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Thermal Field Theory to All Orders in Gradient Expansion Peter Millington p.w.millington@sheffield.ac.uk Astro-Particle Theory and Cosmology Group, University of Sheffield, UK, Consortium for Fundamental Physics arXiv: 1211.3152 PM &

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Thermal Field Theory to All Orders in Gradient Expansion

Peter Millington

p.w.millington@sheffield.ac.uk

Astro-Particle Theory and Cosmology Group, University of Sheffield, UK, Consortium for Fundamental Physics arXiv: 1211.3152 PM & Apostolos Pilaftsis (University of Manchester)

Thursday, 6th December, 2012 Discrete 2012 CFTP, IST, Universidade Tecnica de Lisboa

Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 1 / 15

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Outline

  • 1. Introduction
  • 2. Formalism
  • 3. Master Time Evolution Equations
  • 4. Simple Example
  • 5. Conclusions

Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 2 / 15

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Introduction

Motivation

the density frontier: ultra-relativistic many-body dynamics early Universe:

◮ baryon asymmetry of the Universe ◮ electroweak phase transition ◮ reheating/preheating ◮ relic densities

‘terrestrial:’

◮ quark gluon plasma/glasma/color glass condensates Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 3 / 15

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Introduction

Current Approaches

(semi-classical) Boltzmann transport equations

◮ effective resummation of finite-width effects Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 4 / 15

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Introduction

Current Approaches

(semi-classical) Boltzmann transport equations

◮ effective resummation of finite-width effects

Kadanoff–Baym ⇒ quantum Boltzmann equations

◮ incorporation of off-shell effects ◮ truncated gradient expansion in time derivative ◮ separation of time scales and quasi-particle approximation ◮ varying definitions of physical observables,

e.g. particle number density

Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 4 / 15

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Introduction

Current Approaches

(semi-classical) Boltzmann transport equations

◮ effective resummation of finite-width effects

Kadanoff–Baym ⇒ quantum Boltzmann equations

◮ incorporation of off-shell effects ◮ truncated gradient expansion in time derivative ◮ separation of time scales and quasi-particle approximation ◮ varying definitions of physical observables,

e.g. particle number density

underlying perturbation series contain pinch singularities: δ2(p2 − m2)

Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 4 / 15

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Canonical Quantisation

Boundary Conditions

No assumption of adiabatic hypothesis. QM pictures have a finite microscopic time of coincidence ˜ ti: ΦS(x;˜ ti) = ΦI(˜ ti, x;˜ ti) = ΦH(˜ ti, x;˜ ti)

Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 5 / 15

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Canonical Quantisation

Boundary Conditions

No assumption of adiabatic hypothesis. QM pictures have a finite microscopic time of coincidence ˜ ti: ΦS(x;˜ ti) = ΦI(˜ ti, x;˜ ti) = ΦH(˜ ti, x;˜ ti) ⇒ interactions switched on at ˜ ti ⇒ initial density matrix ρ(˜ ti;˜ ti) specified fully in on-shell Fock states ⇒ finite lower bound on time integrals in path-integral action

Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 5 / 15

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Canonical Quantisation

Canonical Commutation Relations

Interaction-picture creation and annihilation operators satisfy:

  • a(p,˜

t;˜ ti), a†(p′,˜ t′;˜ ti)

  • = (2π)32E(p)δ(3)(p − p′)e−iE(p)(˜

t −˜ t′)

Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 6 / 15

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Canonical Quantisation

Canonical Commutation Relations

Interaction-picture creation and annihilation operators satisfy:

  • a(p,˜

t;˜ ti), a†(p′,˜ t′;˜ ti)

  • = (2π)32E(p)δ(3)(p − p′)e−iE(p)(˜

t −˜ t′)

Ensemble Expectation Value (EEV) at macroscopic time t = ˜ tf − ˜ ti:

  • t = tr ρ(˜

tf ;˜ ti) • tr ρ(˜ tf ;˜ ti)

Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 6 / 15

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Canonical Quantisation

Canonical Commutation Relations

Interaction-picture creation and annihilation operators satisfy:

  • a(p,˜

t;˜ ti), a†(p′,˜ t′;˜ ti)

  • = (2π)32E(p)δ(3)(p − p′)e−iE(p)(˜

t −˜ t′)

