ηc and χc at finite temperature fromQCD Sum Rules

C. A. Dominguez and Yingwen ZhangUniversity of Cape Town, South Africa

M. Loewe Pontificia Universidad Católica de Chile

J. C. RojasUniversidad Católica del Norte

This talk is based on the articles:

Charmonium in the vector channel at finite temperature from QCD

Sum Rules: Physical Review D 81 (2010) 014007

(Pseudo)Scalar Charmonium in Finite Temperature QCD: hep-ph 1010.4172

This work has been supported by: FONDECYT 1095217,Proyecto Anillos ACT119 (CHILE) and NRF (South Africa)

a)Our goal is to discuss the thermal behavior of the hadronic parameters of the pseudoscalar state ηc and the scalar state χc from the perspective of Thermal Hilbert Moment QCD Sum Rules

c) Total width d) Continuum Threshold

Recent Lattice analysis suggests strongly the survivingof these states beyond Tc .

a) Massb) Coupling: Leptonic Decay

Something like an Introduction: Let us consider a typical current correlator in a thermal 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

Going into the details:

Charmonium states (vector, scalar and pseudo scalar channels) seem to have a curious behaviour at finite temperature: Surviving

beyond the critical temperature

A Finite Temperature QCD Sum Rule Discussion

The basic object:

Satisfies a once substracted dispersion relation

where

The Thermal Average is defined as

is any complete set of eigenstates of the QCD Hamiltonian. We will takethe quark-gluon basis

Hilbert moments as Sum Rules

N= 1, 2… and Q0

2 is a free parameter

• The Finite Energy Sum Rules

where

and

The annhilation part for the pseudoscalar case (rest frame)

with

The equivalent expression for the scalar correlator

It is possible to show that in these channels, on the contraryto what happens in the charmonium vector J/Ψ case, the scattering contributions vanish.

• In this way, the leading temperature dependent part in the OPE is the Gluon Condensate.

The moments for this term were calculated by Reinders, Rubinstein and Yazaki (1981). For the scalar correlator we have

F(a,b,c,z) is the hypergeometric function

Is the temperature dependent Gluon condensate

For the pseudoscalar case we have

The T-dependent Gluon Condensate was first estimated in the frame of chiral perturbation theory by Gerber and Leutwyler. To a good approximation it can be written as

However, this is valid only for small temperatures

People from the lattice found a different thermal behavior for the gluon condensate (Boyd and Miller)

T * ≈ 150 MeV is the breakpoint temperature where the condensate begins to decrease and Tc ≈ 250 MeV is the temperature wherethe condensate vanishes.

From data on e+e- annhilation

Going into the hadronic representation we use the groundstate resonance followed by the continuum (PQCD)after the continuum threshold s0 > hadron mass

The leptonic decay constant f is defined by

The finite width extension is constructed as

Beginning with the zero-width approximation we have

This leads to

This allows to determine s0 andQ0

2

For the ηc, s0(0) = 9.1 GeV 2, Q02 = 0 and

We find M(0) = 2.9 GeV compared to Mexp(0) = 2.98 GeV. The result is quite insensible for N= 1 – 6 and Q0

2= 0 – 10 GeV^2

• Something equivalent occurs in the scalar sector

S0(T) evolution for the ηc (a) and the χc (b)

Mass evolution of the ηc

Ratio of the ηc width Γ(T)/Γ(0)

Ratio of the χc width Γ(T)/Γ(0)

Ratio of the leptonic decay constants for ηc (a)and χc (b)

Certainly, the evidence of surviving is not as strong as in the vector channel (J/Ψ state) where we obtainedsolutions to the sum rules beyond Tc

Let us remind the results in this case.

Vector channel

Leptonic decay constant, vector channel

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 understandthe 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.

1) QCD Sum Rules suggest a surviving of

the pseudoscalar and scalar charmonium states

Conclusions:

2) As T approaches Tc the coupling increasesand the width decreases. Masses seem to be stable.

3) In these channels this behavior is determined by the gluon condensate

4) The surviving evidence is not so strongas in the J/Ψ case. This is probably the ideal candidate to be considered in this context.