in-medium spectral functions of vector- and ......in-medium spectral functions of vector- and...

IN-MEDIUM SPECTRAL FUNCTIONS OF VECTOR- AND

AXIAL-VECTOR MESONS

Fabian RenneckeBrookhaven National Laboratory

— RHIC/AGS ANNUAL USERS’ MEETING — BNL 12/06/2018

[Jung, FR, Tripolt, von Smekal, Wambach, hep-ph/1610.08754]

[FR, hep-ph/1504.03585]

Fabian Rennecke, BNL RHIC/AGS Users’ Meeting, 12/06/2018

• Chiral symmetry restoration, dilepton spectra & vector mesons

• Spectral functions from the functional renormalization group

• (Axial) vector meson spectral functions in the medium

• Beyond low-energy modeling: dynamical hadronization

OUTLINE

QCD PHASE DIAGRAM


[Hot QCD White Paper (2015)]

Theorycompute order parameters

L =1

Nc

DTrfP eig

R �0 d⌧A0(⌧)

E• confinement:

• χSB: hq̄qi = hq̄LqR + q̄RqLi

Experiment

measure hadronic final states

phase transitions difficult to grasp!

CHIRAL SYMMETRY BREAKING


0.01

0.1

1

10

100

u d s c b t

m[G

eV]

• 3 (2) (super) light quark flavors

approximate QCD flavor symmetry:

! !!!!!!!!

!!!!!! ! !!!

""

""""

"

""

"""""" "

!!!!!!!

!!!!! !

! ! ! !

##

##

#

#

##!!""!!

ContinuumNt"16Nt"12Nt"10Nt"8

100 120 140 160 180 200 220

0.2

0.4

0.6

0.8

1.0

T !MeV"

#l,s

[WB, hep-lat/1005.3508]

⇠ hq̄qi

• spontaneous symmetry breaking down to

origin of hadron masses

• nucleon masses via coupling to chiral condensate

• 8 light pseudoscalar mesons π, K, η (pseudo Goldstones)

But: no experimental verification of chiral symmetry restoration in QGP yet!

SU(3)L⇥SU(3)R

SU(3)L+R

• mass-splitting of chiral partners …

DILEPTON SPECTRA


)2 (GeV/ceem0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

/GeV

) IN

PH

ENIX

AC

CEP

TAN

CE

2 (c

eedN

/dm

-910

-810

-710

-610

-510

-410

-310

-210

-110 = 200 GeVNNsp+p & Au+Au at

Au+Au min. bias

p+p

)2 (GeV/ceem0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

/GeV

) IN

PH

ENIX

AC

CEP

TAN

CE

2 (c

eedN

/dm

-910

-810

-710

-610

-510

-410

-310

-210

-110

(a)

)2 (GeV/ceem0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

/GeV

) IN

PH

ENIX

AC

CEP

TAN

CE

2 (c

eedN

/dm

-1010

-910

-810

-710

-610

-510

-410

-310

-210

-110

1

10

210 = 200 GeVNNsAu+Au at

210⋅ 5×Au+Au 0-10% 10⋅ 3×Au+Au 10-20%

Au+Au 20-40%-1 10×Au+Au 40-60% -2 10×Au+Au 60-92%

)2 (GeV/ceem0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

/GeV

) IN

PH

ENIX

AC

CEP

TAN

CE

2 (c

eedN

/dm

-1010

-910

-810

-710

-610

-510

-410

-310

-210

-110

1

10

210

(b)

Figure 4.21: (a) shows the invariant mass spectrum in p+p and min. bias Au+Aucollisions. (b) shows the invariant mass spectra of five centrality classes in Au + Aucollisions. The data are shown with statistical (bars) and systematic (shades) errorsseparately. The data are compared to the expected sources from the decays of lighthadron calculated with Exodus and correlated charm decays based on Pythia.

is completely broken, e+e− pairs are generated with a single pT distributionfollowing the measured spectrum of single electrons from semi-leptonic heavy-flavor decays [16], but a random azimuthal opening angle. This distributionis much softer, than the Pythia curve, which would leave room to othercontributions such as thermal radiation via qq annihilation.

