 2 F µν ˜ Anomaly: ∂ µ j µ 5 = F µν A. Florio, Lattice2018, 23/07/18 2 - 3
Model Abelian Instabilities F µν = ∂ µ K µ 16 π 2 F µν ˜ e 2 4 F µν F µν + ¯ L = − 1 Ψ / D Ψ x 3 K 0 = N CS = x 3 � A · � � d � � d � α B 2 π +( D µ φ ) ∗ D µ φ − V ( φ ) Integrate out Ψ F CS = µ N CS 5 = ¯ Ψ γ µ γ 5 Ψ j µ with: V ( φ ) = m 2 | φ | 2 + λ | φ | 4 k 2 AA vs µ kAA e 2 8 π 2 F µν ˜ Anomaly: ∂ µ j µ 5 = F µν A. Florio, Lattice2018, 23/07/18 2 - 3
Model Abelian Instabilities F µν = ∂ µ K µ 16 π 2 F µν ˜ e 2 4 F µν F µν + ¯ L = − 1 Ψ / D Ψ x 3 K 0 = N CS = x 3 � A · � � d � � d � α B 2 π +( D µ φ ) ∗ D µ φ − V ( φ ) Integrate out Ψ F CS = µ N CS 5 = ¯ Ψ γ µ γ 5 Ψ j µ with: V ( φ ) = m 2 | φ | 2 + λ | φ | 4 k 2 AA vs µ kAA e 2 8 π 2 F µν ˜ Anomaly: ∂ µ j µ 5 = F µν Instability if k < α π µ ! A. Florio, Lattice2018, 23/07/18 2 - 3
Abelian Instabilities Comments F µν = ∂ µ K µ 16 π 2 F µν ˜ e 2 x 3 K 0 = N CS = x 3 � A · � � d � � d � α B 2 π • Long-range gauge fields Integrate out Ψ F CS = µ N CS • Symmetric phase • Also at non-zero temperature k 2 AA vs µ kAA Instability if k < α π µ ! A. Florio, Lattice2018, 23/07/18 3 - 4
Comments External Magnetic Field ⇒ � EoM + Flux Cons. = B vac. state • Long-range gauge fields • Symmetric phase • Also at non-zero temperature A. Florio, Lattice2018, 23/07/18 4 - 5
Comments External Magnetic Field ⇒ � EoM + Flux Cons. = B vac. state • Long-range gauge fields • Symmetric phase • Also at non-zero temperature Generating � A cost no energy A. Florio, Lattice2018, 23/07/18 4 - 5
Comments External Magnetic Field ⇒ � EoM + Flux Cons. = B vac. state • Long-range gauge fields Continuum of vac. N CS ∝ � A · � • Symmetric phase B • Also at non-zero temperature Generating � A cost no energy A. Florio, Lattice2018, 23/07/18 4 - 5
External Magnetic Field Sum-Up ⇒ � EoM + Flux Cons. = B vac. state • B = 0: Instability • B � = 0: Non-trivial vac. structure Continuum of vac. N CS ∝ � A · � B Similar to non-abelian! Generating � A cost no energy A. Florio, Lattice2018, 23/07/18 5 - 6
External Magnetic Field Sum-Up ⇒ � EoM + Flux Cons. = B vac. state • B = 0: Instability • B � = 0: Non-trivial vac. structure Continuum of vac. N CS ∝ � A · � B Similar to non-abelian! Generating � A cost no energy A. Florio, Lattice2018, 23/07/18 5 - 6
Overview Lattice Model Results/Outlooks
Lattice Model Real-Time Simulations MC generated thermal ensemble 4 F µν F µν + ( D µ φ ) ∗ D µ φ − V ( φ ) L = − 1 s ( ∂ 0 a ) 2 + 1 1 2 ( ∂ i a ) 2 − 2 c 2 Measure ! n with a a scalar field o i x A + ) 1 ( U t f Reproduce EoM Solve classical EoM A. Florio, Lattice2018, 23/07/18 7 - 8
Real-Time Simulations Technical Comments • Lattice topological F ˜ F MC generated thermal ensemble • External mag. field as twisted BCs Measure • Homogeneous axion much easier t f Refs.: JHEP04(2018)026 Solve classical EoM j.nuclphysb.2017.12.001 A. Florio, Lattice2018, 23/07/18 8 - 9
Real-Time Simulations Technical Comments • Lattice topological F ˜ F MC generated thermal ensemble • External mag. field as twisted BCs Measure • Homogeneous axion much easier This work t f Refs.: JHEP04(2018)026 Solve classical EoM j.nuclphysb.2017.12.001 A. Florio, Lattice2018, 23/07/18 8 - 9
Overview Lattice Model Results/Outlooks
