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Introduction and Motivation Theoretical Framework Flow Equations for the Quark-Meson Model Numerical Results in LPA Summary and Outlook

Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Master’s Thesis Presentation Alexander Stegemann

TU Darmstadt Institute of Nuclear Physics

01.02.2016

Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework Flow Equations for the Quark-Meson Model Numerical Results in LPA Summary and Outlook

Contents

1

Introduction and Motivation

2

Theoretical Framework QCD and the Quark-Meson Model Functional Renormalization Group Analytic Continuation Procedure

3

Flow Equations for the Quark-Meson Model Flow Equation in LPA Flow Equations in LPA′

4

Numerical Results in LPA

5

Summary and Outlook

Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework Flow Equations for the Quark-Meson Model Numerical Results in LPA Summary and Outlook

Contents

1

Introduction and Motivation

2

Theoretical Framework QCD and the Quark-Meson Model Functional Renormalization Group Analytic Continuation Procedure

3

Flow Equations for the Quark-Meson Model Flow Equation in LPA Flow Equations in LPA′

4

Numerical Results in LPA

5

Summary and Outlook

Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework Flow Equations for the Quark-Meson Model Numerical Results in LPA Summary and Outlook

Spectral Functions

Ω ΡΩ Π ΩmΠ Ω ΡΩ Π2Γ ΠΨΨ ΩmΠ Ω2mΨ

Spectral functions contain a multitude of information: Particle masses, decay widths, decay channels, . . .

Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework Flow Equations for the Quark-Meson Model Numerical Results in LPA Summary and Outlook

Analytic Continuation

Ω iΩn

? − →

Ω iΩn

Euclidean QFT at finite temperature: Discrete imaginary energies How go back to real continuous energies? ⇒ Analytic continuation on the level of the FRG flow equations

Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework Flow Equations for the Quark-Meson Model Numerical Results in LPA Summary and Outlook QCD and the Quark-Meson Model Functional Renormalization Group Analytic Continuation Procedure

Contents

1

Introduction and Motivation

2

Theoretical Framework QCD and the Quark-Meson Model Functional Renormalization Group Analytic Continuation Procedure

3

Flow Equations for the Quark-Meson Model Flow Equation in LPA Flow Equations in LPA′

4

Numerical Results in LPA

5

Summary and Outlook

Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework Flow Equations for the Quark-Meson Model Numerical Results in LPA Summary and Outlook QCD and the Quark-Meson Model Functional Renormalization Group Analytic Continuation Procedure

Running Coupling of QCD

[S. Beringer et al., Phys. Rev. D 86, 010001 (2012)] Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework Flow Equations for the Quark-Meson Model Numerical Results in LPA Summary and Outlook QCD and the Quark-Meson Model Functional Renormalization Group Analytic Continuation Procedure

QCD Lagrangian

LQCD =

Nf

• i=1

¯ ψi

• i /

D − mi

• ψi − 1

4F a

µνF aµν

Covariant derivative Dµ = ∂µ − igAµ = ∂µ − ig λa

2 Aa µ

Field strength tensor F a

µν = ∂µAa ν − ∂νAa µ + gfabcAb µAc ν

Chiral symmetry is broken explicitly and spontaneously

Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework Flow Equations for the Quark-Meson Model Numerical Results in LPA Summary and Outlook QCD and the Quark-Meson Model Functional Renormalization Group Analytic Continuation Procedure

Quark-Meson Model

Effective low energy model for QCD with two flavors Mimics the chiral properties of QCD Quarks and mesons as effective degrees of freedom σ ≡ ¯ ψψ

• π ≡ i ¯

ψ τγ5ψ − → φ = (σ, π)T LQM = ¯ ψ

• i/

∂ − h (σ + iγ5 τ π)

• ψ + 1

2(∂µφ)2 − U(φ2) + cσ

Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework Flow Equations for the Quark-Meson Model Numerical Results in LPA Summary and Outlook QCD and the Quark-Meson Model Functional Renormalization Group Analytic Continuation Procedure

Contents

1

Introduction and Motivation

2

Theoretical Framework QCD and the Quark-Meson Model Functional Renormalization Group Analytic Continuation Procedure

3

Flow Equations for the Quark-Meson Model Flow Equation in LPA Flow Equations in LPA′

4

Numerical Results in LPA

5

Summary and Outlook

Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework Flow Equations for the Quark-Meson Model Numerical Results in LPA Summary and Outlook QCD and the Quark-Meson Model Functional Renormalization Group Analytic Continuation Procedure

Renormalization Group

Coarse graining by summarizing degrees of freedom RG equations describe the changing of the couplings ⇒ Macroscopic description based on a microscopic theory

