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Abstract
We discuss the $\{ \beta \}$-expansion for renormalization group invariant quantities tracing this expansion to the different contractions of the corresponding incomplete BPHZ $R$-operation. All of the coupling renormalizations, which follow from these contractions, should be taken into account for the $\{ \beta \}$-expansion. We illustrate this feature considering the nonsinglet Adler function $D^\text{NS}$ in the third order of perturbation. We propose a generalization of the $\{ \beta \}$-expansion for the renormalization group covariant quantities -- the $\{ \beta,\gamma \}$-expansion.
Highlights
E.g., [2, 10,11,12,13])
The second disagreement is related to different interpretations of the {β}-expansion for two related representations of the e+e−-annihilation Adler function DEM which, contrary to the considerations presented in [3, 12, 13], should lead to the identical results, in full agreement with the basis of the incomplete BPHZ R-operation described in detail in [14]
The aim is to demonstrate how the definition of the {β}-expansion proposed in [2] can be realized for the case of the O(a4s) representation of the DEM-function explicitly presented in [17] in terms of the photon anomalous dimension and the polarization function ΠEM(as), which was used in the consideration of [12]
Summary
In our further discussions we shall use the MS-like scheme renormalization prescriptions for the Adler D-function, which were described in detail in [15, 16]. They were used in the process of evaluation of the 2nd order perturbative QCD correction to the D-function and to its spectral density R(s) (in brief the result was published in [22]). To get the renormalized expression for ΠEM(L, as), which determines eventually the renormalized Adler function, one would use ZphaB = a (the Ward identity) for electromagnetic coupling and rewrite eq (3.1) in terms of Z = (Zph − 1)/a: ΠEM(L, as) = Z + ΠEBM(L, asB). In the discussions below we will prove that eqs. (3.6a)–(3.6c), which contain the coefficients of the photon anomalous dimension function of eq (2.11), i.e. the terms −lZl,−1, give the {β}-expanded structure for dNl−S1 coefficients, which is identical to the one formulated in [2] (see, e.g., eqs. (2.9a)–(2.9c) presented above)
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