# Thermal QCD sum rules for mesons

#### Abstract

In their original work extending the vacuum QCD sum rules [1] to those at finite temperature, Bochkarev and Shaposhnikov [2] recognised the importance of multiparticle (branch cuts) contributions in addition to those of single particles (poles) in constructing the spectral representation of thermal two-point functions. Thus for the vector current correlation function, they included not only the ρ-pole with temperature dependent residue and position, but also the ππ continuum. In the same way the spectral representation for the nucleon current correlation function would consist of the nucleon pole with temperature dependent parameters and the πN continuum. Although such a saturation scheme is quite suggestive and had been extensively used in the past [3,4], it lacks a theoretical basis, leaving one to suspect if some equally important contributions are left out. A definitive saturation scheme emerged with the work of Leutwyler and Smilga [5], who calculated the nucleon current correlation function in chiral perturbation theory. Being interested in the shifts of the nucleon pole parameters at low temperature, they considered all the one-loop Feynman diagrams for the correlation function and evaluated them in the vicinity of the nucleon pole. Koike [6] examined the contributions of these diagrams in the context of QCD sum rules. He found a new contribution arising from the nucleon self-energy diagram to the sum rules, not required in the saturation scheme mentioned above. The one-loop Feynman diagrams for the nucleon correlation function were further analysed in Ref. [7]. In this set of diagrams for the correlation function, one has not only diagrams with the (single particle) pole alone and the (two particle) branch point alone, but also other (one particle reducible) diagrams, appearing as product of factors with the pole and the branch cut. As an example, take the case of a vertex correction diagram having this product structure. It may be expressed as the sum of the pole term with constant residue and a remainder, regular at the pole. Clearly to find the pole parameters one may confine oneself to the pole term alone. But if one wishes to evaluate the diagram for large space-like momenta – the region of relevance for the QCD sum rules – both the pole and the remainder become of comparable magnitude. It is these remainder terms which are not included in the earlier saturation scheme. In this paper we write down the thermal QCD sum rules following from the vector current and the axial vector current correlation functions, constructing the spectral side from the set of all one-loop Feynman diagrams. At low temperature only the distribution function of the pions is significant in the heat bath. Thus, of the two particles in the loop, at least one must be a pion, the other being any one of the strongly interacting particles with appropriate quantum numbers in the low mass region. The vertices occurring in the diagrams are obtainable from the chiral perturbation theory [8,9] as well as from other formulations of the effective theory of QCD at low energy [10]. But the former theory alone is based only on the chiral symmetry of QCD, the others assuming one or other unproven symmetries or relations. Thus the chiral perturbation theory is singled out to determine the vertices, if we claim the sum rules to follow from QCD. As with the nucleon sum rules [7], we subtract out the vacuum sum rules from the corresponding full sum rules at finite temperature, equating, in effect, terms of O(T ) on both sides. All our calculations are done in the chiral symmetry limit, though we keep nonvanishing pion mass in intermediate steps. In Sec. II we collect some results to be used later. In Sec. III we analyse the one-loop Feynman diagrams to construct the spectral representation for the correlation functions. In Sec. IV we use the known results of Operator Product Expansion and write the sum rules. Finally our concluding remarks are contained in Sec.V.

### Cite this paper

@inproceedings{MallikThermalQS, title={Thermal QCD sum rules for mesons}, author={Siddhartha Mallik and Sourav Sarkar} }