Functions relevant for obtaining the width Rτ (Adler function D(s), correlation function Π(s)) are not uniquely determined from perturbative expansions. Also, their singularities are only partially known.
Conclusions:
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High ambiguity of functions given by asymptotic expansions. Proving a modification of Watson lemma, we found a class of integration contours having the same expansion. Result: We extended the class of functions with a given expansion by a wide class of curvilinear integrals. [1].
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Expansions in αsn do not share the singularities of D(s). Replacing αsn by special functions Wn(αs) having the singularities like D(s), we determined αs from τ decay. We analyzed Beneke-Jamin model. Result: Strong improvement of convergence properties at all orders. The calculated αs agrees with that obtained with αsn, but converge faster. Series in Wn(αs) have no high-order oscillations, typical for the series in
αsn [2].
[1] I. Caprini, J. Fischer, I. Vrkoč: On the ambiguity of the field correlators represented by asymptotic perturbation expansions. J. Phys. A: Math.Theor. 42 (2009) 395403
[2] I. Caprini, J. Fischer:
αs(s) from tau decays: Contour-improved versus fixed-order summation in a new QCD perturbation expansion. Eur.Phys.J. C 64: 35-45, 2009
