Chiral Symmetry Restoration and Deconfinement in QCD at Finite Temperature

M. LoewePontificia Universidad Católica de Chile

Montpellier, July 2012.

This talk is based on the following article:

Chiral symmetry restoration and deconfinement in QCD at finite temperature: C. A. Dominguez, M. Loewe and Y. Zhang. Hep-ph 1205.336. Submitted to Phys. Rev. D

I acknowledge support form : FONDECYT 1095217 andProyecto Anillos ACT119 (CHILE)

There are two (at least two) phase transitions that may occur in QCD at finite temperature and/or density:

1) Deconfinement due to color screening

2) Chiral symmetry restoration: Moving from a Nambu-Goldstone to a Wigner-Weyl realization

Which are the relevant order parameters in each case?

Both transitions seem to occur approximately at the same temperature

General Aspects:An order parameter is a quantity that vanishes in a certain phase, being finite in a second one.

The relevant physical variables are temperature(T) and baryon chemical potential (μB)

Normally the Polyakov loop (confinement) and the quark condensate (chiral symmetry restoration)are used as order parameters

When μB = 0 and T ≠ 0 lattice results provide a consistente picture, resulting in a similar Tc for both transitions in the range 170 MeV < Tc < 200 MeV (finite quark masses)

However…..

• For finite Baryon Chemical Potential, the fermion determinant becomes complex and lattice simulations are not possible.

So, perhaps we need a new variable, instead of the Polyakov Loop, for discussing deconfinement.

An attractive possibility: the continuum threshold of the hadronic resonance spectral function. Phenomenological order parameter. This discussion can be done in the frame of the extended (finite T and μB) QCD Sum Rules program

Realistic Spectral Function

Im Π

s ≡ E2s0

Realistic Spectral Function (T)

Im Π

s ≡ E2S0(T)

For this purpose we will use QCD Sum Rules.

OPERATOR PRODUCT EXPANSION OF CURRENT CORRELATORS AT SHORT DISTANCES

(BEYOND PERTURBATION THEORY)

CAUCHY’S THEOREM IN THE COMPLEX ENERGY (SQUARED) S-PLANE

CONFINEMENT

• STRONG MODIFICATION TO QUARK & GLUON PROPAGATORS NEAR THE MASS SHELL

• INCORPORATE CONFINEMENT THROUGH A PARAMETRIZATION OF PROPAGATOR

CORRECTIONS

IN TERMS OF QUARK & GLUON VACUUM CONDENSATES

• We reconsider the light quark axial-vector channel, using the first three FESR, together with an improved spectral function

Π0(q2) and Π1(q2) are free of kinematical singularities

• Invoking the OPE

No evidence for d=2 at T=0.

The dimension d=4 is given by

The second term is negligible compared with the gluon condensate

• For d=6 we invoke vacuum saturation

Our set of FESR (leading order in PQCD) are given by

The normalization of the correlator in PQCD

In the hadronic sector we have the pion pole followed by the a1(1260) resonance

A good fit to the ALEPH data is given by

rt

Ma1 = 1.0891 GeV, Γa1 = 568.78 MeV, C fa1 = 0.048326.From the first Weinberg Sum Rule we get f a1 = 0.073→ C = 0.662

r

The pion decay constant is related to the quark Condensate through the GMOR relation

The first three sum rules will be used to determine thePQCD threshold S0 , d=4 and d=6 condensates. Thesemagnitudes will be used later to normalize our finite tempe-rature results.

To leading order in PQCD we get S0 = 0.7 GeV2 from the pionpole. → 1.15 GeV2 when taking the a1 into account.

For the condensates we get the usual results.

Thermal Extension of the QCD Sum Rules

• There are important differences:

• 1) The vacuum is populated (a thermal vacuum)

• 2) A new analytic structure in the complex

s-plane appears, due to scattering. This effect turns out to be very important

New cut associated to a scattering process with quarks (antiquarks) in the populated vacuum (Bochkarev-Schaposnikov)

Finite Temperature Effects 1) Time-like region: ω2 - │q │2 > 0

2) Space-like region: ω2 - │q │2 < 0

Evolution of the quark condensate (equivalent to fπ). The solid line (Schwinger-Dyson approach) is chiral limit. Dottedline is for massive quarks (Lattice data)

Tc = 197 MeV

PreviousInformation

We will concentrate on the chiral limit, as we findthat the FESR have only solutions up to 0.9 Tc

where the quark condensate is essentially unique.

Gluon Condensate

The first three sum rules:

1)

2)

3)

From these Sum Rules, we are able to get:

S0(T);

fa1(T);

Γa1(T)Assumption: ma1 does not depend on T

S0(T) / S0(0): Solid curve. fπ 2(T) / fπ2

(0): Dotted curve.

The widthdefinitelygrows withtemperature

The coupling decreases!!

• Conclusions:

We have confirmed the picture where S0(T) moves to the

left, being a phenomenological order parameter for deconfinement .

The width of the a1 has a divergent behavior as function of T.

The coupling fa1 (T) vanishes at the critical temperature.