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FRG beyond the local potential approximation at finite temperature - - PowerPoint PPT Presentation

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FRG beyond the local potential approximation at finite temperature Alexander Stegemann Cold Quantum Coffee Heidelberg University 23 January 2018 Overview Strong interaction Running coupling, QCD confinement, asymptotic freedom, . . .

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FRG beyond the local potential approximation at finite temperature

Alexander Stegemann

Cold Quantum Coffee Heidelberg University — 23 January 2018

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SLIDE 2

Overview

Strong interaction QCD Running coupling, confinement, asymptotic freedom, . . .

[C. Patrignani et al., Chin. Phys., 2016]

Attempts to solve QCD Lattice QCD Effective theories FRG DSEs

1

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Functional renormalisation group

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Wilsonian renormalisation

  • Euclidean generating functional for a scalar field ϕ

Z[J] =

  • Dϕ exp
  • −S[ϕ] +
  • d4x J(x)ϕ(x)
  • Wilsonian renormalisation

Z[J] =

  • Dϕq≤k
  • Dϕq>k exp
  • −S[ϕ] +
  • d4x J(x)ϕ(x)
  • ≡Zk[J]

→ Integrate out modes successively

2

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SLIDE 5

Flow in theory space (1) c1 c2 c3

b b b

Γk = Λ = S Γk = 0 = Γ

3

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SLIDE 6

Introducing a scale dependence

  • Generating functional

Zk[J] =

  • Dϕ exp
  • −S[ϕ] − ∆Sk[ϕ] +
  • d4x J(x)ϕ(x)
  • Regulator insertion

∆Sk[ϕ] = 1 2

  • d4p

(2π)4 ϕ(p)Rk(p2)ϕ(−p)

  • Effective average action (φ(x) = ϕ(x))

Γk[φ] = sup

J

  • d4x J(x)φ(x) − ln Zk[J]
  • − ∆Sk[φ]

4

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SLIDE 7

Wetterich equation

  • (Exact) Wetterich equation

∂kΓk = 1 2 STr

  • Γ(2)

k

+ Rk −1 ∂kRk

  • Regulator Rk ensures correct integration limits

Γk

k→Λ

− − − → S Γk

k→0

− − − → Γ

  • No sign problem

5

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SLIDE 8

Flow in theory space (2) c1 c2 c3

b b b

Γk = Λ = S R1 Γk = 0 = Γ R2 R3

6

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SLIDE 9

Solving the Wetterich equation

  • (Exact) Wetterich equation

∂kΓk = 1 2 STr

  • Γ(2)

k

+ Rk −1 ∂kRk

  • In practice, truncations are needed
  • Derivative expansion

Γk =

  • d4x
  • Uk
  • φ2

+ 1 2Zk

  • φ2

(∂µφ)2 + . . .

  • 7
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SLIDE 10

Flow in theory space (3) Γk = 0 = Γ Γk = Λ = S R1 R2 R3 c1 c2 c3

b b b

8

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Quark-meson model in LPA

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Quark-meson model in LPA

  • The quark-meson model is a low-energy effective model for

two-flavour QCD

  • Local potential approximation = lowest order derivative expansion

Γk =

  • d4x
  • Uk
  • φ2

+ 1 2✟✟✟ ✟ ✯1 Zk

  • φ2

(∂µφ)2✘✘ ✘ + . . .

  • Ansatz for the effective average action of the quark-meson model

Γk =

  • d4x
  • ¯

ψ / ∂ − µγ0 + hΣ5

  • ψ + 1

2(∂µφ)2 + Uk

  • φ2

− cσ

  • Σ5 = (σ + iγ5

π τ), φ = (σ, π)T

9

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SLIDE 13

Flow equation for Uk

  • Using a 3-dimensional Litim regulator

∂kUk = k4 12π2

  • − 2NcNf

Eψ

  • tanh

Eψ + µ 2T

  • + tanh

Eψ − µ 2T

  • + 1

Eσ coth Eσ 2T

  • + 3

Eπ coth Eπ 2T E2

σ = k2 + m2 σ = k2 + 2U′ k + 4 σ2 U′′ k

E2

π = k2 + m2 π = k2 + 2U′ k

E2

ψ = k2 + m2 ψ = k2 + h2 σ2 10

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Numerical solution

  • Discretise the partial differential equation ∂kUk
  • Tune the UV parameters in the vacuum (mΛ, λΛ, c, h)

UΛ

  • φ2

= 1 2m2

Λφ2 + 1

4λΛ

  • φ22

40 200 400 600 800 1000 k in MeV 100 200 300 400 500 600 700 800 MeV

Σ0 mΠ mΨ mΣ

11

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Inconsistencies in LPA

  • Pion curvature and pole

masses show a large discrepancy in the vacuum mπ,curv ≈ 138 MeV mπ,pole ≈ 100 MeV

[J. Wambach et al., arXiv: 1712.02093v1 [hep-ph]]

  • “Wrong” shape of the

first-order transition line

[R.-A. Tripol et al., arXiv: arXiv:1709.05991 [hep-ph]]

12

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Quark-meson model in LPA′

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Quark-Meson Model in LPA′

  • LPA′ = including a scale dependent wave function renormalisation

Γk =

  • d4x
  • Uk
  • φ2

+ 1 2Zk

  • φ2

(∂µφ)2✘✘ ✘ + . . .