If the yield in the IMR is dominated by open charm, it is expected to in-crease proportional to the number of binary collisions. The yield per numberof binary collision Ncoll in the mass range 1.2 < mee < 2.8 GeV/c2 is shownas function of Npart in Fig. 4.22. The data show no significant centrality de-pendence and are consistent with the expected yield calculated with Pythia.But the apparent scaling with the number of binary collisions may be a co-incidence of two counteracting effects: (i) the suppression of e+e− pairs fromopen charm in the IMR due to modifications of charm which increases withNpart and (ii) an additional contribution due to thermal radiation from qq an-nihilation which is expected to increase faster than proportional to Npart. Asdiscussed in Section 1.3.3, such a coincidence may have been observed at theSPS [74], where a prompt component has been suggested by NA60 [75].

147

[T. Dahms, nucl-ex/0810.3040]

dNl+l�

d4x d4q⇠ ↵2

EM ⇢EMVMD⇠ ↵2

EM

✓⇢⇢ +

1

9⇢! +

2

9⇢�

◆

• dileptons escape fireball essentially without interaction

How to observe chiral symmetry restoration?

• low- / intermediate-mass regime of dilepton spectra dominated by vector meson decays

ideal probes for high T/mu regions formed in early stages of collisions

[Feinberg (`76), McLerran, Toimela (`85), Weldon (`90)][Sakurai (`69)]

• in-medium modification of rho-peak

chiral transition?

• dropping ρ mass at chiral restoration scale [Brown, Rho (`91)]

• suggestions:

• rising ρ mass [Pisarski (`95)]

• melting ρ resonance [Rapp, Wambach (`99)]

(proposed by Pisarski in `81)


DILEPTON SPECTRA

10-4

10-3

10-2

10-1

100

101

0 0.2 0.4 0.6 0.8 1

1/N

mb

evt

dN

ee

acc

. /dM

ee [ (

GeV

/c2)-1

] 19.6 GeV

STAR Preliminary

Au+Au Minimum Bias

dielectron invariant mass, Mee (GeV/c2 )

0 0.2 0.4 0.6 0.8 1

62.4 GeV

sys. uncertainty

cocktail

+ medium modifications

data

0 0.2 0.4 0.6 0.8 1

200 GeV

Figure 9: BES dielectron invariant-mass distributions in the low invariant-mass range from Au+Au collisions at 19.6 GeV(left), 62.4 GeV (middle), and 200 GeV (right panel). The red data points include both statistical and systematic uncer-tainties (boxes). The black curve depicts the hadron cocktail, while the dashed line shows the sum of the cocktail andmodel calculations. The latter includes contributions from both the HG and the QGP phases. The systematic uncertaintyon the former is shown by the light red band.

a broadening of its spectral function. Differential studies of the LMR dielectron distributions asa function of

√sNN, centrality, and pT will allow for a more detailed comparison against these

chiral hadronic models. Further comparisons between these models, lattice QCD calculations,and experimental data could help provide explicit evidence of chiral symmetry restoration inheavy-ion collisions.

5. Summary and Outlook

Measuring dilepton spectra is a challenging task, where very small signals typically sit on topof large backgrounds. With the recent TOF upgrade, STAR has been able to address some of theimportant physics questions regarding the LMR enhancement. It has allowed for cross checkswith previous measurements on both ends of the BES energy spectrum, as well as systematic,large-acceptance measurements at a range of other center-of-mass energies. The measurementsconfirm results at SPS energies in which the LMR enhancement is attributed to the broadeningof the ρ meson. Moreover, model calculations by Rapp [23] appear to be able to describe theBES data at intermediate beam energies. In the future, additional verification of the robustnessof such a description will be allowed by including independent analyses of other BES datasets.Differential measurements will further verify the consistency of such models and advance thestudy of chiral symmetry restoration.

As the typical production rates for dileptons are rather small, large event samples are neededto make significantly accurate measurements at increasingly higher invariant masses. Especiallyat the lower RHIC beam energies, the statistical uncertainties are very large (see Fig. 7). The19.6 GeV measurements have been based on 28 million Au+Au collisions. To achieve statisticaluncertainties at a level of 10% and improve our understanding of the baryonic component to thein-medium effects on the vector mesons, a factor of 10 more statistics would be required. In orderto achieve such an increase total luminosity, RHIC will need the development and installation ofelectron cooling components.

F. Geurts / Nuclear Physics A 904–905 (2013) 217c–224c 223c

[STAR, nucl-ex/1210.5549]

• melting scenario works well for describing dilepton data!

connection to χSR?

• Weinberg sum rule:

vector & axial-vector spectral functions

chiral condensate

need spectral function of ρ and its chiral partner a1

degeneration: χSR!