[T. Herbst, diploma thesis] Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework Flow Equations for the Quark-Meson Model Numerical Results in LPA Summary and Outlook QCD and the Quark-Meson Model Functional Renormalization Group Analytic Continuation Procedure

Wilson’s Approach

Momentum space RG: Successive integration over momentum shells Z =

• Λ

b

D¯ Φ

• Λ

b <|p|≤Λ

D ˜ φ e−S[ ¯

φ, ˜ φ] ≡

• Λ

b

D ¯ φ e−SW[ ¯

φ]

Analogies between position space and momentum space RG Lattice spacing a ← → UV cutoff Λ ∼ 1

a

Blockspin transformation ← → Integration over one momentum a′ = a · b, b > 1 shell: Λ′ = Λ

b < |p| ≤ Λ

Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework Flow Equations for the Quark-Meson Model Numerical Results in LPA Summary and Outlook QCD and the Quark-Meson Model Functional Renormalization Group Analytic Continuation Procedure

Wetterich Equation

Exact one-loop equation for the effective average action ∂kΓk[φ] = 1 2 STr

• Γ(2)

k [φ] + Rk

−1 ∂kRk

• kk 1

2 Interpolation between the bare action Sbare and the full quantum effective action Γ Γk

k→Λ

− − − → Sbare Γk

k→0

− − − → Γ

Γk=Λ = Sbare Γk=0 ≡ Γ

[H. Gies, arXiv:hep-ph/0611146v1] Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework Flow Equations for the Quark-Meson Model Numerical Results in LPA Summary and Outlook QCD and the Quark-Meson Model Functional Renormalization Group Analytic Continuation Procedure

Regulator Functions

p

2

k

2

k

2

Rk (d/dt) Rk

Rk acts as a momentum dependent mass term and ensures the successive integration of fluctuations

[H. Gies, arXiv:hep-ph/0611146v1] Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework Flow Equations for the Quark-Meson Model Numerical Results in LPA Summary and Outlook QCD and the Quark-Meson Model Functional Renormalization Group Analytic Continuation Procedure

Contents

1

Introduction and Motivation

2

Theoretical Framework QCD and the Quark-Meson Model Functional Renormalization Group Analytic Continuation Procedure

3

Flow Equations for the Quark-Meson Model Flow Equation in LPA Flow Equations in LPA′

4

Numerical Results in LPA

5

Summary and Outlook

Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework Flow Equations for the Quark-Meson Model Numerical Results in LPA Summary and Outlook QCD and the Quark-Meson Model Functional Renormalization Group Analytic Continuation Procedure

Analytic Continuation Procedure

Use periodicity of the occupation numbers:

nB(E + ip0) = nB(E) NF(E + ip0) = NF(E)

Replace discrete Euclidean energy by a continous frequency ω:

∂kΓ(2),R

k

(ω) = − lim

ǫ→0 ∂kΓ(2),E k

• p0 = −i(ω+iǫ)

. Ω iΩn

Spectral functions can be written as

ρ(ω, p) = 1 π Im Γ(2),R(ω)

• Re Γ(2),R(ω)2 +

Im Γ(2),R(ω)2

Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework Flow Equations for the Quark-Meson Model Numerical Results in LPA Summary and Outlook Flow Equation in LPA Flow Equations in LPA′

Contents

1

Introduction and Motivation

2

Theoretical Framework QCD and the Quark-Meson Model Functional Renormalization Group Analytic Continuation Procedure

3

Flow Equations for the Quark-Meson Model Flow Equation in LPA Flow Equations in LPA′

4

Numerical Results in LPA

5

Summary and Outlook

Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework Flow Equations for the Quark-Meson Model Numerical Results in LPA Summary and Outlook Flow Equation in LPA Flow Equations in LPA′

Derivative Expansion

Wetterich equation is a functional differential equation ⇒ Use truncation schemes Derivative expansion

Γk[φ] =

• ddx
• Uk(φ2) + 1

2Zk(φ2)(∂µφ)2 + 1 4Yk(φ2)(∂µφ2)2 + O(∂4)

• Quark-meson model in LPA

Γk =

• d4x
• ¯

ψ / ∂ − µγ0 + h (σ + i τ πγ5) ψ + 1 2 (∂µφ)2 + Uk

• φ2

− cσ

• ⇒ Uk is the only scale dependent quantity

Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework Flow Equations for the Quark-Meson Model Numerical Results in LPA Summary and Outlook Flow Equation in LPA Flow Equations in LPA′