  • Ansatz for the purely bosonic part of the quark-meson model

Γk,B =

  • d4x

1 2Zσ (∂µσ)2 + 1 2Zπ (∂µ π)2 + Uk(φ2) − cσ

  • Flow equations for the wave function renormalisations

∂kZα ∼ ∂ ∂p2 δ δφα(−p) δ δφα(p)∂kΓk

  • p=0

α ∈ {σ, π}

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Including a finite temperature

  • The time direction becomes compactified
  • d4x −

→ β dτ

  • d3x

β = 1 T

  • p0 becomes discrete (Matsubara frequencies)

ωn = 2nπT for bosons νn = (2n + 1)πT for fermions

  • d4p

(2π)4 − → T

  • n∈Z
  • d3p

(2π)3

14

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Including a finite temperature

  • Ansatz for the purely bosonic part of the quark-meson model

Γk,B = β dτ

  • d3x
  • 1

2Z

σ (∂0σ)2 + 1

2Z⊥

σ (∂iσ)2

+ 1 2Z

π (∂0

π)2 + 1 2Z⊥

π (∂i

π)2 + Uk(φ2) − cσ

  • Flow equations for the perpendicular wave function renormalisations

∂kZ⊥

α ∼

  • ∂

∂| p|2

  • δ

δφα(−p) δ δφα(p)∂kΓk

  • p0=0
  • p=0
  • Flow equations for the parallel wave function renormalisations

∂kZ

α =

  • ∂kΓ(2)

k,α(p0 = 2πT) − ∂kΓ(2) k,α(p0 = 0)

(2πT)2

  • p=0

15

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Flow equation for Uk

  • Using a 3-dimensional Litim regulator

∂kUk = k4 12π2

  • − 2NcNf

Eψ

  • tanh

Eψ + µ 2T

  • + tanh

Eψ − µ 2T

  • +

Z⊥

σ

Z

σEσ

  • 1 − ησ

5

  • coth

Eσ 2T

  • + 3Z⊥

π

Z

πEπ

  • 1 − ηπ

5

  • coth

Eπ 2T

E2

σ = Z⊥ σ

Z

σ

  • k2 + 2U′

k + 4 σ2 U′′ k

Z⊥

σ

  • E2

π = Z⊥ π

Z

π

  • k2 + 2U′

k

Z⊥

π

  • 16
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SLIDE 21

Numerical solution

  • Discretise the partial differential equation ∂kUk
  • Tune the UV parameters in the vacuum (mΛ, λΛ, c, h)

UΛ

  • φ2

= 1 2m2

Λφ2 + 1

4λΛ

  • φ22

10 200 400 600 800 1000 k in MeV 100 200 300 400 500 600 700 800 MeV

Σ0 mΠ mΨ mΣ

17

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Wave function renormalisations

Zα ∼ ∂ ∂p2 δ δφα(−p) δ δφα(p)Γk

  • p=0

σ=σk

α ∈ {σ, π}

18

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Scale dependence

  • Curvature masses and wave function renormalisations
  • Evaluating at the scale dependent minimum of Uk

T = 10 MeV, µ = 0 MeV

10 200 400 600 800 1000 k in MeV 100 200 300 400 500 600 700 800 MeV

Σ0 mΠ mΨ mΣ

10 200 400 600 800 1000 k in MeV 1 1.5 2 2.5 3 ZΣ

  • ZΠ
  • ZΣ
  • ZΠ
  • 19
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Wave function renormalisations

∂kZα ∼ ∂k ∂ ∂p2 δ δφα(−p) δ δφα(p)Γk

  • p=0

σ=σk

α ∈ {σ, π} → additional term ∼ ∂kσk ∂kZα ∼ ∂k ∂ ∂p2 δ δφα(−p) δ δφα(p)Γk

  • p=0

σ=σfix

α ∈ {σ, π} → no additional term ∂kZα ∼ ∂ ∂p2 δ δφα(−p) δ δφα(p)∂kΓk

  • p=0

α ∈ {σ, π}

20

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Correct IR values

  • Curvature masses and wave function renormalisations
  • Evaluating at the IR minimum

T = 10 MeV, µ = 0 MeV

10 200 400 600 800 1000 k in MeV 100 200 300 400 500 600 700 800 MeV

Σ0 mΠ mΨ mΣ

10 200 400 600 800 1000 k in MeV 1 1.5 2 2.5 ZΣ

  • ZΠ
  • ZΣ
  • ZΠ
  • 21
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Correct scale dependence

  • Curvature masses and wave function renormalisations
  • Iterative procedure to obtain the correct scale dependence

T = 10 MeV, µ = 0 MeV

10 200 400 600 800 1000 k in MeV 100 200 300 400 500 600 700 800 MeV

Σ0 mΠ mΨ mΣ

10 200 400 600 800 1000 k in MeV 1 1.5 2 2.5 ZΣ

  • ZΠ
  • ZΣ
  • ZΠ
  • 22
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Increasing the temperature

10 100 200 300 T in MeV 1 1.5 2 2.5 ZΣ

  • ZΠ
  • ZΣ
  • ZΠ
  • 23
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Summary

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Conclusion and outlook

  • Using truncations beyond LPA is important to get consistent results

in the quark-meson model

  • Using a 3-dimensional regulator causes a large splitting between Z

and Z⊥ at low temperature → Extend the calculations to the whole phase diagram → Calculate mesonic spectral functions → Use a 4-dimensional regulator → Include Zψ and hk

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