η c and χ c at finite temperature from QCD Sum Rules C. A. Dominguez and Yingwen Zhang University of Cape Town, South Africa M. Loewe Pontificia Universidad Católica de Chile J. C. Rojas Universidad 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) (Pseudo)Scalar Charmonium in Finite Temperature QCD: hep-ph This work has been supported by: FONDECYT , 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 η c and the scalar state χ c from the perspective of Thermal Hilbert Moment QCD Sum Rules c) Total width d) Continuum Threshold Recent Lattice analysis suggests strongly the surviving of these states beyond T c. a) Mass b) 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 take the quark-gluon basis Hilbert moments as Sum Rules N= 1, 2… and Q 0 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 contrary to 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 T c ≈ 250 MeV is the temperature where the condensate vanishes. From data on e + e - annhilation

Going into the hadronic representation we use the ground state resonance followed by the continuum (PQCD) after the continuum threshold s 0 > 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 s 0 and Q 0 2 For the η c, s 0 (0) = 9.1 GeV 2, Q 0 2 = 0 and We find M(0) = 2.9 GeV compared to M exp (0) = 2.98 GeV. The result is quite insensible for N= 1 – 6 and Q 0 2 = 0 – 10 GeV ^2

Something equivalent occurs in the scalar sector

S 0 (T) evolution for the η c (a) and the χ c (b)

Mass evolution of the η c

Ratio of the η c width Γ(T)/Γ(0)

Ratio of the χ c width Γ(T)/Γ(0)

Ratio of the leptonic decay constants for η c (a) and χ c (b)

Certainly, the evidence of surviving is not as strong as in the vector channel (J/Ψ state) where we obtained solutions to the sum rules beyond T c 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) ( 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) ; PRD 73 (2006) 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 T c the coupling increases and 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 strong as in the J/Ψ case. This is probably the ideal candidate to be considered in this context.