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

With the use of the world-line formalism, the heavy-quark condensate in theSU(N)-QCD is evaluated for the cases when the next-to-1/r term in thequark-antiquark potential at short distances is either quadratic, or linear. Inthe former case, the standard QCD-sum-rules result is reproduced, while thelatter result is a novel one. Explicitly, it is UV-finite only in less thanfour dimensions. This fact excludes a possibility to have, in four dimensions,very short strings (whose length has the scale of the lattice spacing), andconsequently the short-range linear potential (if it exists) cannot violate theOPE. In any number of dimensions, the obtained novel expression for the quarkcondensate depends on the string tension at short distances, rather than on thegluon condensate, and grows linearly with the number of colors in the same wayas the standard QCD-sum-rules expression. The use of the world-line formalismenables one to generalize further both results to the case of finitetemperatures. A generalization of the QCD-sum-rules expression to the case ofan arbitrary number of space-time dimensions is also obtained and is shown tobe UV-finite, provided this number is smaller than six