Understanding the QGP through Spectral Functions and Euclidean Correlators

BNL April 2008

Angel Gómez Nicola

Universidad Complutense Madrid

IN MEDIUM LIGHT MESON RESONANCES AND CHIRAL SYMMETRY RESTORATION

Spectral properties of light meson resonances in hot and dense matter: Motivation

f0 (600)/ → vacuum quantum numbers, chiral symmetry restoration

Observed in nuclear matter experiments (CHAOS, …) through threshold enhancement?

Any chance for Heavy Ions (finite T)?

→ dilepton spectrum (CERES,NA60) and nuclear matter

Broadening vs Mass shift (scaling?)

What can medium effects tell about the nature of these states?

DILEPTONS

NA45/CERES (e+e-)

Compatible with both broadening and dropping-mass scenarios

NA60 (+-)

Broadening favored, dropping massalmost excluded

Rapp-WambachBrown/Rho

meson cocktail

2000 data

→ e+e- IN NUCLEAR MATTER

Signals free of T≠0 complications

Linear decrease of vector meson masses from scaling&QCD sum rules:

0

1)0(

)(

M

M Brown, Rho ‘91Hatsuda, Lee ‘922.01.0

30 fm 17.0 normal nuclear matter density

Experiments not fully compatible:

KEK-E325 (C,Fe-Ti): = 0.0920.002

Jlab-CLAS (C,Cu): = 0.020.02

Cabrera,Oset,Vicente-Vacas ‘02Urban, Buballa, Rapp,Wambach ‘98

Chanfray,Schuck ‘98Other many-body approaches give negligible mass shift

production in Nuclear Matter: threshold enhancement in the (I=J=0) channel

A → A’

A → A’

Crystal BallCHAOS

MAMI-B

Threshold enhancement as a signal of chiral symmetry restoration:

M <> decreases, so that when M 2m , phase space is squeezed 0 and the pole reaches the real axis

Hatsuda, Kunihiro ‘85

O(N) models at finite T show that the remains broad when M 2m

Further in-medium strength causes 2nd-sheet pole to move into 1st sheet bound state.

Hidaka et al ‘04

Patkos et al ‘02

Finite density analysis compatible with threshold enhancement ofcross section

Davesne, Zhang, Chanfray ‘00Roca et al ‘02

models in state O(N)qq

Narrow resonance argument !

CHIRAL SYMMETRY UNITARITY+

Inverse Amplitude Method

““Thermal” polesThermal” poles

Dynamically generated (no explicit resonance fields)

OUR APPROACH: UNITARIZED CHIRAL PERTURBATION THEORY

AGN, F.J.Llanes-Estrada, J.R.Peláez PLB550, 55 (2002), PLB606:351-360,2005 A.Dobado, AGN, F.J.Llanes-Estrada, J.R.Peláez, PRC66, 055201 (2002)D.Fernández-Fraile,AGN, E.Tomás-Herruzo, PRD76:085020,2007

scattering amplitude and form factors in T > 0 SU(2)

one-loop ChPT

Chiral Perturbation Theory:

Relevant for low and moderate temperatures below Chiral SSB

Weinberg’s chiral power counting:

) MeV200below ( GeV 14 ,, cTTfTmEp

Dp /

NLSM

43214

0222

2

,,,2

llll

UfmUUf T

20

0

||1 ,

)2(),(

UUf

U

SUUUU

Most general derivative and mass expansion of NGB mesons compatible with the SSB pattern of QCD model-independent low-energy predictions.

42

Perturbative Unitarity22

24 )()();(Im EtETEt IJT

IJ )2( mE

In the two-pion c.om. frame: (static resonaces):

ChPT does not reproduce resonances due to the lack of exact unitarity (resonances saturate unitarity bounds).

Unitarization: The Inverse Amplitude Method

Two-pion thermal phase spaceenhancement

)2/(214

1)(2

2

EnE

mE BT

22)1( BB nn

Enhancement Absorption

IJIJIJ ttt 42

Exact unitarity

TIAMIAM

TIAM ttt

12Im Im

+ ChPT matching at low energies

42 ttt IAM

* At T>0, valid for dilute gas (only two-pion states).

);()(

)();(

24

22

2 22

TEtEt

EtTEt IAM

Thermal and poles(2nd Riemann sheet)

*

Very sucessful at T=0 for scattering data up to 1 GeVand low-lying resonance multiplets, also for SU(3)

Dobado, Peláez, Oset, Oller, AGN.

300 400 500 600 700 800Mp MeV

150

125

100

75

50

25

0

p

2VeM

2m

T0

T200 MeVIJ1

= 20 MeV

Thermal phase space enhancement + Increase of effective vertex, small mass reduction up to Tc.

