An important problem of perturbative QCD is an accurate determination of the strong coupling
αs from hadronic
τ decays. The decay width
Rτ can be determined from the two-point correlation functions
Π(
s) which are the Fourier transforms of vacuum expectation values of current products.
One example is the Adler function
D(
s), expressed through the 1st derivative of
Π(
s). These functions are expanded in perturbation series; it is useful to know the location and nature of their singularities. These are known only partially, in the complex planes (1) of energy s, (2) of the coupling
αs(
s) , and (3) in the Borel plane. Information from the different planes can be combined, but the relations between them are complicated.
Results:
-
The determination of the function given by an asymptotic series is highly ambiguous. A modification of the Watson lemma allowed us to find a class of contours of integration along which the integrals of the Borel-Laplace type possess the same asymptotic expansion. By this we were able to extend the class of functions with a given asymptotic expansion by curvilinear contour integrals. We discuss possible physical consequences and possibilities of reducing the ambiguities [1].
- Perturbative expansions in powers of the strong coupling αs are not useful because the expanded function D(s) is singular in αs. We replace the powers of αs by a special set of functions Wn(αs), whose Borel transforms possess singularities at the same points at that the Borel transform of D(s). The properties of
Wn(αs) as well as conditions of convergence are known. The new expansion was used to determine αs from τ decay, using two different methods of summation. The Beneke-Jamin model was also analyzed.
The analysis resulted in a significant improvement of convergence properties both at finite and at very high orders. The calculation of αs gives a good agreement with the value which is obtained by standard procedures in powers of αs, but with a faster convergence. Moreover, our expansions in Wn(αs) have no violent oscillations at high orders, which are typical for the standard expansions in powers of αs .
Important publications:
[1] I. Caprini, J. Fischer, I. Vrkoč: On the ambiguity of the field correlators represented by asymptotic perturbation expansions. arXiv: 0909.0110 [hep-th];
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. arXiv: 0906.5211 [hep-ph];
Eur.Phys.J. C 64: 35-45, 2009
[3] I. Caprini, J. Fischer: Comment on “Infrared freezing of Euclidean observables”. Phys.Rev. D76:018501, 2007
[4] I. Caprini, J. Fischer: On the infrared freezing of perturbative QCD in the Minkowskian region. Phys.Rev. D71:094017, 2005
Researcher:
Jan Fischer
