ar X iv : h ep - p h / 99 07 40 7 v 1 1 9 Ju l 1 99 9 Gluon condensate term in heavy quark mass

We investigate a connection between a renormalon ambiguity of heavy quark mass and the gluon condensate contribution into the quark dispersion law related with a virtuality, being a displacement of heavy quark from the perturbative mass-shell, which happens inside a hadron. Operator Product Expansion is among the most powerful tools in the heavy quark physics. In this respect it is usually applied in the form of series in the inverse heavy quark mass, determining the characteristic energy scale, say, in sum rules or for decays etc. [1]. It is well recognized that the Wilson coefficients standing in front of quark-gluon operators can contain the uncertainty caused by the factorization of perturbative contribution and the nonperturba-tive matrix elements of composite operators. In this case the restriction on internal virtualities in Feynman diagrams has to be introduced to control the dependence on the infrared energy scale λ. Usually, the gluon propagator is modified by replacement: 1/k 2 → 1/(k 2 − λ 2 g) or the cut off the gluon momenta is performed as k 2 > λ 2 [2]. The calculation results depend on these parameters. Say, a peculiar behaviour at λ 2 g → 0 appears in physical quantities. For example, a perturbative correlator of two heavy quark currents acquires a power correction like λ 4 /m 4 , where m is the heavy quark mass [3]. Physically, it means that the OPE expansion can be valid if we sum the perturbative and nonperturbative parts with the vacuum expectation of gluon operator which has the same low energy scale dependence: the gluon condensate ∼ λ 4. Then the λ-dependent term can be eaten due to the appropriate definition of OPE with the condensates. Another case takes place for the uncertainty in the heavy quark mass, where the perturbative calculation of self-energy with the gluon virtuality cut off leads to the linear term in λ. However, there is no appropriate operator whose vacuum expectation is proportional to the first power of low energy scale [1]. It was shown that the mentioned uncertainty proportional to the powers of factorization scale λ can be related with the perturbative summation of higher order diagrams, which in the limit of infinitely large number of flavors has the divergency of series in β 0 α s , where β 0 denotes the first coefficient of Gell-Mann–Low function in QCD. The Borel transform of such series has …