QCD Thermal Sum Rules: General

Aspects and Applications

M. Loewe

Pontificia Universidad Católica de Chile

Morelia, Mexico, April 2011

This talk is based on several articles written along

many years in collaboration with C. A. Dominguez

(University of Cape Town, South Africa) and

different coworkers.

I acknowledge support form : FONDECYT 1095217 and

Proyecto Anillos ACT119 (CHILE) and NRF (South Africa)

• The plan is the following:

• 1) A general introduction to QCD Sum Rules

• 2) The extension to a finite temperature scenario

• 3) Comments on the light-light and heavy-light quarks systems.

• 4) Surviving of Charmonium states

The Idea of QCD Sum Rules is to establish a connection between fundamental aspects of QCD and the hadronic world.

It is based on:

1) The analytic structure of correlators (Cauchy Theorem) which implies global duality

2) The validity of the OPE (also in the presence

of non perturbative effects)

It involves both perturbative as non perturbative

contributions.

The usual resumation implies a

shift in the pole of the propagator

Let us consider a two point correlator

Confinement effects are parametrized through quark and

gluon condensates

The red area denotes

the quark condensate

The red area denotes

the gluon condensate

(infrared dynamics)

There are many other

Condensates…..

Here we have the

four-quark condensate

Dimension 6

In principle there are several condensates of increasing

order. In this way we will get an expansion of the form

For example (light quark sector) for the

Axialvector-Axialvector

correlator we have the current Aμ

The Fourier-transform of the correlator produces the tensor

such that

where

Notice that the operators are ordered by increasing

dimensionality and the coefficients fall down by powers

of Q2 (i.e. non-perturbative contributions). In most

applications higher condensates may be ignored

The correlator can also be expressed in the hadronic sector.

Bound states correspond to poles on the real axis and

resonances lie on the second Riemann sheet. (Complex

s-plane)

In principle we have also cuts associated to non resonant

multiple particle production.

By integrating along the contour, Cauchy’s theorem

produces immediately a set of FESR.

On the cut we have the hadronic resonances.

On the circle a description in terms of QCD

degrees of freedom is assumed to be valid.

• hadron

Since the Hadronic spectral function has the form

We immediatelly find

And this provides us with the following set of Finite

Energy Sum Rules

Where the dimensionless perturbative PQCD

moments M N are given by

If PQCD reduces only to the quark loop we

have (axial-axial correlator)

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

Let us consider a typical current correlator in a

thermal vacuum (a populated vacuum)

We can write the standard spectral

representation

Normal analytic structure of the spectral function

for current correlators in the s plane (at zero

temperature):

New effect in the pressence of a populated vacuum:

Annihilation + Scattering contributions to the spectral

function

Annihilation term (survives when T → 0)

New Effect:The current may scatter off particles in

the populated vacuum. Notice that this term

vanishes when T→0 (Bochkarev and Schaposnikov, Nucl.

Phys. B268 (1986) 220)

New cut associated to a scattering process with

quarks (antiquarks) in the populated vacuum

• Let us consider a toy model: The non

linear O(N) σ model

Such that f is the coupling

constant

In the Large N limit the theory is asymptotically free

There are condensates

All other condensates factorize.

Where the α field is given by

S

Supose we are interested in the Green function

associated to the propagation of the α field. At the one

Loop order

Expanding

• Using the scalar current

the OPE provides exactly the same

answer.

For the finite temperature corrections we can do

the calculation exactly

with

• Our results for the light-light and heavy light quark systems provides us with the following picture

1) The hadronic mass remains almost constant as function of T up to T → Tc .

2) The hadronic widths grow with T, having a divergent behavior when T → Tc.

3) In the case of effective coupling constants like gρππ ,

gπNN, the axial vector coupling of the nucleon gA, as well as form factors like Fπ, it turns out that the couplings

vanish and the mean square associated radius diverge

when T→Tc

In all cases, the continuum threshold S0(T) moves

to the left, going to the threshold where the

resonances start to appear. In fact Tc is defined

as the temperature where this occurs.

As an example, and without going into details let

us consider now the three point πNN correlator.

Here we have to deal with the three point correlator

The nucleon (η) and pion (J 5) interpolating

currents are defined as

We have the following definition

Relevant

diagrams

Thermal behavior of the coupling constant

I will present our last results in the

Charmonium sector

1)The vector J/Ψ channelC. A. Dominguez, M. Loewe, J. C. Rojas and Y. Zhang

Phys. Rev D 81 (2010) 014007

2) The pseudo- and scalar sectors (ηc, χ)C. A. Dominguez, M. Loewe, J. C. Rojas and Y. Zhang

Phys. Rev D 83 (2011) 034033

a)Our goal is to discuss the thermal behavior of the

hadronic parameters of Charmonium (vector channel

J/Ψ, pseudoscalar ηc and scalar ) from the

perspective of

Thermal Hilbert Moment QCD Sum Rules

c) Total width

d) Continuum Threshold

Thermal behavior of J/Ψ is an important

signal in the Physics of heavy Ion collisions.

