Heavy-quark condensate at zero and finite temperatures for various forms of the short-distance potential

Abstract

With the use of the world-line formalism, the heavy-quark condensate in the SU(N)-QCD is evaluated for the cases when the next-to-1/ r term in the quark-antiquark potential at short distances is either quadratic, or linear. In the former case, which takes place in the stochastic vacuum model, the standard QCD-sum-rules result is reproduced, while the latter result is a novel one. Explicitly, it is UV-finite only in less than four dimensions. This fact excludes a possibility to have, in four dimensions, very short strings, whose length is smaller than the vacuum correlation length. Accordingly, the linear term in the potential also vanishes at such distances, and OPE is not violated. In any number of dimensions, the obtained novel expression for the quark condensate depends on the string tension at short distances, rather than on the gluon condensate, and grows linearly with the number of colors in the same way as the standard QCD-sum-rules expression. The use of the world-line formalism enables one to generalize further both results to the case of finite temperatures. A generalization of the QCD-sum-rules expression to the case of an arbitrary number of space-time dimensions is also obtained and is shown to be UV-finite, provided this number is smaller than six.