Remark on the effect of renormalization scheme dependence on the determination ofΛQCDfrom τ-lepton decay

Abstract

The perturbative QCD corrections to the semileptonic decay width of the r lepton are analyzed in various renormalization schemes. Significant differences are found between a priori admissible schemes, which indicates that it is impossible to use these corrections to obtain a precise determination of the QCD scale parameter A.PACS number(s): 13.35. + s, 12.38.Aw, 12.38.QkRecently there has been some interest in the quantum where k = p / m , is a constant scale parameter, p is the chromodynamic effects in r-lepton decay [l-171, which renormalization point, and a ( /~) =~~( p ) / 4 . r r~ is the runare represented by the quantity called R .:ning coupling constant. The numerical value of a ( p ) is determined by the implicit equation

Strong renormalization scheme dependence in τ-lepton decay: fact or fiction?

“…In view of the arbitrariness in the selection of the RRS, no absolute meaning can, however, be given to the scale µ or the parameter k and thus no arguments on the existence of a "natural" scale, given for instance by Q, are sufficient to fix the RS. I have repeated these simple facts as the author of [1] starts his criticism of the "conventional" approach with a correct but obvious and well-known observation just made, i.e. that the same µ, or equaivalently k, implies different r i 's and a(µ) in different RRS's.…”

“…In a recent article [1] P. Raczka has questioned the optimism of several earlier papers [2,3,4] concernig the possibility to use the semileptonic decay rate of the τ -lepton and namely the quantity r τ appearing in the measured ratio R τ ≡ Γ(τ − → ν τ + hadrons) Γ(τ − → ν τ e − ν e ) = 3 (1 + r τ ) (1 + O(α em ))…”

“…There is no problem in this respect and there are various distinct, though in principle equivalent, ways how to do it. The lack of clear distinction between these two separate issues is, however, a frequent phenomenon and can be detected also in [1].…”

“…In the notation of [1] (with the exception of Q which is used below to denote a generic external momentum variable instead m τ , appropriate to (1)), I shall discuss QCD predictions for any physical quantity depending on Q and admitting perturbative expansion of the form (in massless QCD)…”

“…It can be redrawn (as in [1]) for any particular physical quantity expressible in the form (2) as a function of b ln (Q/Λ MS ) by simply shifting the origin of Fig.2a by an appropriate value of r 1 and taking into account different value of external momentum Q. The content of Fig.2a, with curves corresponding to r 1 ∈ (−3.83, 8.32) has been interpreted in [1] as an evidence for the strong RS dependence of the τ decay rate (2). This interval of r 1 values correspond to the overlap between those obtained within the MS RRS varying k in (4) between 1/2 and 2 and between 1/3 and 1 within the symmetric MOM RRS.…”