Z 1

0ds⇥⇢V (s)� ⇢A(s)

⇤= �2mqhq̄qi

Mass

Spec

tral

Func

tion

"a1"

"ρ"

pert. QCD

Dropping Masses ?

Mass

Spec

tral

Func

tion

"a1""ρ" pert. QCD

Melting Resonances ?

[Rapp, Wambach, van Hees, hep-ph/0901.3289]


WHAT WE NEED

We need a framework to compute thermodynamic properties / the phase structure and spectral properties on the same footing!

Requirements:

• non-perturbative method to describe QCD at low energies

• capture symmetries and their breaking patterns

• direct access to spectral properties (avoid analytic continuation of numerical data)

• quantum fluctuations (beyond mean-field approximations)

• access to finite μ (no sign problem)

FUNCTIONAL RG


• introduce regulator to partition function to suppress momentum modes below energy scale k (Euclidean space):

k2

k2

k2

k2

Rk(p2)

12@tRk(p2)

p2

• scale dependent effective action: � = h'iJ�k[�] = supJ

⇢Z

xJ(x)�(x)� lnZk[J ]

��Sk[J ]

• evolution equation for Γk: [Wetterich 1993]

@t�k = 12STr

⇣�(2)k +Rk

⌘�1@tRk

�@t = k

d

dk

successively integrate out fluctuations from UV to IR (Wilson RG)

full quantum effective action(generates 1PI correlators)

Γk is eff. action that incorporates all fluctuations down to scale k

lowering k: zooming out / coarse graining

�⇤ �k �0 = �

Zk[J ] =

ZD' e�S[']��Sk[']+

Rx J'

�Sk['] =1

2

Zd4q

(2⇡)4'(�q)Rk(q)'(q)

ANALYTIC CONTINUATION


• one-loop like structure of the FRG allows for straightforward analytic continuation! [Floerchinger, hep-ph/1112.4374][Tripolt, Strodthoff, von Smekal, J. Wambach, hep-ph/1311.0630][Pawlowski, Strodthoff, hep-ph/1508.01160]

The analytic continuation problem

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

w

ip0

?

?[R.-A. Tripolt]

@t�(2),Ek (p0, ~p )

@t�(2),Rk (!, ~p ) = � lim

✏!0@t�

(2),Ek

�p0 = �i(! + i✏), ~p

�

1. start from Euclidean spacetime (Matsubara formalism for finite T)

Euclidean 2-pt function

⇠ n 2 Z

2. exploit the periodicity of the occupation numbers

nB/F (E + ip0) = nB/F (E)

3. rotate frequency to the real axis retarded 2-pt function

4. spectral function

⇢(!, ~p ) = � 1

⇡ImGR(!, ~p ) =

1

⇡

Im�(2),R(!, ~p )��(2),R(!, ~p )

��2

direct non-perturbative computation of spectral functions (without sign-problem)

p p

p+q

q

A LOW ENERGY MODEL


• need hadronic electromagnetic current for dilepton spectrum:

• vector meson dominance:

dNl+l�

d4x d4p⇠ ⇢EM ⇠ h jEM jEMi

jµEM

��hadrons

=e

g

�m2

⇢⇢µ +m2

��µ +m2

!!µ�

• VMD can be realized by promoting the flavor symmetry to a gauge symmetry; vector mesons emerge as gauge bosons

• 2 flavor (no Φ) quark-meson model based on VMD (only iso-triplet vector mesons: no ω):


� = (~⇡,�)

V µ = ~⇢µ ~T + ~aµ1~T 5

�k =

Zd4x

nq̄⇥�µ@

µ � µ�0 + hS(� + i�5~⌧~⇡) + ihV (�µ~⌧~⇢µ + �µ�5~⌧~a

µ1 )

⇤q

+ Uk(�2)� c� +

1

2

�Dµ�

�2+

1

8trVµ⌫V

µ⌫ +1

4m2

V trVµVµo+��⇡a1

• covariant derivative:

• field strength:

[Sakurai (`60), Kroll, Lee, Zumino (`68), Lee, Nieh (`68)]

e+

e�⇢0 �⇤

q̄

q

⇡+

⇡�

q̄

q

⇡+

⇡� ⇢0 �⇤

e+

e�

1

2(Dµ�)

2 =1

2(@µ�)

