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

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, . . . Attempts to solve QCD [C. Patrignani et al. , Chin. Phys. , 2016] Lattice QCD Effective theories FRG DSEs 1

Functional renormalisation group

Wilsonian renormalisation • Euclidean generating functional for a scalar field ϕ � � � � d 4 x J ( x ) ϕ ( x ) Z [ J ] = D ϕ exp − S [ ϕ ] + • Wilsonian renormalisation � � � � � d 4 x J ( x ) ϕ ( x ) Z [ J ] = D ϕ q ≤ k D ϕ q > k exp − S [ ϕ ] + � �� � ≡ Z k [ J ] → Integrate out modes successively 2

b b b Flow in theory space (1) c 1 Γ k = Λ = S Γ k = 0 = Γ c 2 c 3 3

Introducing a scale dependence • Generating functional � � � � d 4 x J ( x ) ϕ ( x ) Z k [ J ] = D ϕ exp − S [ ϕ ] − ∆ S k [ ϕ ] + • Regulator insertion � d 4 p ∆ S k [ ϕ ] = 1 ( 2 π ) 4 ϕ ( p ) R k ( p 2 ) ϕ ( − p ) 2 • Effective average action ( φ ( x ) = � ϕ ( x ) � ) �� � d 4 x J ( x ) φ ( x ) − ln Z k [ J ] Γ k [ φ ] = sup − ∆ S k [ φ ] J 4

Wetterich equation • (Exact) Wetterich equation �� � ∂ k Γ k = 1 � − 1 Γ ( 2 ) 2 STr + R k ∂ k R k k • Regulator R k ensures correct integration limits k → Λ k → 0 Γ k − − − → S Γ k − − − → Γ • No sign problem 5

b b b Flow in theory space (2) c 1 Γ k = Λ = S R 1 R 2 R 3 Γ k = 0 = Γ c 2 c 3 6

Solving the Wetterich equation • (Exact) Wetterich equation �� � ∂ k Γ k = 1 � − 1 Γ ( 2 ) 2 STr + R k ∂ k R k k • In practice, truncations are needed • Derivative expansion � � � + 1 ( ∂ µ φ ) 2 + . . . � φ 2 � � φ 2 � d 4 x Γ k = U k 2 Z k 7

b b b Flow in theory space (3) c 1 Γ k = Λ = S R 1 R 2 R 3 Γ k = 0 = Γ c 2 c 3 8

Quark-meson model in LPA

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 � � ✯ 1 ✟ � + 1 2 ✟✟✟ ✘ ( ∂ µ φ ) 2 ✘✘ � φ 2 � � φ 2 � d 4 x Γ k = U k Z k + . . . • Ansatz for the effective average action of the quark-meson model � � � ψ + 1 � / � 2 ( ∂ µ φ ) 2 + U k � φ 2 � ¯ d 4 x Γ k = ψ ∂ − µγ 0 + h Σ 5 − c σ π ) T Σ 5 = ( σ + i γ 5 � π� τ ) , φ = ( σ,� 9

Flow equation for U k • Using a 3-dimensional Litim regulator � � � E ψ + µ � � E ψ − µ �� k 4 − 2 N c N f ∂ k U k = tanh + tanh 12 π 2 E ψ 2 T 2 T � E σ � E π � � � + 1 + 3 coth coth E σ 2 T E π 2 T σ = k 2 + m 2 σ = k 2 + 2 U ′ k + 4 � σ � 2 U ′′ E 2 k π = k 2 + m 2 π = k 2 + 2 U ′ E 2 k ψ = k 2 + m 2 ψ = k 2 + h 2 � σ � 2 E 2 10

Numerical solution • Discretise the partial differential equation ∂ k U k • Tune the UV parameters in the vacuum ( m Λ , λ Λ , c , h ) = 1 Λ φ 2 + 1 � φ 2 � � φ 2 � 2 2 m 2 U Λ 4 λ Λ MeV 800 700 600 m Σ 500 400 m Ψ 300 200 m Π 100 Σ 0 0 k in MeV 40 200 400 600 800 1000 11

Inconsistencies in LPA • Pion curvature and pole masses show a large discrepancy in the vacuum m π, curv ≈ 138 MeV m π, 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

Quark-meson model in LPA ′

Quark-Meson Model in LPA ′ • LPA ′ = including a scale dependent wave function renormalisation � � � � φ 2 � + 1 ✘ � φ 2 � ( ∂ µ φ ) 2 ✘✘ � d 4 x Γ k = U k 2 Z k + . . . � • Ansatz for the purely bosonic part of the quark-meson model � 1 � � 2 Z σ ( ∂ µ σ ) 2 + 1 π ) 2 + U k ( φ 2 ) − c σ d 4 x Γ k , B = 2 Z π ( ∂ µ � • Flow equations for the wave function renormalisations � ∂ � δ δ ∂ k Z α ∼ δφ α ( p ) ∂ k Γ k α ∈ { σ, π } ∂ p 2 δφ α ( − p ) p = 0 13

