Reinhart Koegerler

We present a generalisation of our previous approach of a renormalon-motivated resummation of the QCD observables. Previously it was applied to the spacelike observables whose perturbation expansion was D ( Q 2 ) = a ( Q 2 ) + O ( a 2 ) , where a ( Q ² ) ≡ α _s ( Q ² )/ π is the running QCD coupling. Now we generalise the resummation to spacelike quantities D ( Q 2 ) = a ( Q 2 ) ν 0 + O ( a ν 0 + 1 ) and timelike quantities F ( σ ) = a ( σ ) ν 0 + O ( a ν 0 + 1 ) , where ν ₀ is in general a noninteger number (0 < ν ₀ ≤ 1). We evaluate with this approach a timelike quantity, namely the scheme-invariant factor of the Wilson coefficient of the chromomagnetic operator in the heavy-quark effective Lagrangian, and related quantities.

A method of resummation of truncated perturbation series, related to diagonal Pad\'e approximants but giving results independent of the renormalization scale, was developed more than ten years ago by us with a view of applying it in perturbative QCD. We now apply this method in analytic QCD models, i.e., models where the running coupling has no unphysical singularities, and we show that the method has attractive features such as a rapid convergence. The method can be regarded as a generalization of the scale-setting methods of Stevenson, Grunberg, and Brodsky-Lepage-Mackenzie. The method involves the fixing of various scales and weight coefficients via an auxiliary construction of diagonal Pad\'e approximant. In low-energy QCD observables, some of these scales become sometimes low at high order, which prevents the method from being effective in perturbative QCD where the coupling has unphysical singularities at low spacelike momenta. There are no such problems in analytic QCD.

Analytic QCD models are those versions of QCD in which the running coupling parameter a(Q^2) has the same analytic properties as the spacelike physical quantities, i.e., no singularities in the complex Q^2 plane except on the timelike semiaxis. In such models, a(Q^2) usually differs from its perturbative analog by power terms ~(Lambda^2/Q^2)^k for large momenta, introducing thus nonperturbative terms in spacelike physical quantities whose origin is the UV regime. Consequently, it contradicts the ITEP-OPE philosophy which states that such terms can come only from the IR regimes. We investigate whether it is possible to construct analytic QCD models which respect the ITEP-OPE philosophy and, at the same time, reproduce not just the high-energy QCD observables, but also the low-energy ones, among them the well-measured semihadronic tau decay ratio.

We construct models of analytic QCD (i.e.,with the running coupling parameter free of Landau singularities) which address several problems encountered in previous analytic QCD models, among them their incompatibility with the ITEP-OPE philosophy (due to UV power terms) and too low values of the semihadronic tau decay ratio. The starting point of the approach is the construction of appropriate nonperturbative beta functions.

The values of the presently available truncated perturbative expressions for the pressure of the quark-gluon plasma at finite temperatures and finite chemical potential are trustworthy only at very large energies. When used down to temperatures close to the critical one Tc, they suffer from large uncertainties due to the renormalization scale freedom. In order to reduce these uncertainties, we perform resummations of the pressure by applying Pade-related approximants to the available perturbation series for the short-distance and for the long-distance contributions. In the two contributions, we use two different renormalization scales which reflect different energy regions contributing to the different parts. Application of the obtained expressions at low temperatures is made possible by replacing the usual four-loop barMS beta function for alpha_s by its Borel-Pade resummation, eliminating thus the unphysical Landau singularities of alpha_s. The obtained results are remarkably insensitive to the chosen renormalization scale and can be compared with lattice results -- for the pressure (p), the chemical potential contribution (delta p) to the pressure, and various susceptibilities. A good qualitative agreement with the lattice results is revealed down to temperatures close to Tc.