Spectral Functions from the Functional Renormalization Groupebratkov/Conferences/NeD-TURIC-2014/talks/0… · Spectral Functions from the Functional Renormalization Group Ralf-Arno

Spectral Functions from theFunctional Renormalization Group

Ralf-Arno Tripolta, Nils Strodthoffb, Lorenz von Smekala,c , Jochen Wambacha,d

a Theoriezentrum, Institut für Kernphysik, TU Darmstadtb Institut für Theoretische Physik, Ruprecht-Karls-Universität Heidelberg

c Institut für Theoretische Physik, Justus-Liebig-Universität Giessend GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt

NeD-2014 & TURIC-2014, Hersonissos, Crete, Greece

June 9th, 2014 | Ralf-Arno Tripolt | Spectral Functions from the FRG | 1

Outline

I) Introduction and motivation

II) Theoretical setup

I functional renormalization group (FRG)

I quark-meson model

I analytic continuation procedure

III) Results for finite temperature and chemical potential

I phase diagram

I spectral functions

IV) Summary and outlook


I) Introduction and Motivation

[courtesy L. Holicki]


What is a Spectral Function?

A spectral function ρ(ω) can be defined as:

I ρ(ω) ≡ i2π

(DR(ω)− DA(ω)

)For a free scalar field with mass m,the retarded and advanced propagators are:

I DR(ω) =(

(ω + iε)2 −m2)−1

I DA(ω) =(

(ω − iε)2 −m2)−1

and the spectral function is, for ε→ 0:

I ρ(ω) = sgn(ω) δ(ω2 −m2)

Re DR(ω) Im DR(ω)

0.0 0.5 1.0 1.5 2.0

-60

-40

-20

0

20

40

0.0 0.5 1.0 1.5 2.0

-60

-40

-20

0

20

40

0.0 0.5 1.0 1.5 2.0

-10

-5

0

5

10

15

20

ω = m ω = m

ρ(ω)

ω = m[J. Kapusta and C. Gale, Finite-Temperature Field Theory, Cambridge Univ. Press (2006)]


Why are Spectral Functions interesting?

Spectral functions give information on the particlespectrum and determine both real-time andimaginary-time propagators,

I DR(ω) = −∫

dω′ρ(ω′)

ω′ − ω − iε

I DA(ω) = −∫

dω′ρ(ω′)

ω′ − ω + iε

I DE (ωn) =∫

dω′ρ(ω′)ω′ + iωn

and thus allow access to many observables,e.g. transport coefficients like the shear viscosity:

I η = limω→0

1ω

∫d4x eiωt

〈[Tij (x), T

ij (0)]〉

[B. Mueller, arXiv: 1309.7616]


The Analytic Continuation Problem

In Euclidean QFT, energies are discrete for T > 0and there are infinitely many analytic continuationsD(z) from the imaginary to the real axis with:

I D(z)|z=iωn = DE (iω)

Ambiguity is resolved by imposing physical(Baym-Mermin) boundary conditions:

I D(z) goes to zero as |z| → ∞

I D(z) is analytic outside the real axis

Im z

Re z

When based on numeric results for the discrete frequencies ωn, cf. lattice QCD,this analytic reconstruction is an ill-posed problem and very difficult.

[N. Landsman, C. v. Weert, Physics Reports 145, 3&4 (1987) 141][G. Baym, N. Mermin, J. Math. Phys. 2 (1961) 232]


II) Theoretical Setup



Functional Renormalization Group

Flow equation for the effective average action Γk :

∂kΓk =12

STr(∂k Rk

[Γ(2)k + Rk

]−1)

I Γk interpolates between bare action S at k = Λand full quantum effective action Γ at k = 0

I regulator Rk acts as a mass term and suppressesfluctuations with momenta smaller than k

[C. Wetterich, Phys. Lett. B301 (1993) 90]

Γk=Λ = S

k

k −∆k

Γk=0 = Γ


Quark-Meson Model

Ansatz for the scale-dependent effective average action:

Γk [ψ̄,ψ,φ] =∫

d4x{ψ̄(∂/ + h(σ + i~τ~πγ5)− µγ0

)ψ + 12 (∂µφ)

2 + Uk (φ2)− cσ

}

I effective low-energy model for QCD with two flavors

I describes spontaneous and explicit chiral symmetry breaking

I flow equation for the effective average action:


Flow Equations for 2-Point Functions

Quark-meson vertices given by Yukawa interaction:

Γ(2,1)ψ̄ψσ

= h, Γ(2,1)ψ̄ψ~π

= ihγ5~τ

Mesonic vertices obtained from scale-dependent effective potential:

Γ(0,3)k ,φiφjφm =δ3Uk

δφmδφjδφi, Γ(0,4)k ,φiφjφmφn =

δ4Ukδφnδφmδφjδφi


Decay Channels for Sigma Meson

Possible processes affecting the sigma 2-point function Γ(2)σ,k :

I σ∗ → σ σ, ω ≥ 2 mσ

I σ∗ → π π, ω ≥ 2 mπ

I σ∗ → ψ̄ ψ, ω ≥ 2 mψ

Diagrams involving 4-point vertices only give rise to ω-independent contributions.


Decay Channels for Pions

Possible processes affecting the pion 2-point function Γ(2)π,k :

I π∗ → σ π, ω ≥ mσ + mπ

I π∗ π → σ, ω ≤ mσ −mπ

I π∗ → ψ̄ ψ, ω ≥ 2 mψ

Diagrams involving 4-point vertices only give rise to ω-independent contributions.


Analytic Continuation Procedure

Two-step procedure on the level of flow equations to obtain Γ(2),R(ω):

1) Use periodicity in external energy p0 = n 2πT :

I nB,F (E + ip0)→ nB,F (E)

2) Substitute p0 by continuous real frequency:

I Γ(2),R(ω) = − lim�→0

Γ(2),E (p0 = iω − �)

Spectral function is then given by:

I ρ(ω) =1π

Im Γ(2),R(ω)(Re Γ(2),R(ω)

)2 + (Im Γ(2),R(ω))2

p0 p0

[N. Landsman, C. v. Weert, Physics Reports 145, 3&4 (1987) 141]

[A. Das, R. Francisco, J. Frenkel, Phys. Rev. D 86 (2012) 047702]


III) Results for finite T and µ



Phase Diagram

Phase diagram of the quark-meson model:

I chiral order parameter σ0 ≡ fπ ,decreases with darker color

I critical endpoint at µ = 293 MeVand T = 10 MeV

I we will study spectral functionsalong µ = 0 and T = 10 MeV line

100 200 300 400Μ @MeVD

50

100

150

200

250T @MeVD

270 290 310

5

10

15

20

[R.-A. T., N. Strodthoff, L. v. Smekal, J. Wambach, Phys. Rev. D 89, 034010 (2014)]


Masses and Order Parameter

Screening masses and order parameter invacuum, T = 0 and µ = 0:

I σ0 = 93.5 MeV

I mπ = 138 MeV

I mσ = 509 MeV

I mψ = 299 MeV

Screening masses determine thresholds inspectral functions, e.g. at T = 10 MeV, µ = 0:

I σ∗ → π π, ω ≥ 2 mπ ≈ 280 MeVI σ∗ → ψ̄ ψ, ω ≥ 2 mψ ≈ 600 MeV

mΣ

mΨ

mΠ

fΠ

0 50 100 150 200 250 300T @MeVD0

100

200

300

400

500

@MeVDΜ = 0 MeV

mΣ

mΨ

mΠ

fΠ

200 250 300 350 400Μ @MeVD0

100

200

300

400

500

@MeVDT = 10 MeV



Spectral Functions along µ = 0

0 100 200 300 400 500 600 700Ω @MeVD10-4

0.01

1

100

@LUV-2 DT=10 MeV

ΡΣ

ΡΠ

2

3

456

1: σ∗ → σσ

2: σ∗ → ππ

3: σ∗ → ψ̄ψ

4: π∗ → σπ

5: π∗π → σ

6: π∗ → ψ̄ψ




0 100 200 300 400 500 600 700Ω @MeVD10-4

0.01

1

100

@LUV-2 DT=100 MeV

ΡΣ

ΡΠ

2

3

45

6

1: σ∗ → σσ

2: σ∗ → ππ

3: σ∗ → ψ̄ψ

4: π∗ → σπ

5: π∗π → σ

6: π∗ → ψ̄ψ




0 100 200 300 400 500 600 700Ω @MeVD10-4

0.01

1

100

@LUV-2 DT=150 MeV

ΡΣ

ΡΠ

12 3

4

5

6

1: σ∗ → σσ

2: σ∗ → ππ

3: σ∗ → ψ̄ψ

4: π∗ → σπ

5: π∗π → σ

6: π∗ → ψ̄ψ




0 100 200 300 40010-4

0.01

1

100

@LUV-2 DT=200 MeV

ΡΣΡΠ

3

5

6

0 100 200 300 400Ω @MeVD10-4

0.01

1

100

@LUV-2 DT=250 MeV

ΡΣΡΠ

3

6

1: σ∗ → σσ

2: σ∗ → ππ

3: σ∗ → ψ̄ψ

4: π∗ → σπ

5: π∗π → σ

6: π∗ → ψ̄ψ



Spectral Functions along µ = 0 - Animation

(Loading movie...)


Lavf55.25.101

spectral_functions.mp4Media File (video/mp4)

Spectral Functions along T = 10 MeV

0 100 200 300 400 500 600 700Ω @MeVD10-4

0.01

1

100

@LUV-2 DΜ=200 MeV

ΡΣ

ΡΠ

2

3

46

1: σ∗ → σσ

2: σ∗ → ππ

3: σ∗ → ψ̄ψ

4: π∗ → σπ

5: π∗π → σ

6: π∗ → ψ̄ψ




0 100 200 300 400 500 600 700Ω @MeVD10-4

0.01

1

100

@LUV-2 DΜ=292 MeV

ΡΣΡΠ

123

46

1: σ∗ → σσ

2: σ∗ → ππ

3: σ∗ → ψ̄ψ

4: π∗ → σπ

5: π∗π → σ

6: π∗ → ψ̄ψ




0 100 200 300 400 500 600 700Ω @MeVD10-4

0.01

1

100

@LUV-2 DΜ=292.8 MeV

ΡΣΡΠ

1

23

46

1: σ∗ → σσ

2: σ∗ → ππ

3: σ∗ → ψ̄ψ

4: π∗ → σπ

5: π∗π → σ

6: π∗ → ψ̄ψ




0 50 100 150 200 25010-4

0.01

1

100

@LUV-2 DΜ=292.97 MeV

ΡΣ ΡΠ

1

0 100 200 300 400Ω @MeVD10-4

0.01

1

100

@LUV-2 DΜ=400 MeV

ΡΣΡΠ

3

6

1: σ∗ → σσ

2: σ∗ → ππ

3: σ∗ → ψ̄ψ

4: π∗ → σπ

5: π∗π → σ

6: π∗ → ψ̄ψ



Summary and Outlook

A new method to obtain spectral functions at finite T and µ from the FRGhas been presented:

I involves analytic continuation from imaginary toreal frequencies on the level of the flow equations

I feasibility of the method demonstrated by calculatingmesonic spectral functions in the quark-meson model

I allows to iteratively take into account full momentum dependenceof 2-point functions without assumptions on structure of propagator

I inclusion of finite external spatial momenta will allow forcalculation of transport coefficients like the shear viscosity


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Spectral Functions from the Functional Renormalization Groupebratkov/Conferences/NeD-TURIC-2014/talks/0… · Spectral Functions from the Functional Renormalization Group Ralf-Arno