Chern-Simons Density CS Evolution • µ = 0 , B = 0 10 − 4 � A · � ⇒ � Q 2 ( t ) � → cst B costs E = � Q 2 � 10 − 5 • µ = 0 , B � = 0 RD walk A · � � ⇒ � Q 2 ( t ) � → Γ Vt B costs E = 10 − 6 10 − 1 10 1 10 3 10 5 t A. Florio, Lattice2018, 23/07/18 9 - 10
Chern-Simons Density CS Evolution 10 − 2 • µ = 0 , B = 0 10 − 4 � A · � ⇒ � Q 2 ( t ) � → cst B costs E = � Q 2 � • µ = 0 , B � = 0 10 − 6 RD walk A · � � ⇒ � Q 2 ( t ) � → Γ Vt B costs E = e 2 = 0 . 125 10 − 8 10 − 1 10 1 10 3 10 5 t A. Florio, Lattice2018, 23/07/18 9 - 11
Chern-Simons Density CS Evolution 10 − 2 • µ = 0 , B = 0 10 − 4 � A · � ⇒ � Q 2 ( t ) � → cst B costs E = � Q 2 � • µ = 0 , B � = 0 10 − 6 RD walk � A · � ⇒ � Q 2 ( t ) � → Γ Vt B costs E = e 2 = 0 . 125 e 2 = 0 . 5 10 − 8 10 − 1 10 1 10 3 10 5 t A. Florio, Lattice2018, 23/07/18 9 - 11
Results CS Evolution e 6 B 2 ≈ 2 . 5 · 10 − 5 Γ th 10 − 2 Γ pred. by MHD: Γ exp e 6 B 2 ≈ 1 . 5 ± 0 . 2 · 10 − 3 Measured Γ : 10 − 4 � Q 2 � Agree on parametric 10 − 6 but e 2 = 0 . 125 Γ exp e 2 = 0 . 5 Γ th ≈ 60! 10 − 8 10 5 10 − 1 10 1 10 3 t A. Florio, Lattice2018, 23/07/18 11 - 12
Results Next: Chemical Potential e 6 B 2 ≈ 2 . 5 · 10 − 5 Γ th Γ pred. by MHD: Γ exp e 6 B 2 ≈ 1 . 5 ± 0 . 2 · 10 − 3 Measured Γ : • µ � = 0 , B = 0 α ∝ 1 Agree on parametric Lat. art.: µ min = k min π N but Γ exp Γ th ≈ 60! A. Florio, Lattice2018, 23/07/18 13 - 14
Next: Chemical Potential Chemical Potential 30 20 • µ � = 0 , B = 0 � µ � α ∝ 1 Lat. art.: µ min = k min π N 10 N = 16 0 0 200 400 t A. Florio, Lattice2018, 23/07/18 13 - 14
Next: Chemical Potential Chemical Potential 30 N = 16 20 • µ � = 0 , B = 0 � µ � α ∝ 1 Lat. art.: µ min = k min π N 10 0 10 − 3 10 − 1 10 1 10 3 10 5 t A. Florio, Lattice2018, 23/07/18 13 - 14
Next: Chemical Potential Chemical Potential 30 N = 16 20 N = 224 • µ � = 0 , B = 0 � µ � α ∝ 1 Lat. art.: µ min = k min π N 10 0 10 − 3 10 − 1 10 1 10 3 10 5 t A. Florio, Lattice2018, 23/07/18 13 - 14
Next: Chemical Potential Chemical Potential 30 N = 16 20 N = 224 • µ � = 0 , B = 0 N = 420 � µ � α ∝ 1 Lat. art.: µ min = k min π N 10 0 10 − 3 10 − 1 10 1 10 3 10 5 t A. Florio, Lattice2018, 23/07/18 13 - 14
Next: Chemical Potential Chemical Potential 30 • µ � = 0 , B = 0 N = 16 α ∝ 1 Lat. art.: µ min = k min π 20 N = 224 N N = 420 � µ � Question: 10 µ independent rate at small µ ? 0 10 − 3 10 − 1 10 1 10 3 10 5 t A. Florio, Lattice2018, 23/07/18 13 - 14
Further Outlooks Chemical Potential 30 N = 16 • B � = 0 , µ � = 0 20 N = 224 N = 420 � µ � • Hard Thermal Loops 10 • Non-homogeneous axion 0 10 − 3 10 − 1 10 1 10 3 10 5 t A. Florio, Lattice2018, 23/07/18 14 - 15
Further Outlooks Chemical Potential 30 N = 16 • B � = 0 , µ � = 0 20 N = 224 N = 420 � µ � • Hard Thermal Loops 10 • Non-homogeneous axion 0 10 − 3 10 − 1 10 1 10 3 10 5 t A. Florio, Lattice2018, 23/07/18 14 - 15
Further Outlooks Chemical Potential 30 N = 16 • B � = 0 , µ � = 0 20 N = 224 N = 420 � µ � • Hard Thermal Loops 10 • Non-homogeneous axion 0 10 − 3 10 − 1 10 1 10 3 10 5 t A. Florio, Lattice2018, 23/07/18 14 - 15
Further Outlooks Take Away • B � = 0 , µ � = 0 • Discrepancies between MHD and full simulations • Hard Thermal Loops • Exciting outlooks • Non-homogeneous axion A. Florio, Lattice2018, 23/07/18 15 - 16
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