Spectral Functions and Transport Coefficients with the Functional ...

Spectral Functions and Transport Coefficientswith the Functional Renormalization Group

Ralf-Arno Tripolt1, Lorenz von Smekal2, Jochen Wambach1,3

1 ECT*, Trento, Italy2 Justus-Liebig-Universitt Gieen, Germany

3 TU Darmstadt, Germany

International Workshop XLIV on Gross Properties of Nuclei andNuclear Excitations - QCD matter: dense and hot

Hirschegg, January 21, 2016

January 21, 2016 | Ralf-Arno Tripolt | Spectral Functions and Transport Coefficients with 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

I quark and meson spectral functions

I mesonic contributions to the shear viscosity and to /s

IV) Summary and outlook


I) Introduction and motivation

[courtesy L. Holicki]


What is a spectral function?

HL



HL

= m



HL

= m

2



HL

= m

2

+

m+m


Why are spectral functions interesting?

Spectral functions determine bothreal-time and imaginary-time propagators,

I DR() =

d()

i

I DA() =

d()

+ i

I DE (p0) =

d() + ip0

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

I =124

lim0

lim|~p|0

1

d4x eipx

[Tij (x), T

ij (0)]

[B. Mueller, arXiv: 1309.7616]


Why are spectral functions interesting?

Calculation of dilepton excess spectrarequires in-medium spectral function:

dNlld4xd4q

= 2

33L(M)M2

ImEM (M, q) fB(q0; T ),

fB(q0; T ). . . thermal Bose function, = e2/4. . . EM coupling constant,L(M). . . final-state lepton phase space factor,M =

q20 ~q 2. . . dilepton invariant mass,

and the EM spectral function (M 1GeV):

ImEM ImD +

19

ImD +29

ImD

10-11

10-10

10-9

10-8

10-7

10-6

10-5

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

In-In +

+X

Ti=235 MeV

Tc=170 MeV

Dim

uo

n Y

ield

dN

/d

M

[

(20

MeV

)-1]

Dimuon Invariant Mass M (GeV/c2)

NA60 Datahadronic

QGPtotal thermal

[R. Rapp, H. van Hees, Phys.Lett. B 753 (2016) 586-590]


II) Theoretical setup



Functional Renormalization Group

Flow equation for the effective average action k :

kk =12

STr(k Rk

[(2)k + Rk

]1)[C. Wetterich, Phys. Lett. B 301 (1993) 90]

kGk12

[wikipedia.org/wiki/Functional_renormalization_group]

I k interpolates between bare action S at k = and effective action at k = 0

I regulator Rk acts as a mass term and suppresses fluctuationswith momenta smaller than k

I the use of 3D regulators allows for a simple analytic continuation procedure


Quark-meson model

Ansatz for the scale-dependent effective average action:

k [ ,,] =

d4x{(/ + h( + i~~5) 0

) +

12

()2 + Uk (

2) c}

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:

kGk12


Flow of the Effective Potentialat = 0 and T = 0

(Loading movie...)


Lavf55.44.100

pot_broken_k.mp4Media File (video/mp4)

Flow equations for two-point functions

kGk, sH2L s ss

s3

s sp

p2

s sy

y

1

2

s s

s 3

2

s s

p

kGk, pH2L p ps

p

p pp

s2

p py

y

1

2

p p

s 5

2

p p

p

kGk, yH2L y ys

y, y

y yy, y

s3

y yp

y, y3

y yy, y

p

I quark-meson vertices are given by (3)

= h, (3)~

= ih5~

I mesonic vertices from scale-dependent effective potential: U (3)k ,ijm , U(4)k ,ijmn

I one-loop structure and 3D regulators allow for a simple analytic continuation!

[R.-A. Tripolt, L. von Smekal, and J. Wambach, Phys. Rev. D 90, 074031 (2014)]


The analytic continuation problem

Calculations at finite temperature are often performed using imaginary energies:

ip0

ip0


The analytic continuation problem

Analytic continuation problem: How to get back to real energies?

ip0

?

