Advances in High Energy Physics

Volume 2017 (2017), Article ID 9291623, 24 pages

https://doi.org/10.1155/2017/9291623

## Finite Temperature QCD Sum Rules: A Review

^{1}Instituto de Ciencias Nucleares, Universidad Nacional Autonoma de Mexico, Apartado Postal 70-543, 04510 Mexico City, Mexico^{2}Centre for Theoretical and Mathematical Physics and Department of Physics, University of Cape Town, Rondebosch 7700, South Africa^{3}Instituto de Fisica, Pontificia Universidad Catolica de Chile, Casilla 306, Santiago 22, Chile^{4}Centro Cientfico-Tecnologico de Valparaiso, Casilla 110-V, Valparaiso, Chile

Correspondence should be addressed to C. A. Dominguez

Received 24 August 2016; Revised 16 November 2016; Accepted 14 December 2016; Published 5 February 2017

Academic Editor: Lokesh Kumar

Copyright © 2017 Alejandro Ayala et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

The method of QCD sum rules at finite temperature is reviewed, with emphasis on recent results. These include predictions for the survival of charmonium and bottonium states, at and beyond the critical temperature for deconfinement, as later confirmed by lattice QCD simulations. Also included are determinations in the light-quark vector and axial-vector channels, allowing analysing the Weinberg sum rules and predicting the dimuon spectrum in heavy-ion collisions in the region of the rho-meson. Also, in this sector, the determination of the temperature behaviour of the up-down quark mass, together with the pion decay constant, will be described. Finally, an extension of the QCD sum rule method to incorporate finite baryon chemical potential is reviewed.

#### 1. Introduction

The purpose of this article is to review progress over the past few years on the thermal behaviour of hadronic and QCD matter obtained within the framework of QCD sum rules (QCDSR) [1, 2] extended to finite temperature, . These thermal QCDSR were first proposed long ago by Bochkarev and Shaposhnikov [3], leading to countless applications, with the most recent ones being reviewed here. The first step in the thermal QCDSR approach is to identify the relevant quantities to provide information on the basic phase transitions (or crossover), that is, quark-gluon deconfinement and chiral-symmetry restoration. This is done below, to be followed in Section 2 by a brief description of the QCD sum rule method at , which relates QCD to hadronic physics by invoking Cauchy’s theorem in the complex square-energy plane. Next, in Section 3 the extension to finite will be outlined in the light-quark axial-vector channel, leading to an intimate relation between deconfinement and chiral-symmetry restoration. In Section 4 the thermal light-quark vector channel is described, with an application to the dimuon production rate in heavy-ion collisions at high energies, which can be predicted in the -meson region in excellent agreement with data. Section 5 is devoted to the thermal behaviour of the Weinberg sum rules and the issue of chiral-mixing. In Section 6 very recent results on the thermal behaviour of the up-down quark mass will be shown. Section 7 is devoted to the thermal behaviour of heavy-quark systems, that is, charmonium and bottonium states, which led to the prediction of their survival at and above the critical temperature for deconfinement, confirmed by lattice QCD (LQCD) results. In Section 8 we review an extension of the thermal QCDSR method to finite baryon chemical potential. Finally, Section 9 provides a short summary of this review.

Figure 1 illustrates a typical hadronic spectral function, , in terms of the square-energy, , in the time-like region, , at (curve (a)). First, there could be a delta-function corresponding to a stable particle present as a pole (zero-width state) in the spectral function. This could be, for example, the pion pole entering the axial-vector or the pseudoscalar correlator, with the spectral functionwhere is the pion (weak-interaction) decay constant, defined as , and is its mass. This is followed by resonances of widths increasing in size with increasing and corresponding to poles in the second Riemann sheet in the complex -plane. For instance, for narrow resonances the Breit-Wigner parametrization is normally adequate:where is the coupling of the resonance to the current entering a correlation function, being its mass and being its (hadronic) width. At high enough squared-energy, , the spectral function becomes smooth and should be well approximated by perturbative QCD (PQCD). In the sequel, this parameter will be indistinctly referred to as the perturbative QCD threshold, the continuum threshold, or the deconfinement parameter. At finite this spectrum gets distorted. The pole in the real axis moves down into the second Riemann sheet, thus generating a finite width. The widths of the rest of the resonances increase with increasing , and some states begin to disappear from the spectrum, as first proposed in [4]. Eventually, close to, or at, the critical temperature for deconfinement, , there will be no trace of the resonances, as their widths would be very large, and their couplings to hadronic currents would approach zero. At the same time, would approach the origin. Thus becomes a phenomenological order parameter for quark-deconfinement, as first proposed by Bochkarev and Shaposhnikov [3]. This order parameter, associated with QCD deconfinement, is entirely phenomenological and quite different from the Polyakov-loop used by LQCD. Nevertheless, and quite importantly, qualitative and quantitative conclusions regarding this phase transition (or crossover) and the behaviour of QCD and hadronic parameters as obtained from QCDSR and LQCD should agree. It is reassuring that this turns out to be the case, as will be reviewed here. In this scenario, whatever happens to the mass is totally irrelevant; it could either increase or decrease with temperature, providing no information about deconfinement. The crucial parameters are the width and the coupling, but not the mass. In fact, if a particle mass would approach the origin, or even vanish with increasing temperature, this in itself is not sufficient to signal deconfinement, as a massless particle with a finite coupling and width would still contribute to the spectrum. What is required is that the widths diverge, and the couplings vanish. In all applications of QCD sum rules at finite , the hadron masses in some channels decrease, and in other cases they increase slightly with increasing . At the same time, for all light- and heavy-light-quark bound states the widths are found to diverge and the couplings to vanish close to or at , thus signalling deconfinement. However, in the case of charmonium and bottonium hadronic states, after an initial surge, the widths decrease considerably with increasing temperature, while the couplings are initially independent of and eventually grow sharply close to . This survival of charmonium states was first predicted from thermal QCD sum rules [5, 6] and later extended to bottonium [7], in qualitative agreement with LQCD [8, 9].