2 � igV @µ� · Vµ�� g2V2(Vµ�) · (Vµ�)

1

8trVµ⌫Vµ⌫ =

1

8tr (@µV⌫ � @⌫Vµ)

2 � igV2

tr @µV⌫ [Vµ, V⌫ ]�g2V8tr [Vµ, V⌫ ]

2

A LOW ENERGY MODEL


phase diagram: 7

FIG. 5: (color online) Phase diagram of the quark-meson

model as a contour plot of the order parameter for chiral

symmetry �0(T, µ). The value of �0(T, µ) decreases with in-

creasing temperature and chemical potential as indicated with

a darker color. The CEP is indicated as a red dot, whereas

the first-order phase boundary is indicated by a black line.

quark-meson model on the level of the e↵ective poten-tial, is depicted in Fig. 5, see also [27, 30] for earlierstudies on the quark-meson model. It is obtained by thelocation of the global minimum of the e↵ective poten-tial at the IR scale �0 ⌘ �0(T, µ). With the parame-ters given in Tab. I we find a critical endpoint at around(µCEP, TCEP) ⇡ (298, 10) MeV, which divides a crossoverregion from a first-order phase transition at lower temper-atures. We note that the slope, dT/dµ, of the first-orderline is very di↵erent than the one observed in mean-fieldstudies, see e.g. [50]. In fact, the regime to the right ofthe first-order line, i.e. at large chemical potentials and

FIG. 6: (color online) Euclidean curvature masses of

mesons and constituent quark mass vs. chemical potential at

T = 10 MeV.

low temperatures, is likely to be dominated by an inho-mogeneous ground state which leads to unphysical e↵ectslike a negative entropy density in the present truncation.We therefore avoid this regime in the following and referto [51] for further details.The same Euclidean curvature masses of the mesons,

plotted together with the constituent quark mass overtemperature at µ = 0 MeV in Fig. 4, are shown along theµ-axis at a constant temperature of T = 10 MeV acrossthe CEP in Fig. 6. They behave as expected in a modelbased on chiral symmetry. For vanishing chemical poten-tial, the Euclidean curvature masses of the chiral partnersm�, m⇡ and m⇢, ma1 become degenerate at high temper-atures, T & 200 MeV. The quark mass m decreases, in-dicating the gradual restoration of chiral symmetry. Fora fixed temperature of T = 10 MeV the masses do notreally change over a wide range of chemical potential, asexpected from the Silver Blaze property [52]. Near theCEP at around µCEP ⇡ 298 MeV, the sigma mass dropssignificantly as expected at this second-order phase tran-sition. In addition, the chiral condensate as well as thevector-meson masses decrease when crossing the CEP.For very high chemical potentials the masses of the chi-ral partners coincide again and the quark mass decreases,similar to the case of high temperature and vanishingchemical potential.

C. In-medium spectral functions at |~p| = 0

Before turning to the ⇢ and a1 spectral functions at fi-nite temperature and chemical potential, in this subsec-tion for vanishing external spatial momentum, |~p| = 0,we will discuss the temperature dependence of the phys-ically relevant vector-meson pole masses. They are ob-tained from the zero-crossing of the real part of the 2-point functions and are shown in Fig. 7 vs. T at µ = 0.At T = 0 chiral symmetry is broken and the pole

masses assume the vacuum values of mp⇢ = 789.3 MeV

and mpa1

= 1274.7 MeV for the UV-parameters of Tab. I.With increasing temperature the di↵erence between thepole masses decreases until they become degenerate atT ⇡ 200 MeV, i.e. at about the same temperature as theEuclidean curvature masses. The observed behavior sup-ports the “melting-⇢-scenario”, where the ⇢ meson massremains almost stable and the a1 mass shifts towards themass of the ⇢ meson [11, 39].

