Flow Eqns. for Spectral Functions Including Wave Function - PowerPoint PPT Presentation

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 Introduction and Motivation 1 Theoretical Framework 2 QCD and the Quark-Meson Model Functional Renormalization Group Analytic Continuation Procedure Flow Equations for the Quark-Meson Model 3 Flow Equation in LPA Flow Equations in LPA ′ Numerical Results in LPA 4 Summary and Outlook 5 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 Introduction and Motivation 1 Theoretical Framework 2 QCD and the Quark-Meson Model Functional Renormalization Group Analytic Continuation Procedure Flow Equations for the Quark-Meson Model 3 Flow Equation in LPA Flow Equations in LPA ′ Numerical Results in LPA 4 Summary and Outlook 5 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 Ρ � Ω � Ρ � Ω � Π� 2 Γ Π�ΨΨ Π Ω Ω Ω � 2 m Ψ Ω� m Π Ω� m Π 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 i Ω n i Ω n ? − → Ω Ω 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 QCD and the Quark-Meson Model Flow Equations for the Quark-Meson Model Functional Renormalization Group Numerical Results in LPA Analytic Continuation Procedure Summary and Outlook Contents Introduction and Motivation 1 Theoretical Framework 2 QCD and the Quark-Meson Model Functional Renormalization Group Analytic Continuation Procedure Flow Equations for the Quark-Meson Model 3 Flow Equation in LPA Flow Equations in LPA ′ Numerical Results in LPA 4 Summary and Outlook 5 Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework QCD and the Quark-Meson Model Flow Equations for the Quark-Meson Model Functional Renormalization Group Numerical Results in LPA Analytic Continuation Procedure Summary and Outlook 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 QCD and the Quark-Meson Model Flow Equations for the Quark-Meson Model Functional Renormalization Group Numerical Results in LPA Analytic Continuation Procedure Summary and Outlook QCD Lagrangian N f ψ i − 1 � ¯ 4 F a µν F a µν � i / � L QCD = ψ i D − m i i = 1 Covariant derivative D µ = ∂ µ − i gA µ = ∂ µ − i g λ a 2 A a µ Field strength tensor F a µν = ∂ µ A a ν − ∂ ν A a µ + gf abc A b µ A c ν Chiral symmetry is broken explicitly and spontaneously Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework QCD and the Quark-Meson Model Flow Equations for the Quark-Meson Model Functional Renormalization Group Numerical Results in LPA Analytic Continuation Procedure Summary and Outlook 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 ¯ π ) T � ψ� τγ 5 ψ − → φ = ( σ,� ψ + 1 � � 2 ( ∂ µ φ ) 2 − U ( φ 2 ) + c σ ¯ i / L QM = ψ ∂ − h ( σ + i γ 5 � τ� π ) Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework QCD and the Quark-Meson Model Flow Equations for the Quark-Meson Model Functional Renormalization Group Numerical Results in LPA Analytic Continuation Procedure Summary and Outlook Contents Introduction and Motivation 1 Theoretical Framework 2 QCD and the Quark-Meson Model Functional Renormalization Group Analytic Continuation Procedure Flow Equations for the Quark-Meson Model 3 Flow Equation in LPA Flow Equations in LPA ′ Numerical Results in LPA 4 Summary and Outlook 5 Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework QCD and the Quark-Meson Model Flow Equations for the Quark-Meson Model Functional Renormalization Group Numerical Results in LPA Analytic Continuation Procedure Summary and Outlook 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 QCD and the Quark-Meson Model Flow Equations for the Quark-Meson Model Functional Renormalization Group Numerical Results in LPA Analytic Continuation Procedure Summary and Outlook Wilson’s Approach Momentum space RG: Successive integration over momentum shells � � � φ e − S [ ¯ φ, ˜ φ ] ≡ φ e − S W [ ¯ D ¯ D ˜ D ¯ φ ] Z = Φ Λ Λ Λ b < | p |≤ Λ b b Analogies between position space and momentum space RG UV cutoff Λ ∼ 1 Lattice spacing a ← → 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 QCD and the Quark-Meson Model Flow Equations for the Quark-Meson Model Functional Renormalization Group Numerical Results in LPA Analytic Continuation Procedure Summary and Outlook Wetterich Equation Exact one-loop equation for the effective average action ∂ k Γ k [ φ ] = 1 �� � − 1 � Γ ( 2 ) � k � k � 1 2 STr k [ φ ] + R k ∂ k R k 2 Interpolation between the Γ k = Λ = S bare bare action S bare and the full quantum effective action Γ k → Λ Γ k − − − → S bare Γ k =0 ≡ Γ k → 0 Γ k − − − → Γ [H. Gies, arXiv:hep-ph/0611146v1] Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework QCD and the Quark-Meson Model Flow Equations for the Quark-Meson Model Functional Renormalization Group Numerical Results in LPA Analytic Continuation Procedure Summary and Outlook Regulator Functions (d/dt) R k 2 k R k 2 k 2 p R k 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 QCD and the Quark-Meson Model Flow Equations for the Quark-Meson Model Functional Renormalization Group Numerical Results in LPA Analytic Continuation Procedure Summary and Outlook Contents Introduction and Motivation 1 Theoretical Framework 2 QCD and the Quark-Meson Model Functional Renormalization Group Analytic Continuation Procedure Flow Equations for the Quark-Meson Model 3 Flow Equation in LPA Flow Equations in LPA ′ Numerical Results in LPA 4 Summary and Outlook 5 Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization

Introduction and Motivation Theoretical Framework QCD and the Quark-Meson Model Flow Equations for the Quark-Meson Model Functional Renormalization Group Numerical Results in LPA Analytic Continuation Procedure Summary and Outlook Analytic Continuation Procedure Use periodicity of the occupation numbers: i Ω n n B ( E + i p 0 ) = n B ( E ) N F ( E + i p 0 ) = N F ( E ) Ω Replace discrete Euclidean energy by a continous frequency ω : ∂ k Γ ( 2 ) , R ǫ → 0 ∂ k Γ ( 2 ) , E � p 0 = − i ( ω + i ǫ ) � ( ω ) = − lim . k k Spectral functions can be written as Im Γ ( 2 ) , R ( ω ) p ) = 1 ρ ( ω,� Re Γ ( 2 ) , R ( ω ) � 2 + � Im Γ ( 2 ) , R ( ω ) � 2 π � Alexander Stegemann Flow Eqns. for Spectral Functions Including Wave Function Renormalization