Two-loop calculations in the large quark mass expansion in the framework of the QCD sum rules method

Two-loop gluon-condensate contributions to heavy-quark current correlators: exact results and approximations

“…After correcting this inconsistency, we find agreement between our new results and the partial results of [5,6] for the first 7 moments. In addition to studying this z → 0 limit in Section 3, we also compare our results with those obtained in [9], as z → −∞, and in [10], as z → 1, finding agreement with the leading term, in each case.…”

“…Renormalization of the resulting bare two-loop terms amounts to no more than mass renormalization of the d-dimensional one-loop contributions, which we perform in the onshell scheme, expressing our results in terms of the pole-mass, m. Doing this, we discover that a term was missed in [5,6], when performing the limit ε → 0 in the MS scheme, with the effect that the results given in [3,5,6] do not hold in the MS scheme (nor in any scheme that respects the Ward identity relating axial and pseudoscalar correlators). After correcting this inconsistency, we find agreement between our new results and the partial results of [5,6] for the first 7 moments.…”

“…This work is therefore an extension of that in [5,6], where the first 7 moments of these coefficient functions were analytically computed, by reducing the problem to the evaluation of vacuum scalar two-loop diagrams with one massless and two massive lines and arbitrary integer indices. Here we apply the more general strategy of [4,7], reducing the problem to evaluation of two-point diagrams, of the type needed for the two-loop photon propagator [8], and using the programs developed for [4] to obtain explicit analytical expressions for such diagrams, in d = 4 − 2ε spacetime dimensions.…”

“…In Section 4, we assess numerical approximations, developed in [2,3,4,11,12], by comparing predictions, made on the basis of previously available input [5,9,10], with exact new two-loop results. The outcome is most satisfactory: with 7 moments as input, the methods achieve very high accuracy; with 4 moments, they fare almost as well; most remarkably, we shall show that a single moment, combined with only the leading behaviours as z → 1 and z → −∞, gives the two-loop terms in the next 10 moments to better than 1% accuracy.…”

“…Here we apply the more general strategy of [4,7], reducing the problem to evaluation of two-point diagrams, of the type needed for the two-loop photon propagator [8], and using the programs developed for [4] to obtain explicit analytical expressions for such diagrams, in d = 4 − 2ε spacetime dimensions. As input to these programs, we use the intermediate results of [5,6], obtained after the calculation of traces and the action of the projector for the operator G 2 .…”