Remark on the perturbative component of inclusiveτdecay

Abstract

In the context of the inclusive τ -decay, we analyze various forms of perturbative expansions which have appeared as modifications of the original perturbative series. We argue that analytic perturbation theory, which combines renormalization-group invariance and Q 2 -analyticity, has significant merits favoring its use to describe the perturbative component of τ -decay.PACS Numbers: 11.10.Hi, 11.55.Fv, 12.38.Cy, 13.35.Dx A perturbative approximation in quantum chromodynamics as a rule cannot be exhaustive in the low energy region of a few GeV and a nonperturbative component has to be included. The reliability of extracting nonperturbative parameters from data is connected with uncertainties in the perturbative description of a process arising from the inevitable truncation of the perturbation theory (PT) series. The initial perturbative series that is obtained after the renormalization procedure is not the final product of the theory. This series can be modified and its properties can be improved on the basis of additional information coming from general properties of the quantity under consideration. In this note we consider various descriptions of the perturbative component in the context of an analysis of the inclusive decay of the τ lepton. We will discuss merits and drawbacks of the series expansion for the R τ -ratio in terms of powers of the parameter α s (M 2 τ ) [1], the prescription of Ref.[2] which uses a contour representation, and the approximation based on the analytic approach proposed in Ref. [3].The main object in a description of the hadronic decay of the τ -lepton and of many other physical processes is the correlator Π(q 2 ) or the corresponding Adler function D(−q 2 ) = −q 2 dΠ(q 2 )/dq 2 . The analytic properties of the D-function are contained within the relationwhere R(s) = ImΠ(s)/π. According to this equation, the D-function is an analytic function in the complex Q 2 -plane with a cut along the negative real axis. After renormalization, the perturbative expansion of the D-function has the form of a power series in the expansion parameter a µ = α s (µ 2 )/π. In the massless case the series has the formAs is well known, this expression is unsatisfactory both from theoretical and practical viewpoints. Any partial sum of it is not renormalization-group invariant and logarithms in the coefficients lead to an ill-defined behavior in both infrared and ultraviolet regions. The modification of the initial representation (2) based on renormalization-group invariance readswhereā(Q 2 ) is the running coupling. This commonly used modification removes some of the undesirable features of the expansion (2). A partial sum of the series (3) is now µ-independent. The log-terms in the coefficients of Eq, (2) have been summed into the running coupling and the series (3) can now be used in the ultraviolet region. However, the correct analytic properties of the partial sum of Eq. (2), the principal merit of this expansion, are no longer valid due to unphysical singularities of the perturbative running coup...

Adler function for light quarks in analytic perturbation theory

“…in the framework of the APT approach there is a little sensitivity to the approximation in solving the renormalization group equation for the running coupling [7,8]. In the following, we use an explicit form for the analytic running coupling in the timelike region,ã an (s), that is derived by using the formula (8) in Eq.…”

“…This investigation together with other results (see, for example, Refs. [8,29,30]) allows us to formulate the following features of the APT method: (i) this approach maintains the correct analytic properties and leads to a self-consistent procedure of analytic continuation from the spacelike to the timelike region; (ii) it has much improved convergence properties and turns out to be stable with respect to higher-loop corrections; (iii) renormalization scheme dependence of the results obtained within this method is reduced dramatically.…”

“…The analytic approach to QCD [6], the so-called analytic perturbation theory (APT) [7,8], maintains needed analytic properties and allows one to give meaning to the initial expression. The APT description can be equivalently phrased either on the basis of the expression (1) or on the contour representation (2) [7].…”

Summation of threshold singularities in correlators of quark currents

“…Expansion of function (12) in the singular perturbation charge demonstrates the occurrence of the so-called 2 π -terms in the highest orders. These terms are logarithmic and arise due to the special analytical continuation of the perturbation charge to the Minkowski region [29,30,31].…”

“…Obviously, this method breaks smoothness of the invariant charge function and hence does not support its analyticity. The APT method opens a new opportunity of describing different number of active quarks [50][51][52]. As a result of new joining which uses the physical region of time-like momenta, the same as in PT and quite natural for these purposes, a relationship between the scale parameters .…”