THE THERMAL POLE

02

2

00 0M

M

g

gTTT T

(for a narrow Breit-Wigner resonance)

2)2/( pppole iMs (2nd Riemann sheet)

MeV151

MeV756

:0

p

pM

T

The unitarized EM pion form factor shows also broadening compatible with dilepton data and VMD analysis:

THE THERMAL f0(600)/ POLE

Strong pole mass reduction (chiral restoration) means phase space squeezing, which overcomes low-T thermal enhancement

2)2/( pppole iMs (2nd Riemann sheet)

MeV466

MeV441

:0

p

pM

T

275 300 325 350 375 400 425 450Mp MeV

250

200

150

100

50

0

p2

VeM

2m

T0T200 MeV

IJ0

T=100 MeV

= 20 MeV

However, the pole remains wide even for M~2 m

(spectral function not peaked around the mass for broad resonances)

Narrow vs Broad Resonances

NARROW: pp M (s) strongly peaked around2pMs

)4( 4 2222 mMmM ppp Phase space squeezing

Threshold enhancement MM p 2

(R “particle” at rest)12)(2

1

2

1d dM

MD

2-particle differential phase space

differential decay rate

8

)4()( 2

12 12

Mssd

32

pD

Narrow vs Broad Resonances

BROAD: pp M (s) broadly distributed

pole away from the real axis ChPT approach valid at threshold no enhancement

Generalized decay rate:

H.A.Weldon, Ann.Phys.228 (1993) 43

)()()()(

224s

s

sssMdsMD

NO phase-squeezingfor wide enough s !

Narrow vs Broad Resonances:

REAL AXIS POLES AND ADLER ZEROS

Require extra terms in the IAM to account properly for Adler zeros t(sA)=0.

Otherwise, spurious real poles below threshold in the 1st,2nd Riemann sheets.

Preserving chiral symmetry+unitarity: )(11

sgtt IAMIAM

No difference away from sA

No additional poles for T0 with theredefined amplitudes.

Alternatively derived with dispersionrelations.

)('/)( ;

)]([

)(')('))((

)()(

2224442

22

242222

24

ststssss

st

ststss

ssssst

sg

A

A

A

AGN,J.R.Peláez,G.Ríos PRD77, 056006 (2008)

No problem for I=J=1 24 MsA

2s

4s

THE NATURE OF THERMAL RESONANCES: f0(600)/

Does not behave as a (thermal) state, not even near the chiral limitqq

Consistent with not- scalar nonet (tetraquark,glueball,meson-meson…) “molecule” picture

qq

J.R.Peláez ‘04M.Alford,R.L.Jaffe ‘00

THE NATURE OF THERMAL RESONANCES:

No BR-like scaling with condensate.

Mass dropping only very near “critical” (too high) T0 , as in BR-HLS models

Harada&Sasaki ‘06Brown&Rho ‘05

Nature of our thermal dominated by non-restoring effects (broadening)

8.0

60

6

1

T

T

NUCLEAR CHIRAL RESTORING EFFECTS

Chiral restoring effects at T=0 and finite nuclear density approx. encoded in f

0222

2

35.01)0(

1)0(

)(

)0(

)(

fmqq

qq

f

f N

Meissner,Oller,Wirzba ‘02

Thorsson,Wirzba ‘95

MeV45N

)2.8at 0( 0 qq

Justified by approximate validity of GOR (0,T=0) qqmfm q22

Non chiral-restoring many-body effects not included (p-h, p-wave self-energy, …)

Cabrera,Oset,Vicente-Vacas ‘05

Chiral restoring expected to be important in the -channel as densityapproachesthe transition . No broadening to compete with now !

bound state (“molecule” behaviour)

02.1

07.1

09.1

Mass linear fits:0

1)0(

)(

M

M2.00 toup

Compatible with some theoretical estimates and KEK experiment.

However, additional medium effects (important in this channel!) might lead to negligible mass shift

No threshold enhancement for reasonably high densities.

Compatible with BR-like scalingBrown,Rho ‘04

qq“non-molecular” ( )

In-medium light meson resonances studied through scattering poles in Unitarized ChPT provide chiral symmetry predictions for their spectral properties and nature.

The f0(600)/ shows chiral symmetry restoration features but remains as a T0 wide not- state no threshold enhancement at finite T.qq

The finite-T behaviour is dominated by thermal broadening in qualitative agreement with dilepton data. Mass dropping does not scale with the condensate.

Nuclear density chiral-restoring effects encoded in fdrive the poles to the realaxis giving threshold enhancement in the -channel and BR-like scaling in the -channel. bound states of different nature formed near the transition.

Full finite-density analysis, SU(3) extension (,K*,a0,…)