Ensemble Expectation Value (EEV) at macroscopic time t = ˜ tf − ˜ ti:

  • t = tr ρ(˜

tf ;˜ ti) • tr ρ(˜ tf ;˜ ti)

Most general EEVs permitted:

a(p,˜ tf ;˜ ti)a†(p′,˜ tf ;˜ ti)t = (2π)32E(p)δ(3)(p − p′) + 2E1/2(p)E1/2(p′)f (p, p′, t) a†(p′,˜ tf ;˜ ti)a(p,˜ tf ;˜ ti)t = 2E1/2(p)E1/2(p′)f (p, p′, t)

Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 6 / 15

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Schwinger–Keldysh CTP Formalism

Z[ρ, J±, t] = tr

¯

Te

−i

Ωt d4x J−(x)ΦH(x)

ρH

˜

tf ;˜ ti

  • Te

i

Ωt d4x J+(x)ΦH(x)

x0 ∈

˜

ti = − t

2,˜

tf = + t

2

  • b

b b

Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 7 / 15

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Schwinger–Keldysh CTP Formalism

Z[ρ, J±, t] = tr

¯

Te

−i

Ωt d4x J−(x)ΦH(x)

ρH

˜

tf ;˜ ti

  • Te

i

Ωt d4x J+(x)ΦH(x)

x0 ∈

˜

ti = − t

2,˜

tf = + t

2

  • b

b b

Re t Im t ˜ z(0) = ˜ ti ˜ z(1/2) = ˜ tf − iǫ/2 ˜ z(1) = ˜ ti − iǫ C+ C− macroscopic time t = Re ˜ z(u) − ˜ ti initial conditions: macroscopic time t = 0

  • bservation:

macroscopic time t = ˜ tf − ˜ ti

⇒ finite upper and lower bounds on time integrals in path-integral action.

Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 7 / 15

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Non-Homogeneous Free Propagators

Propagator Double-Momentum Representation Feynman (Dyson) i∆0

F(D)(p, p′,˜

tf ;˜ ti) = (−)i p2 − M 2 + (−)iǫ(2π)4δ(4)(p − p′) +2π|2p0|1/2δ(p2 − M 2)˜ f (p, p′, t)ei(p0−p′

0)˜

tf 2π|2p′ 0|1/2δ(p′2 − M 2)

+(−)ve- freq. Wightman i∆0

>(<)(p, p′,˜

tf ;˜ ti) = 2πθ(+(−)p0)δ(p2 − M 2)(2π)4δ(4)(p − p′) +2π|2p0|1/2δ(p2 − M 2)˜ f (p, p′, t)ei(p0−p′

0)˜

tf 2π|2p′ 0|1/2δ(p′2 − M 2)

Retarded (Advanced) i∆0

R(A)(p, p′) =

i (p0 + (−)iǫ)2 − p2 − M 2 (2π)4δ(4)(p − p′) Pauli- Jordan i∆0(p, p′) = 2πε(p0)δ(p2 − M 2)(2π)4δ(4)(p − p′) Hadamard i∆0

1(p, p′,˜

tf ;˜ ti) = 2πδ(p2 − M 2)(2π)4δ(4)(p − p′) +2π|2p0|1/2δ(p2 − M 2)2˜ f (p, p′, t)ei(p0−p′

0)˜

tf 2π|2p′ 0|1/2δ(p′2 − M 2)

Principal- part i∆0

P(p, p′) = P

i p2 − M 2 (2π)4δ(4)(p − p′)

Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 8 / 15

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Diagrammatics

L =

1 2∂µΦ∂µΦ − 1 2M 2Φ2 + ∂µχ†∂µχ − m2χ†χ − gΦχ†χ k1 χ k′

1

k2 χ k′

2

q q′ Φ Φ a b

Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 9 / 15

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Diagrammatics

L =

1 2∂µΦ∂µΦ − 1 2M 2Φ2 + ∂µχ†∂µχ − m2χ†χ − gΦχ†χ k1 χ k′

1

k2 χ k′

2

q q′ Φ Φ a b

  • 1. time-dependent, energy-non-conserving vertices:

∼ −ig t 2πsinc

i

p0

i

  • t

2

  • δ(3)

i

pi

  • Peter Millington (University of Sheffield)

TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 9 / 15

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Diagrammatics

L =

1 2∂µΦ∂µΦ − 1 2M 2Φ2 + ∂µχ†∂µχ − m2χ†χ − gΦχ†χ k1 χ k′

1

k2 χ k′

2

q q′ Φ Φ a b

  • 1. time-dependent, energy-non-conserving vertices:

∼ −ig t 2πsinc

i

p0

i

  • t

2

  • δ(3)

i

pi

  • 2. momentum-non-conserving, non-homogeneous free propagators

Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 9 / 15

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Physically Meaningful Observables

Construct from EEVs of field operators: Φ(x;˜ ti)Φ(y;˜ ti)

t = tr ρ(˜

tf ;˜ ti)Φ(x;˜ ti)Φ(y;˜ ti) tr ρ(˜ tf ;˜ ti)

Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 10 / 15

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SLIDE 19

Physically Meaningful Observables

Construct from EEVs of field operators: Φ(x;˜ ti)Φ(y;˜ ti)

t = tr ρ(˜

tf ;˜ ti)Φ(x;˜ ti)Φ(y;˜ ti) tr ρ(˜ tf ;˜ ti) Physically meaningful observables must be equal-time and picture-independent.

Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 10 / 15

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Physically Meaningful Observables

Construct from EEVs of field operators: Φ(x;˜ ti)Φ(y;˜ ti)

t = tr ρ(˜

tf ;˜ ti)Φ(x;˜ ti)Φ(y;˜ ti) tr ρ(˜ tf ;˜ ti) Physically meaningful observables must be equal-time and picture-independent. Particle number density: count charges not quanta of energy ⇒ no need for quasi-particle approximation.

Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 10 / 15

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Physically Meaningful Observables

Construct from EEVs of field operators: Φ(x;˜ ti)Φ(y;˜ ti)

t = tr ρ(˜

tf ;˜ ti)Φ(x;˜ ti)Φ(y;˜ ti) tr ρ(˜ tf ;˜ ti) Physically meaningful observables must be equal-time and picture-independent. Particle number density: count charges not quanta of energy ⇒ no need for quasi-particle approximation. By writing the Noether charge in terms of a charge density, we may define the particle number density:

n(p, X, t) = lim

X0 → t 2

dp0 2π d4P (2π)4 e−iP·X θ(p0)p0i∆<(p + P

2 , p − P 2 , t; 0)

Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 10 / 15

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Master Time Evolution Equations

Partially inverting the CTP Schwinger–Dyson equation: ∂tf (p + P

2 , p − P 2 , t) −

  • dp0

2π dP0 2π e−iP0t 2 p · P θ(p0)∆<(p + P

2 , p − P 2 , t; 0)

+

  • dp0

2π dP0 2π e−iP0t θ(p0)

  • F(p + P

2 , p − P 2 , t; 0) + F ∗(p − P 2 , p + P 2 , t; 0)

  • =
  • dp0

2π dP0 2π e−iP0t θ(p0)

  • C (p + P

2 , p − P 2 , t; 0) + C ∗(p − P 2 , p + P 2 , t; 0)

  • Peter Millington (University of Sheffield)

TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 11 / 15

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Master Time Evolution Equations

Partially inverting the CTP Schwinger–Dyson equation: ∂tf (p + P

2 , p − P 2 , t) −

  • dp0

2π dP0 2π e−iP0t 2 p · P θ(p0)∆<(p + P

2 , p − P 2 , t; 0)

+

  • dp0

2π dP0 2π e−iP0t θ(p0)

  • F(p + P

2 , p − P 2 , t; 0) + F ∗(p − P 2 , p + P 2 , t; 0)

  • =
  • dp0

2π dP0 2π e−iP0t θ(p0)

  • C (p + P

2 , p − P 2 , t; 0) + C ∗(p − P 2 , p + P 2 , t; 0)

  • Force and collision terms:

F(p + P

2 , p − P 2 , t; 0) ≡ −

  • d4q

(2π)4 iΠP(p + P

2 , q, t; 0) i∆<(q, p − P 2 , t; 0),

C (p + P

2 , p − P 2 , t; 0) ≡ 1

2

  • d4q

(2π)4

  • iΠ>(p + P

2 , q, t; 0) i∆<(q, p − P 2 , t; 0)

− iΠ<(p + P

2 , q, t; 0)

  • i∆>(q, p − P

2 , t; 0) − 2i∆P(q, p − P 2 , t; 0)

No nested Poisson brackets as in gradient expansion of Kadanoff–Baym equations.

Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 11 / 15

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Simple Example

Time-Dependent Width L =

1 2∂µΦ∂µΦ − 1 2M 2Φ2 + ∂µχ†∂µχ − m2χ†χ − gΦχ†χ

t < 0: Φ’s and χ’s in non-interacting equilibria at same temperature t = 0: interactions switched on

Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 12 / 15

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Simple Example

Time-Dependent Width L =

1 2∂µΦ∂µΦ − 1 2M 2Φ2 + ∂µχ†∂µχ − m2χ†χ − gΦχ†χ

t < 0: Φ’s and χ’s in non-interacting equilibria at same temperature t = 0: interactions switched on

1 2 3 0.8 0.9 1.0 1.1 1.2 1.3 Mt 1000

1q ,t

q 1 GeV q 10 GeV q 100 GeV Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 12 / 15

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SLIDE 26

Simple Example

Time-Dependent Width L =

1 2∂µΦ∂µΦ − 1 2M 2Φ2 + ∂µχ†∂µχ − m2χ†χ − gΦχ†χ

t < 0: Φ’s and χ’s in non-interacting equilibria at same temperature t = 0: interactions switched on

1 2 3 0.8 0.9 1.0 1.1 1.2 1.3 Mt 1000

1q ,t

q 1 GeV q 10 GeV q 100 GeV 1 2 3 4 5 0.990 0.995 1.000 1.005 1.010 Mt 1000

1 1 2q ,t

q 1 GeV Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 12 / 15

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Simple Example

Evanescent Processes

1 2 0.500 1.000 1.500 Mt1000

1 1 2

q 1 GeV 1 2 0.10 0.00 0.10 0.20 Mt1000

1 Landau

q 1 GeV 1 2 0.10 0.00 0.10 Mt1000

1 3 0

q 1 GeV 1 2 0.500 1.000 1.500 Mt1000

1 1 2

q 10 GeV 1 2 0.10 0.00 0.10 0.20 Mt1000

1 Landau

q 10 GeV 1 2 0.01 0.00 0.01 Mt1000

1 3 0

q 10 GeV 1 2 0.500 1.000 1.500 Mt1000

1 1 2

q 100 GeV 1 2 0.10 0.00 0.10 0.20 Mt1000

1 Landau

q 100 GeV 1 2 0.01 0.00 0.01 Mt1000

1 3 0

q 100 GeV q E1 E2 u/t χ χ Φ

1 → 2 decay (left)

q E1 E2 u/t χ Φ χ

2 → 1 Landau damping (center)

E2 q E1 u/t χ χ Φ E2 q E1 u/t χ Φ χ

3 → 0 total annihilation (right)

Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 13 / 15

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Simple Example

Time Evolution Equations Truncating the master time evolution equations in a loopwise sense: ∂tfΦ(|p|, t) = −g2 2

  • {α}=±1
  • d3k

(2π)3 1 2EΦ(p) 1 2Eχ(k) 1 2Eχ(p − k) × t 2π sinc

  • αEΦ(p) − α1Eχ(k) − α2Eχ(p − k)

t

2

  • ×
  • π + 2Si
  • αEΦ(p) + α1Eχ(k) + α2Eχ(p − k)

t

2

  • ×

θ(−α) + fΦ(|p|, t) θ(α1) 1 + fχ(|k|, t) + θ(−α1)f C

χ (|k|, t)

× θ(α2) 1 + f C

χ (|p − k|, t)

+ θ(−α2)fχ(|p − k|, t) − θ(α) + fΦ(|p|, t) θ(α1)fχ(|k|, t) + θ(−α1) 1 + f C

χ (|k|, t)

× θ(α2)f C

χ (|p − k|, t) + θ(−α2)

1 + fχ(|p − k|, t) Still valid to all orders in gradient expansion.

Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 14 / 15

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Conclusions

Obtain master time evolution equations valid to all orders in gradient expansion and to all orders in perturbation theory.

Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 15 / 15

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SLIDE 30

Conclusions

Obtain master time evolution equations valid to all orders in gradient expansion and to all orders in perturbation theory. Loopwise truncation of time evolution equations resum all loop insertions and remain valid to all orders in gradient expansion.

Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 15 / 15

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SLIDE 31

Conclusions

Obtain master time evolution equations valid to all orders in gradient expansion and to all orders in perturbation theory. Loopwise truncation of time evolution equations resum all loop insertions and remain valid to all orders in gradient expansion. Underlying non-equilibrium field theory free of pinch singularities.

Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 15 / 15

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Conclusions

Obtain master time evolution equations valid to all orders in gradient expansion and to all orders in perturbation theory. Loopwise truncation of time evolution equations resum all loop insertions and remain valid to all orders in gradient expansion. Underlying non-equilibrium field theory free of pinch singularities. Non-homogeneous free propagators and time-dependent vertices break space-time translational invariance from tree-level.

Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 15 / 15

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Conclusions

Obtain master time evolution equations valid to all orders in gradient expansion and to all orders in perturbation theory. Loopwise truncation of time evolution equations resum all loop insertions and remain valid to all orders in gradient expansion. Underlying non-equilibrium field theory free of pinch singularities. Non-homogeneous free propagators and time-dependent vertices break space-time translational invariance from tree-level. Early-time dynamics consistently describe energy-violating processes, leading to non-Markovian evolution of memory effects.

Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 15 / 15

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Backup Slides

Particle Number Density

Charge operator:

Q(x0;˜ ti) = i

  • d3x
  • Φ†(x;˜

ti)π†(x;˜ ti) − π(x;˜ ti)Φ(x;˜ ti)

  • ?

≡

  • d3X

d3p (2π)3 Q(p, X, X0;˜ ti)

Insert unity and symmetrise in x and y:

1 =

  • d4y

d3p (2π)3 e−ip·(x−y)δ(x0 − y0)

Charge-density operator:

Q(p, X, X0;˜ ti) = i 2

  • d4R e−ip·R δ(R0)

×

  • Φ†(X − R

2 ;˜

ti)π†(X + R

2 ;˜

ti) − π(X − R

2 ;˜

ti)Φ(X + R

2 ;˜

ti) + (R → −R)

  • Peter Millington (University of Sheffield)

TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 16 / 15

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Backup Slides

Particle Number Density

Take EEV in equal-time limit: Q(p, X,˜ tf ;˜ ti)

t =

lim

X0 →˜ tf

i

  • d4R e−ip·R

× δ(R0) ∂R0

  • i∆<(R, X,˜

tf ;˜ ti) − i∆<(−R, X,˜ tf ;˜ ti)

  • Separate particles (+ve freq.) and anti-particles (−ve freq.):

δ(R0) = i 2π lim

ǫ → 0+

  • 1

R0 + iǫ − 1 R0 − iǫ

  • +ve freq. part of i∆<(R, X,˜

tf ;˜ ti) and −ve freq. part of i∆<(−R, X,˜ tf ;˜ ti) ⇒ particle number density

Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 17 / 15

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Backup Slides

Particle Number Density

Fourier transform w.r.t. R and shift ˜ tf → ˜ tf − ˜ ti = t: ⇒ particle number density: n(p, X, t) = lim

X0 → t

dp0

2π p0 ×

  • θ(p0)i∆<(p, X, t; 0) − θ(−p0)i∆<(−p, X, t; 0)
  • Also counts off-shell contributions.

Inserting equilibrium propagators: n(p, X, t) = f B

E(p) =

1 eβE(p) − 1

Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 18 / 15

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SLIDE 37

Backup Slides

Pinch Singularities: δ2(p2 − M 2)

early times: δ2(p2 − M 2) → δ(p2 − M 2)δt(p0 − p′

0)δ(p′2 − M 2)

intermediate times:

◮ pinch singularities grow: tδ(p2 − M 2) ◮ equilibration occurs: f (t) − feq = δf (t) = δf (0)e−Γt

late times: f → feq and pinch singularities cancel ⇐ finite time domain ⇐ f ’s in free propagators evaluated at time of observation

Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 19 / 15

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Backup Slides

Phase-Space Evolution

0 5 101520 0.0 0.1 0.2 0.3 0.4 0.5 k GeV Θ Mt 0 0 5 101520 0.0 0.1 0.2 0.3 0.4 0.5 k GeV Θ Mt 1 0 5 101520 0.0 0.1 0.2 0.3 0.4 0.5 k GeV Θ Mt 2 0 5 101520 0.0 0.1 0.2 0.3 0.4 0.5 k GeV Θ Mt 4 0 5 101520 0.0 0.1 0.2 0.3 0.4 0.5 k GeV Θ Mt 8 0 5 101520 0.0 0.1 0.2 0.3 0.4 0.5 k GeV Θ Mt 16 0 5 101520 0.0 0.1 0.2 0.3 0.4 0.5 k GeV Θ Mt 32 0 5 101520 0.0 0.1 0.2 0.3 0.4 0.5 k GeV Θ Mt 64 0 5 101520 0.0 0.1 0.2 0.3 0.4 0.5 k GeV Θ Mt 128 5 4 3 2 1 1 2 3 4 5 104 x d 2

1q,Mt,k,Θ d kdΘ

Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 20 / 15

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SLIDE 39

Backup Slides

Non-Markovian Oscillations

200 400 600 800 1000 20 10 10 20 30 Mt Ω1q,u,t M q 10 GeV

u1, b1 u1, b1 u2, b1 u2, b1 u3, b1 u3, b1 Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 21 / 15

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SLIDE 40

Backup Slides

Kadanoff–Baym Equations Kinetic equation: q · ∂X∆≷(q, X) −

  • d4Q

(2π)4 (2π)4δ(4)

t

(Q) sin

  • Q · X + ♦−

q,X + 2♦+ Q,X

  • {Π≷(q + Q

2 , X)}{∆P(q − Q 2 , X)} + {ΠP(q + Q 2 , X)}{∆≷(q − Q 2 , X)}

  • =

i 2

  • d4Q

(2π)4 (2π)4δ(4)

t

(Q) cos

  • Q · X + ♦−

q,X + 2♦+ Q,X

  • {Π>(q + Q

2 , X)}{∆<(q − Q 2 , X)} − {Π<(q + Q 2 , X)}{∆>(q − Q 2 , X)}

  • Diamond operators:

♦±

p,X{A}{B} = 1

2{A, B}±

p,X ≡ 1

2

  • ∂A

∂pµ ∂B ∂Xµ ± ∂A ∂X µ ∂B ∂pµ

  • Peter Millington (University of Sheffield)

TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 22 / 15

slide-41
SLIDE 41

Backup Slides

Not Just a Complicated Zero

0.00 0.02 0.04 0.06 0.08 0.10 0.200 0.100 0.000 0.100 0.200 Mt 1000 M tf p ,t g 2fB p p 1 GeV 0.00 0.02 0.04 0.06 0.08 0.10 0.002 0.001 0.000 0.001 0.002 Mt 1000 M tf p ,t g 2fB p p 10 GeV

Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 23 / 15

slide-42
SLIDE 42

Backup Slides

Inclusion of Thermal Masses

Local self-energy −λ

χ†χ 2:

Πloc(1)

χ

(p, p′,˜ tf ;˜ ti) = −(2π)4δ(4)

t

(p − p′)ei(p0 − p′

0)˜

tf m2 th(˜

tf ;˜ ti) Thermal mass:

m2

th(˜

tf ;˜ ti) = λ 2

  • d3k

(2π)3 1

  • 2Eχ(k)

d3k′ (2π)3 1

  • 2Eχ(k′)
  • fχ(k, k′, t)ei[E(k)−E(k′)]˜

tf + f C∗ χ (−k, −k′, t)e−i[E(k)−E(k′)]˜ tf

  • Quasi-particle approximation: m2 → m2

th(˜

tf ;˜ ti) Coupling to system of ODEs (spatially homogeneous case): ∂tmth(t) = λ 2mth(t)

  • d3k

(2π)3 1 2Eχ(k) 1 2

  • ∂tfχ(|k|, t) + ∂tf C

χ (|k|, t)

  • Peter Millington (University of Sheffield)

TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 24 / 15