In order to exhibit the T - and µ-induced modifica-tions more clearly, Fig. 8 shows logarithmic plots ofthe ⇢ and a1 spectral functions for vanishing chemicalpotential (left column) and for a fixed temperature ofT = 10 MeV along the µ-axis towards the CEP (rightcolumn). At T = 0 MeV, the ⇢⇤ ! ⇡ + ⇡ threshold givesrise to a non-vanishing value of the ⇢ spectral function for! & 280 MeV. For ! & 600 MeV the decay into quark-antiquark pairs becomes energetically possible and givesrise to another threshold in the spectral function. Thespectral function of the a1 meson exhibits the first thresh-

RG flow of the two-point functions:

∂kΓ(2)ρ,k =

ρ ρ

π

π π−

1

2

ρ ρ

π π − 2ρ ρ

ψ

ψ ψ

∂kΓ(2)a1,k

=a1 a1

π

σ σ+

a1 a1

σ

π π−

1

2

a1 a1

π π −

1

2

a1 a1

σ σ − 2a1 a1

ψ

ψ ψ

approx.: no internal vector mesons


SPECTRAL FUNCTIONS


0 500 1000 1500 20000.0

0.1

0.2

0.3

0.4

0.5

0.6

energy @MeVD

spectralfunction@GeV-2 D r , T = 10 MeV

a1 , T = 10 MeV

0 500 1000 1500 20000.0

0.1

0.2

0.3

0.4

0.5

0.6

energy @MeVD


a1 , T = 100 MeV

0 500 1000 1500 20000.0

0.1

0.2

0.3

0.4

0.5

0.6

energy @MeVD


a1 , T = 150 MeV

0 500 1000 1500 20000.0

0.1

0.2

0.3

0.4

0.5

0.6

energy @MeVD


a1 , T = 300 MeV


SPECTRAL FUNCTIONS


0 500 1000 1500 20000.0

0.1

0.2

0.3

0.4

0.5

0.6

energy @MeVD


a1 , T = 10 MeV

0 500 1000 1500 20000.0

0.1

0.2

0.3

0.4

0.5

0.6

energy @MeVD


a1 , T = 100 MeV

0 500 1000 1500 20000.0

0.1

0.2

0.3

0.4

0.5

0.6

energy @MeVD


a1 , T = 150 MeV

0 500 1000 1500 20000.0

0.1

0.2

0.3

0.4

0.5

0.6

energy @MeVD


a1 , T = 300 MeV

model spectral functions in favor of

"melting resonance scenario"

0 50 100 150 200 250 3000

500

1000

1500

T [MeV]

[Me

V]

� = 0 MeV

m�p

ma1

p

pole masses


SPECTRAL FUNCTIONS


0 200 400 600 800 1000 1200 140010-5

10-4

0.001

0.010

0.100

1

� [MeV]

�[G

eV-

2]

T = 10 MeV, � = 290 MeV

0 200 400 600 800 1000 1200 140010-5

10-4

0.001

0.010

0.100

1

� [MeV]

�[G

eV-

2]

T = 10 MeV, � = 297.4 MeV

0 200 400 600 800 1000 1200 140010-5

10-4

0.001

0.010

0.100

1

� [MeV]

�[G

eV-

2]

T = 50 MeV, � = 600 MeV

1

2

3

4

34

1 2

3 4

• some decay channels:

• manifestation of CEP in dilepton spectra:

• a1 couples to critical mode σ• include vector meson fluctuations

∂kΓ(2)ρ,k = −

1

2

ρ ρ

π π −

1

2

ρ ρ

σ σ −

1

2

ρ ρ

ρ ρ−

1

2

ρ ρ

a1 a1 − 2ρ ρ

ψ

ψ ψ

+ρ ρ

a1

π π+

ρ ρ

π

a1 a1+

ρ ρ

σ

ρ ρ+

ρ ρ

ρ

σ σ

+ρ ρ

π

π π+

ρ ρ

ρ

ρ ρ+

ρ ρ

a1

a1 a1

feedback on ρ spectral function

• similar contribution from π’s• fine-tuning of T and μ to CEP necessary


DYNAMICAL HADRONIZATION


Up to now only low-energy model; what happens if gluon fluctuations are included?

• effective four-quark interaction channels are generated: ⇠ ↵2

s,k �T,k(q̄ T q)2

• grows with decreasing scale

• RG-flow controlled by IR-attractive fixed point for small g

JHEP06(2006)024

λi

∂tλi g = 0

g ! 0

g > gcr

T > 0, g = 0

Figure 5: Sketch of a typical β function for the fermionic self-interactions λi: at zero gaugecoupling, g = 0 (upper solid curve), the Gaußian fixed point λi = 0 is IR attractive. For smallg ! 0 (middle/blue solid curve), the fixed-point positions are shifted on the order of g4. Forgauge couplings larger than the critical coupling g > gcr (lower/green solid curve), no fixed pointsremain and the self-interactions quickly grow large, signaling χSB. For increasing temperature, theparabolas become broader and higher, owing to thermal fermion masses; this is indicated by thedashed/red line.

ing larger than the regulator scale k, these functions approach zero, which reflects the

decoupling of massive modes from the flow.