a) Mass

b) Coupling: Leptonic Decay

Going into the details:

Charmonium (vector channel) seems to have a curious

bahaviour at finite temperature: Surviving beyond the

critical temperatureA Finite Temperature QCD Sum Rule Discussion

The basic object:

Satisfies a once substracted dispersion relation

The Thermal Average is defined as

The vector current is

The imaginary part involves two pieces:

a) Time like region (q2 ≥ 4mQ2)

In the rest frame

b) Space-like region (q2 ≤ 0). Im ΠS(q2,T) Vanishes for T=0:

In the rest frame of the thermal bath reduces to

The only non-perturbative relevant object is the Gluon Condensate

Going into the hadron representation of our correlator (cero width)

The leptonic decay constant is

In the finite width case

To complete the hadronic parametrization we need the hadronic scattering

term due to the current scattering off heavy-light pseudoscalar D-mesons

This term is 2-3 orders of magnitude smaller than it’s QCD counterpart

To eliminate the substraction we use Hilbert moments:

The Sum Rules

Cauchy Theorem implies quark-hadron duality.

The Hilbert Moments become Finite Energy Sum Rules (FESR)

where

and

The non perturbative term (dimension 4)

where and

The thermal dependence (low T) of the gluon condensate

can be approximate as

Hypergeometric

function

A subtle point

A gluonic twist-two term in the OPE has been considered

by Klingl et al and computed on the lattice. This implies

a modification of the non perturbative QCD moments

For small N the corrections are of the order 2-6% independent

of Q2

Lattice QCD is the right tool to go beyond the low

temperature regime (Boyd and Miller)

T* ≈ 150 MeV T*C ≈ 250

MeVFirst we determine s0 and Q2 at T=0 using as imput: MV = 3.097 GeV,

fV= 196 MeV and ΓV = 93.2 KeV, mQ= 1.3 GeV

In the zero width approximation we find

Given the extreme small total width of the J/Ψ this relation also holds

at finite width

This relation depends on s0 and Q2. Together with

the duality relation, N=1 we get

Reproduces the experimantal

values of mass coupling and

width within 1%.

Valid up to Q2~20 GeV2

as N up to ~ 10The different choices do not

produce a qualitative change at finite T

other than 10% of Tc (Tc = 140 MeV)

Let us go to our results……

The mass, in the zero

width follows from

The width can be

determined from

Finally, the leptonic decay

constant

can be determined from e.g.

The rise of the coupling beyond Tc is the result of a subtle

balance between different contributions:

If T increases, s0(T) decreases while the QCD scattering

moment increases ≈ annhilation moment at T ~ 160 MeV.

The non perturbative moment stays approx. constant and

negative mostly cancelling the annhilation moment.

The hadronic moment dividing the QCD moment is a

decreasing function of T implying an increase in the

coupling.

This behavior (coupling and width) is stable if Q2 ~ 10-20

GeV2 and N ~ 1-10

• The pseudoscalar and scalar channels

(ηc and χ)

with

We follow almost the same procedure.

The main difference comes from the absence of

scattering terms in both channels.

Our results:

• S0(T)/S0(0) as function of T/Tc for Tc= 180 MeV

Curve (a) is for ηc and curve (b) for χc.

Evolution of the ratio of the ηc mass M(t)/M(0)

for Tc= 180 MeV (Similar results for χc)

The ratio of the ηc width Γ(T)/Γ(0) for Tc= 180MeV

The ratio of the χc widths as function of T/Tc for

Tc= 180 MeV

The ratio of the leptonic decay constants f as function

of T/Tc. Curve (a) for ηc and curve (b) for χc

Other Approaches to this Problem

• 1) The Lattice community has found strong evidence

supporting the surviving of 1S charmonium sates beyond

the critical temperature. (Umeda, Nomura, Matsufuru: Eur.

Phys.J.C 39S1 (2005) 9; S. Datta, F. Karsch, P. Petreczky,

Wetzorke: Phys. Rev. D 69 (2004) (094507

With the kernel

There has been many attempts to understand

the problem form the perspective of Potential

Models

• A, Mócsy and P. Petreczky: PRL 99 (2007) 211602;

PRD 73 (2006) 074007. Several screened potentials are

used

• However, this approach fails to reproduce the results

from the lattice!! Probably this is related to the analytic

structure. The central cut here is absent.

Conclusions:

At low T the J/ψ behaves as other light-light and heavy-light

quark resonances: s0 and fV decrease and the width

increases with increasing T. However…..

As T approaches Tc the coupling increases and the

width decreases.

This scenario is new: the scattering

QCD contribution to the spectral function as well

as the peculiar Gluon condensate thermal behavior

are responsible for this.

A strong indication for surviving of

Charmonium beyond Tc

•Thank You