?


Two-step analytic continuation procedure

1) Use periodicity in external imaginary energy ip0 = i2nT :

nB,F (E + ip0) nB,F (E)

2) Substitute p0 by continuous real frequency :

(2),R(,~p) = lim0

(2),E (ip0 i,~p)

ip0

?

?

Spectral function is then given by

(,~p) = Im(1/(2),R(,~p))/

Quark spectral function is parametrized as

k ,(,~p) = i~~p (A)k ,(,~p) (B)k ,(,~p) 0

(C)k ,(,~p)

[R.-A. T., N. Strodthoff, L. v. Smekal, and J. Wambach, Phys. Rev. D 89, 034010 (2014)][J. M. Pawlowski, N. Strodthoff, Phys. Rev. D 92, 094009 (2015)][N. Landsman and C. v. Weert, Physics Reports 145, 3&4 (1987) 141]


III) Results



Phase diagram of the quark-meson model

I chiral order parameter 0decreases towards higher T and

I a crossover is observed atT 175 MeV and = 0

I critical endpoint (CEP) at 292 MeV and T 10 MeV

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

100 200 300 400 @MeVD

50

100

150

200

250T @MeVD

270 290 310

5

10

15

20

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


Masses and Order Parameter vs. T

m

m

m

0

0 50 100 150 200 250 300T @MeVD0

100

200

300

400

500

@MeVD = 0 MeV

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

Screening masses determine thresholds in spectral functions, e.g. at T = 10 MeV:

+ , 2 m 280 MeV


Masses and Order Parameter vs.

m

m

m

0

200 250 300 350 400 @MeVD0

100

200

300

400

500

@MeVDT = 10 MeV

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

Screening masses determine thresholds in spectral functions, e.g. at = 0:

+ , 2 m 600 MeV


Decay channels of the sigma mesons

*

+

(2m)2 + ~p 2

*

+

(2m)2 + ~p 2

*

+

(2m)2 + ~p 2



Decay channels of the pions

*

+

(m + m)2 + ~p 2

*

,

,

+ +

(m m)

1 + ~p2

m2

(m m)

1 + ~p2

m2

*

+

(2m)2 + ~p 2



Decay channels of the (anti-)quarks

y*

s, p

y

+ +

(m + m)2 + ~p 2

(m + m)2 + ~p 2

y*

s, p

y

+

+

(m m)

1 + ~p2

m2

(m m)

1 + ~p2

m2

y*

y

s, p

+ +

(m m)

1 + ~p2

m2

(m m)

1 + ~p2

m2


Flow of Sigma and Pion Spectral Functionat = 0, T = 0 and ~p = 0

(Loading movie...)


Lavf56.40.101

spectral_flow_vacuum_eps0.mp4Media File (video/mp4)

Flow of Quark Spectral Function (C)at = 0, T = 0 and ~p = 0

(Loading movie...)


Lavf57.11.100

quark_spectral_flow_T0.mp4Media File (video/mp4)

Sigma and Pion Spectral Functionwith increasing T at = 0 and ~p = 0

(Loading movie...)


Lavf57.11.100

spectral_T_eps0.mp4Media File (video/mp4)

Sigma and Pion Spectral Functionwith increasing at T 10 MeV and ~p = 0

(Loading movie...)


Lavf57.11.100

spectral_mu_eps0.mp4Media File (video/mp4)

Towards the shear viscosity

Applying the Green-Kubo formula for the shear viscosity

=1

24lim0

lim|~p|0

1

d4x eipx

[Tij (x), T

ij (0)]

to the quark-meson model with energy-momentum tensor

T ij (x) =i2

( i j j i

)+ j i + j~ i~

gives (dominant contribution)

d2

d3p

(2)3p2x p

2y nB()

(2(,~p) + 3

2(,~p)

)


