Analytic perturbation theory for QCD practitioners and upsilon decay

Abstract

Within the ghost-free Analytic Perturbation Theory (APT), devised in the last decade for low energy QCD, simple approximations are proposed for 3-loop analytic couplings and their effective powers, in both the space-like (Euclidean) and time-like (Minkowskian) regions, accurate enough in the large range (1-100 GeV) of current physical interest.Effectiveness of the new Model is illustrated by the example of Υ(1S) decay where the standard analysis gives α s (M Υ ) = 0.170 ± 0.004 value that is inconsistent with the bulk of data for α s . Instead, we obtain α M od s (M Υ ) = 0.185 ± 0.005 that corresponds to α M od s (M Z ) = 0.120 ± 0.002 that is close to the world average. The issue of scale uncertainty for Υ decay is also discussed.Later on, both the constructions were joined [9] by suitable integral transformations into the so-called Analytic Perturbation Theory (APT). An essential feature of the APT scheme is that Minkowskian and Euclidean counterparts for powers of usual QCD coupling (α s (Q 2 )) k form nonpower sets {A k (s)} and {A k (Q 2 )} . For the fresh reviews of APT see [10,11].Meanwhile, ghost free APT expressions for effective couplings in the Euclidean α E (Q 2 ) and Minkowskian α M (s) regions, as well as for their "effective powers" A k (Q 2 ) and A k (s), are presented by simple analytic expressions only in the one-loop case (see eqs.(6)-(9)), which are not accurate enough for practical goals.Higher-loop APT expressions are more intricate involving a special Lambert function. A few years ago the first three of them A k (Q 2 ) , A k (s); (k = 1, 2, 3) , sufficient for most of applications, were tabulated by Magradze and Kourashev [12] in the 3-loop case.Here, we propose simple model expressions for 3-loop APT couplings α E (Q 2 ) = A 1 , α M (s) = A 1 and for higher expansion functions A k (Q 2 ) , A k (s) connected by simple iterative relations. The Model depends on one additional parameter and is valid in the region above 1 GeV .The paper is organized as follows. Sections 1 and 2 contain a brief review of the main ideas, technique and some results of APT. In Section 3, we present our Model. Section 4 contains revised analysis of Υ decay data with α s value extracted anew by means of APT and our Model. Special attention is paid to scale uncertainty. The last, Section 5, is devoted to the summary of the results.

On the running coupling constant in QCD

“…As yet noted the first function in both sets coincides with the relative analytic coupling, respectively (105) and (125), while A k≥2 and A k≥2 start with an IR zero and oscillate in the IR domain around k − 1 zeros; furthermore they all obey the UV asymptotic 1/ ln k z resembling the corresponding powers α k s (z) [114]. The differential recursion relations [89] …”

“…Note that in principle the quantity V PT (Q 2 , x) is well defined, since α SDG has no unphysical singularities, due to its definition (89). To be consistent up two-loop one should use equations for α NP SDG of the type considered in sec.…”

“…[117] for different initial Λ (n f =3) ), as a result of an interpolation procedure; this yields an accuracy within 1% in the whole space-like region. A "one-loop-like" model employing the scaling constant as a fitting parameter, suitable from 2 to 100 GeV, has been recently developed in [89]. Increasing difficulties arise when dealing with even higher-loop level and it becomes prohibitive to achieve useful explicit formulas.…”

“…[89]). Meanwhile, A k and A k for k = 1, 2, 3 have been tabulated in [90] up to three-loop, starting from the exact two-loop solution (23), and the Pade' approximant technique in the three-loop case; further improved approximations as eq.…”

“…(89) means "single gluon dressing". From (89), if Π is evaluated in the MS scheme, we can write, setting µ = Q,…”