Flow Equation in LPA

Flow equation for the effective potential

∂kUk = k4 12π2

• − 2NcNf

Ek,ψ

• tanh

Ek,ψ + µ

2T

• + tanh

Ek,ψ − µ

2T

• +

1 Ek,σ coth

Ek,σ

2T

• +

3 Ek,π coth

Ek,π

2T

Energies

E 2

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

E 2

k,π = k2 + m2 k,π = k2 + 2U′ k

E 2

k,ψ = k2 + m2 k,ψ = k2 + h2σ2

Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework Flow Equations for the Quark-Meson Model Numerical Results in LPA Summary and Outlook Flow Equation in LPA Flow Equations in LPA′

Contents

1

Introduction and Motivation

2

Theoretical Framework QCD and the Quark-Meson Model Functional Renormalization Group Analytic Continuation Procedure

3

Flow Equations for the Quark-Meson Model Flow Equation in LPA Flow Equations in LPA′

4

Numerical Results in LPA

5

Summary and Outlook

Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework Flow Equations for the Quark-Meson Model Numerical Results in LPA Summary and Outlook Flow Equation in LPA Flow Equations in LPA′

LPA′

Quark-meson model in LPA′

Γk =

• d4x
• ¯

ψ/ ∂ − µγ0 + h(σ + iγ5 τ π) ψ + 1 2Zk(∂µφ)2 + 1 8Yk

• ∂µφ22 + Uk
• φ2

At finite temperature

ΓB

k =

• d4x

1

2 Zk,

• ∂0φ2 + 1

2 Zk,⊥

• ∂iφ2 + 1

8 Yk,

• ∂0φ22 + 1

8 Yk,⊥

• ∂iφ22 + Uk
• φ2

Notation ˜ Z ≡ Zσ ≡ Zk + σ2Yk Z ≡ Zπ ≡ Zk

Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework Flow Equations for the Quark-Meson Model Numerical Results in LPA Summary and Outlook Flow Equation in LPA Flow Equations in LPA′

Flow Equations in LPA′ (1)

Flow equation for the effective potential

∂kUk = k4 12π2

• − 2NcNf

Ek,ψ

• tanh

Ek,ψ + µ

2T

• + tanh

Ek,ψ − µ

2T

• +

˜ Z⊥ ˜ ZEk,σ

• 1 − ˜

η 5

• coth

Ek,σ

2T

• +

3Z⊥ ZEk,π

• 1 − η

5

• coth

Ek,π

2T

• Anomalous dimensions

˜ η ≡ ησ ≡ −k d dk ln ˜ Z⊥ η ≡ ηπ ≡ −k d dk ln Z⊥

Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework Flow Equations for the Quark-Meson Model Numerical Results in LPA Summary and Outlook Flow Equation in LPA Flow Equations in LPA′

Flow Equations in LPA′ (2)

Flow equations for the mesonic two-point functions kk,Σ

2

Σ Σ Σ Σ

• Σ

Π Π Σ

1 2

Σ Σ Σ

1 2

Π Σ Σ

2

Σ Ψ Ψ Σ

kk,Π

2

Π Σ Π Π

• Π

Π Σ Π

1 2

Σ Π Π

1 2

Π Π Π

2

Π Ψ Ψ Π

Mesonic three- and four-point vertices are extracted from Uk Quark-meson three-point vertices are given by Γ(1,1,1)

k, ¯ ψψσ = h

Γ(1,1,1)

k, ¯ ψψπi = ihγ5τi

Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework Flow Equations for the Quark-Meson Model Numerical Results in LPA Summary and Outlook Flow Equation in LPA Flow Equations in LPA′

Flow Equations in LPA′ (3)

Flow equations for the wave function renormalizations

Γ(2)

k,φaφb(p) =

• ˜

Zp2

0 + ˜

Z⊥ p2 + 2U′

k + 4σ2U′′ k

• δa1δb1

+

• Zp2

0 + Z⊥

p2 + 2U′

k

• (δa2δb2 + δa3δb3 + δa4δb4)
• ⇒ ∂k ˜

Z ∼

• ∂

∂p2

• δ

δφ1(−p) δ δφ1(p)∂kΓk

• p=0
• p0=0

and similar for ˜ Z⊥, Z and Z⊥

Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework Flow Equations for the Quark-Meson Model Numerical Results in LPA Summary and Outlook Flow Equation in LPA Flow Equations in LPA′

Flow Equations in LPA′ (4)

Flow equations for the wave function renormalizations

∂k ˜ Z =

• ∂

∂p2 ∂kΓ(2)

k,σ(p0, 0)

• p0=0

∂k ˜ Z⊥ =

• ∂

∂ p2 ∂kΓ(2)

k,σ(0,

p)

• p=0

∂kZ =

• ∂

∂p2 ∂kΓ(2)

k,π(p0, 0)

• p0=0

∂kZ⊥ =

• ∂

∂ p2 ∂kΓ(2)

k,π(0,

p)

• p=0

Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework Flow Equations for the Quark-Meson Model Numerical Results in LPA Summary and Outlook