Including a finite temperature • The time direction becomes compactified � β � � β = 1 d 4 x − d 3 x → d τ T 0 • p 0 becomes discrete (Matsubara frequencies) ω n = 2 n π T for bosons ν n = ( 2 n + 1 ) π T for fermions � � d 4 p d 3 p � ( 2 π ) 4 − → T ( 2 π ) 3 n ∈ Z 14

Including a finite temperature • Ansatz for the purely bosonic part of the quark-meson model � β � � 1 σ ( ∂ 0 σ ) 2 + 1 d 3 x 2 Z � 2 Z ⊥ σ ( ∂ i σ ) 2 Γ k , B = d τ 0 � + 1 π ) 2 + 1 π ) 2 + U k ( φ 2 ) − c σ 2 Z � 2 Z ⊥ π ( ∂ 0 � π ( ∂ i � • Flow equations for the perpendicular wave function renormalisations � � � � ∂ δ δ ∂ k Z ⊥ α ∼ δφ α ( p ) ∂ k Γ k ∂ | � p | 2 δφ α ( − p ) p 0 = 0 � p = 0 • Flow equations for the parallel wave function renormalisations � � ∂ k Γ ( 2 ) k ,α ( p 0 = 2 π T ) − ∂ k Γ ( 2 ) k ,α ( p 0 = 0 ) ∂ k Z � α = ( 2 π T ) 2 � p = 0 15

Flow equation for U k • Using a 3-dimensional Litim regulator � E ψ + µ � E ψ − µ � k 4 − 2 N c N f � � �� ∂ k U k = tanh + tanh 12 π 2 E ψ 2 T 2 T � E σ � E π � � Z ⊥ + 3 Z ⊥ � 1 − η σ � � � 1 − η π � σ π + coth coth Z � Z � 5 2 T 5 2 T σ E σ π E π � � k + 4 � σ � 2 U ′′ k 2 + 2 U ′ σ = Z ⊥ E 2 σ k Z � Z ⊥ σ σ � � k 2 + 2 U ′ π = Z ⊥ E 2 π k Z � Z ⊥ π π 16

Numerical solution • Discretise the partial differential equation ∂ k U k • Tune the UV parameters in the vacuum ( m Λ , λ Λ , c , h ) = 1 Λ φ 2 + 1 � φ 2 � � φ 2 � 2 2 m 2 U Λ 4 λ Λ MeV 800 700 600 500 m Σ 400 m Ψ 300 200 m Π 100 Σ 0 0 k in MeV 10 200 400 600 800 1000 17

Wave function renormalisations � ∂ � δ δ Z α ∼ δφ α ( p )Γ k α ∈ { σ, π } ∂ p 2 δφ α ( − p ) p = 0 σ = σ k 18

Scale dependence • Curvature masses and wave function renormalisations • Evaluating at the scale dependent minimum of U k T = 10 MeV, µ = 0 MeV MeV 800 3 Z Σ � 700 Z Π � 600 2.5 500 m Σ 400 2 m Ψ Z Π � 300 200 1.5 m Π Z Σ � 100 Σ 0 0 k in MeV 1 k in MeV 10 200 400 600 800 1000 10 200 400 600 800 1000 19

Wave function renormalisations � ∂ � δ δ ∂ k Z α ∼ ∂ k δφ α ( p )Γ k α ∈ { σ, π } ∂ p 2 δφ α ( − p ) p = 0 σ = σ k → additional term ∼ ∂ k σ k � ∂ � δ δ ∂ k Z α ∼ ∂ k δφ α ( p )Γ k α ∈ { σ, π } ∂ p 2 p = 0 δφ α ( − p ) σ = σ fix → no additional term � ∂ � δ δ ∂ k Z α ∼ δφ α ( p ) ∂ k Γ k α ∈ { σ, π } ∂ p 2 δφ α ( − p ) p = 0 20

Correct IR values • Curvature masses and wave function renormalisations • Evaluating at the IR minimum T = 10 MeV, µ = 0 MeV MeV 800 2.5 Z Π 700 � 600 Z Σ � 500 m Σ 2 400 m Ψ Z Π � 300 1.5 200 m Π Z Σ � 100 Σ 0 0 k in MeV 1 k in MeV 10 200 400 600 800 1000 10 200 400 600 800 1000 21

Correct scale dependence • Curvature masses and wave function renormalisations • Iterative procedure to obtain the correct scale dependence T = 10 MeV, µ = 0 MeV MeV 2.5 Z Π 800 � 700 600 Z Σ � 2 500 m Σ 400 Z Π � m Ψ 300 1.5 200 m Π Z Σ � 100 Σ 0 0 k in MeV 1 k in MeV 10 200 400 600 800 1000 10 200 400 600 800 1000 22

Increasing the temperature 2.5 Z Π � Z Σ � 2 Z Π � 1.5 Z Σ � 1 T in MeV 10 100 200 300 23

Summary

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 Z ψ and h k 24