Space-like processes of the sigma mesons

*

+

0 |~p|

*

+

0 |~p|

, ,

*

+

+

0 |~p|



Space-like processes of the pions

*

+

0 |~p|

*

+

0 |~p|

, ,

*

+

+

0 |~p|



Space-like processes of the quarks

s, p y

y*

+ +

0 |~p|

y s, p

y*

+

+

0 |~p|


Sigma Spectral Functionvs. and ~p at = 0 and T = 0 MeV

I time-like region ( > ~p)is Lorentz-boosted tohigher energies

I space-like region( < ~p) is non-zero atfinite T due tospace-like processes


Sigma Spectral Functionvs. and ~p at = 0 and increasing T



(Loading movie...)


Lavf57.11.100

sigma_mu0_T_3D.mp4Media File (video/mp4)

Pion Spectral Functionvs. and ~p at = 0 and T = 0 MeV


I capture process + issuppressed at large ~p



Pion Spectral Functionvs. and ~p at = 0 and increasing T


I capture process + issuppressed at large ~p


(Loading movie...)


Lavf57.11.100

pion_mu0_T_3D.mp4Media File (video/mp4)

Sigma Spectral Functionvs. and ~p at T 10 MeV and increasing



I sigma becomes stablenear the criticalendpoint for smallmomenta

(Loading movie...)


Lavf57.11.100

sigma_mu_T9_3D.mp4Media File (video/mp4)

Pion Spectral Functionvs. and ~p at T 10 MeV and increasing



I + thresholdmoves to smallerenergies due todecreasing sigma mass

(Loading movie...)


Lavf57.11.100

pion_mu_T9_3D.mp4Media File (video/mp4)

Shear viscosity at = 0

I ,LKW: result from chiralperturbation theory

[Lang, Kaiser, and Weise, Eur. Phys. J. A 48, 109 (2012)]

I large shear viscosity at lowtemperatures due to smallwidth of the pion peak 4 processes missing

I stable-particle deltafunctions are regularized bya Breit-Wigner shape

=1

2(2 2 20)2 + 422

0 50 100 150 200T @MeVD

10-11

10-7

0.001

10

105

h @GeV3D hs+hphs

hp

hp, LKW


Entropy density at = 0

I entropy density can beextracted from the effectivepotential:

s = p/T = Uk0/T

I it has been UV-corrected bytaking quark fluctuations fromhigher scales into account

I Stefan-Boltzmann value isreproduced at high T :

sSB/T 3 = 142/15

0 100 200 300 4000

2

4

6

8

10

sT3

sL=1 GeV

sL

sSB

T @MeVD


Shear viscosity over entropy density /s at= 0

I ,LKW: result from chiralperturbation theory

[Lang, Kaiser, and Weise, Eur. Phys. J. A 48, 109 (2012)]

I entropy density s containsquarks and mesons

I ( + )/s large at low Tdue to large and small s

I ( + )/s is always largerthan the AdS/CFT limitingvalue of /s 1/4

0 50 100 150 200T @MeVD

0.01

10

104

107

1010

hs

Hhs+hpLshsshps1H4pL

hp, LKWs

[Kovtun, Son, and Starinets, Phys. Rev. Lett. 94, 111601 (2005)]


Shear viscosity near the CEP

I stable-particle delta functionis regularized by aBreit-Wigner shape

=1

2(2 2 20)2 + 422

I shear viscosity stronglydepends on the chosenvalue for

I at the CEP, shear viscosityof the sigma mesons diverges due to themassless excitation

291.0 291.2 291.4 291.6 291.8m @MeVD10-10

0.001

104

1011

1018

1025

h @GeV3D

hs


Summary and outlook

We presented a new method to obtain real-time quantities like spectral functionsand transport coefficients at finite T and from the FRG:

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

I it is thermodynamically consistent and symmetry-structure preserving

I feasibility of the method demonstrated by calculatingquark and meson spectral functions and /s for the quark-meson model

Outlook:

I calculation of the shear viscosity of the quarks

I extending the model by including vector and axial-vector mesons andimproving the truncations


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