Within this set of degrees of freedom, a simple picture for the chiral dynamics arises: for

vanishing gauge coupling, the flow is solved by vanishing λi’s, which defines the Gaußian

fixed point. This fixed point is IR attractive, implying that these self-interactions are

RG irrelevant for sufficiently small bare couplings, as they should be. At weak gauge

coupling, the RG flow generates quark self-interactions of order λ ∼ g4, as expected for

a perturbative 1PI scattering amplitude. The back-reaction of these self-interactions on

the total RG flow is negligible at weak coupling. If the gauge coupling in the IR remains

smaller than a critical value g < gcr, the self-interactions remain bounded, approaching

fixed points in the IR. These fixed points can simply be viewed as order-g4 shifted versions

of the Gaußian fixed point, being modified by the gauge dynamics. At these fixed points,

the fermionic subsystem remains in the chirally invariant phase which is indeed realized at

high temperature.

If the gauge coupling increases beyond the critical coupling g > gcr, the above-

mentioned IR fixed points are destabilized and the quark self-interactions become critical.

This can be visualized by the fact that ∂tλi as a function of λi is an everted parabola;

see figure 5; for g = gcr, the parabola is pushed below the λi axis, such that the (shifted)

Gaußian fixed point annihilates with the second zero of the parabola. In this case, the

gauge-fluctuation-induced λ̄’s have become strong enough to contribute as relevant opera-

tors to the RG flow. These couplings now increase rapidly, approaching a divergence at a

finite scale k = kχSB. In fact, this seeming Landau-pole behavior indicates χSB and, more

specifically, the formation of chiral condensates. This is because the λ̄’s are proportional

dimensions which were shown to influence the quantitative results for the present system only on the

percent level, if at all [50].

– 17 –

[Gies, Wetterich, hep-th/0209183]

�T,k

k

⇤ � 1GeV

• strong coupling exceeds critical value: chiral symmetry breaking

four-quark interaction diverges

• introduce mesons via bosonization: (similar: baryonization from 6-quark interactions)

�T,k =h2T,k

m2�,k

Yukawa coupling

meson mass parameter

• initial action: gauge fixed QCD

DYNAMICAL HADRONIZATION


• bosonize 4-quark interaction in each RG step: dynamical hadronization

`mesons´ become scale dependent fields

low-energy parameters are predicted; only αS and quark masses as input (intermediate fixed point in 4-quark interaction!)

[Gies, Wetterich, Phys.Rev. D '02, Pawlowski, Ann.Phys. '07, Floerchinger, Wetterich, Phys.Lett. B '09]

[Braun, Fister, Pawlowski, FR, hep-ph/1412.1045]

0.1 0.2 0.5 1.0 2.0 5.0 10.0 20.00

5

10

15

20

25

k @GeVD

g 1g 4g 2

g 3

g 5

@t�k ⇠ @t�T,k

hT,k(q̄ T q)

• no VMD assumed here: vector meson couplings different a priori

• VMD would imply

gV = g1 =pg2 = g4 =

pg5

g3 = 0

[FR, hep-ph/1504.03585]

e.g. for ρT = �µ~⌧

• including gauge fields: dynamical hadronization

• low-energy model based on VMD

• direct, non-perturbative computation

• degeneration of vector and axial-vector spectral functions above the chiral transition

• consistent with melting resonance scenario

• `marry´ dynamical hadronization and analytic continuation• vector meson fluctuations• baryons & more decay channels• confinement• dilepton rates

Outlook / work in progress:

SUMMARY & OUTLOOK

• in-medium vector- and axial-vector meson spectral functions

• no model parameter dependence• `test´ of VMD


• including gauge fields: dynamical hadronization

• low-energy model based on VMD

• direct, non-perturbative computation

• degeneration of vector and axial-vector spectral functions above the chiral transition

• consistent with melting resonance scenario

• `marry´ dynamical hadronization and analytic continuation• vector meson fluctuations• baryons & more decay channels• confinement• dilepton rates

Outlook / work in progress:

SUMMARY & OUTLOOK

• in-medium vector- and axial-vector meson spectral functions

• no model parameter dependence• `test´ of VMD


thank you

in-medium spectral functions of vector- and ......in-medium spectral functions of vector- and...

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