Contents

1

Introduction and Motivation

2

Theoretical Framework QCD and the Quark-Meson Model Functional Renormalization Group Analytic Continuation Procedure

3

Flow Equations for the Quark-Meson Model Flow Equation in LPA Flow Equations in LPA′

4

Numerical Results in LPA

5

Summary and Outlook

Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework Flow Equations for the Quark-Meson Model Numerical Results in LPA Summary and Outlook

Numerical Procedure

Functional differential equation has been turned into a PDE ∂kUk = f (k, φ2, U′

k, U′′ k )

Uk is discretized in the direction of φ2 ⇒ Solve coupled ordinary differential equations at each grid point Number of grid points used: 60 to 100 Ansatz for Uk in the UV Uk=Λ(φ2) = b0 + b1 · φ2 + b3 · φ4

Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework Flow Equations for the Quark-Meson Model Numerical Results in LPA Summary and Outlook

Phase Diagram

Chiral order parameter σ0 decreases with increasing T and µ Critical endpoint at µ ≈ 293 MeV, T ≈ 10 MeV

Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework Flow Equations for the Quark-Meson Model Numerical Results in LPA Summary and Outlook

Curvature Masses

Curvature masses determine thresholds for the decay processes Masses and order parameter in the vacuum: σ0 = 93.5 MeV mψ = 299.1 MeV mσ = 509.4 MeV mπ = 136.9 MeV

50 100 150 200 250 300T in MeV 100 200 300 400 500 MeV

Μ0 MeV 100 grid points

Σ0 mΠ mΨ mΣ 100 200 300 400Μ in MeV 100 200 300 400 500 MeV

T10 MeV 100 grid points

Σ0 mΠ mΨ mΣ

Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework Flow Equations for the Quark-Meson Model Numerical Results in LPA Summary and Outlook

Contents

1

Introduction and Motivation

2

Theoretical Framework QCD and the Quark-Meson Model Functional Renormalization Group Analytic Continuation Procedure

3

Flow Equations for the Quark-Meson Model Flow Equation in LPA Flow Equations in LPA′

4

Numerical Results in LPA

5

Summary and Outlook

Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework Flow Equations for the Quark-Meson Model Numerical Results in LPA Summary and Outlook

Summary

Short review of QCD, QM model and FRG Analytic continuation procedure on the level of the flow equations Flow equation in LPA for the effective potential

Study chiral symmetry breaking

Flow equations in LPA′ for . . .

. . . the effective potential . . . the mesonic two-point functions . . . the wave function renormalizations

Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework Flow Equations for the Quark-Meson Model Numerical Results in LPA Summary and Outlook

Outlook (1)

Calculate mesonic spectral functions for finite temperature and chemical potential Problem in LPA at µ = 0 MeV and T = 10 MeV: mπ,curv ≈ 138 MeV mπ,pole ≈ 100 MeV

100 200 300 400 500 600 700 Ω MeV 104 0.01 1 100 UV

2

T 10 MeV ΡΣ ΡΠ

2 3 4 5 6 100 200 300 400 500 600 700

• 4

0.01 1 100

• MeV

100 200 300 400 500 600 700

• 4

0.01 1 100

• MeV

100 200 300 400

4

0.01 1 100

• MeV

100 200 300 400

• 4

0.01 1 100

• MeV

[R.-A. Tripolt, N. Strodthoff, L. von Smekal, and J. Wambach, arXiv:hep-ph/1311.0630] Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework Flow Equations for the Quark-Meson Model Numerical Results in LPA Summary and Outlook

Outlook (1)

Calculate mesonic spectral functions for finite temperature and chemical potential Problem in LPA at µ = 0 MeV and T = 10 MeV (PQM): mπ,curv ≈ 137 MeV mπ,pole ≈ 100 MeV

200 400 600 800 1000 104 0.01 1 100 Ω MeV GeV2

T 10 MeV

ΡΠ ΡΣ 1 3 6 5 200 400 600 800 1000

4

0.01 1 100

• MeV

200 400 600 800 1000

4

0.01 1 100

• MeV

[C. Jung, master’s thesis] Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework Flow Equations for the Quark-Meson Model Numerical Results in LPA Summary and Outlook

Outlook (2)

Pion mass in LPA′

100 200 300 400 50 100 150 mπ [MeV] T [MeV]

mcur mpol mscr

[A. J. Helmboldt, J. M. Pawlowski, and N. Strodthoff, arXiv:1409.8414v1] Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework Flow Equations for the Quark-Meson Model Numerical Results in LPA Summary and Outlook

Outlook (3)

PQM model in LPA′ Examine a possible regulator dependence Include spatial momentum Include fermionic wave function renormalizations Calculate quark spectral functions

Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization