Critical exponents and equation of state from series summation

Universal quantities near the phase transition of O(N) symmetric vector models, can be determined, in the framework of the (f2 )2 field theory, and the corresponding renormalization group (RG), in the form of perturbative series. The O(N) symmetric field theories describe, in particular for N = 0, the universal properties of the statistics of long polymers, for N = 1, the liquid–vapour transition, for N = 2, superfluid helium transition, and so on. Universal quantities have been calculated within two different schemes, the Wilson-Fisher ϵ = 4 − d expansion, and perturbative expansion at fixed dimensions 2 and 3 (as suggested by Parisi). In both cases, the series are divergent, and the expansion parameters are not small. In fixed dimensions smaller than 4, the series are proven to be Borel summable. For the ϵ expansion, there are reasons that the property is equally true, but a proof is lacking. With this assumption, in both cases, although the series are divergent, they define unique functions. Since the expansion parameters are not small, summation methods are then required to determine these functions. A specific summation method, based on a parametric Borel transformation and mapping, in which the knowledge of the large order behaviour has been incorporated, has been successfully applied to the series, and has led to a precise evaluation of critical exponents and other universal quantities.

Field theory: Perturbative expansion and summation methods

Chapter 23 examines perturbative expansion and summation methods in field theory. In quantum field theory, all perturbative expansions are divergent series in the mathematical sense. This leads to a difficulty when the expansion parameter is not small. In the case of Borel summable series, using the knowledge of the large order behaviour, a number of summation techniques have been developed to derive convergent sequences from divergent series. Some methods apply directly on the series like Padé approximants or order–dependent mapping (the ODM method). Others involve first a Borel transformation, like the Padé–Borel method. The method of Borel transformation, suitably modified, followed by a conformal mapping, has been applied to renormalization group (RG) functions of the phi4 3 field theory and has led to precise values of critical exponents.

Renormalization Group: From a general concept to numbers

Chapter 5 first recalls the importance of the concept of scale decoupling in physics. It then emphasizes that quantum field theory and the theory of critical phenomena have provided two examples where this concepts fails. To deal with such a situation, a new tool has been invented: the renormalization group. In the framework of effective quantum field theory, a perturbative renormalization group has been formulated. Its implementation has led to the discovery of fixed points as zeros of beta functions, and calculations of critical exponents of a class of macroscopic phase transitions in the form of Wilson–Fisher epsilon or fixed dimension expansions. These expansions being divergent, they could summed by methods based on the Borel transformation and the determination of the large order behaviour of perturbation theory.

DETERMINATION OF THE STRONG COUPLING FROM HADRONIC TAU DECAYS USING RENORMALIZATION GROUP SUMMED PERTURBATION THEORY

We determine the strong coupling constant αs from the τ hadronic width using a renormalization group summed (RGS) expansion of the QCD Adler function. The main theoretical uncertainty in the extraction of αs is due to the manner in which renormalization group invariance is implemented, and the as yet uncalculated higher order terms in the QCD perturbative series. We show that new expansion exhibits good renormalization group improvement and the behavior of the series is similar to that of the standard CIPT expansion. The value of the strong coupling in [Formula: see text] scheme obtained with the RGS expansion is [Formula: see text]. The convergence properties of the new expansion can be improved by Borel transformation and analytic continuation in the Borel plane. This is discussed elsewhere in these issues.

RτAND HIGGS DECAY WIDTH IN CORGI APPROACH, BASED ON THE LEADING-b APPROXIMATION

We consider Adler D-function for the vector and scalar correlators with recent QCD analysis at N4LO approximation. The portion of perturbative coefficients of these observable containing the leading power of b, the first beta-function coefficient, is resummed to all-orders. The factorial behavior of coefficients allows us to use the Borel transformation to reveal the existent singularities as renormalons. CORGI approach is employed to avoid the renormalization dependence of the observable. In addition to the CORGI approach, the calculations of the standard perturbative QCD approach which uses the [Formula: see text] scheme with a physical choice of renormalization scale, are also presented. A comparison between the results of the standard and CORGI approaches for Rτand Higgs decay width is done. The comparison anticipates a better result for the CORGI approach. As an adjunct to these studies, the difference between the all order and fixed order result at N4LO approximation is used to estimate the uncertainty in [Formula: see text] extracted from Rτmeasurements. We find at N4LO approximation a smaller uncertainty with respect to the previous result which has been done at N4LO approximation.