Using improved Operator Product Expansion in Borel-Laplace Sum Rules with ALEPH $τ$ decay data, and determination of pQCD coupling

Abstract

We use improved truncated Operator Product Expansion (OPE) for the Adler function, involving two types of terms with dimension D = 6, in the double-pinched Borel-Laplace Sum Rules and Finite Energy Sum Rules for the V+A channel strangeless semihadronic τ decays. The generation of the higher order perturbative QCD terms of the D = 0 part of the Adler function is carried out using a renormalon-motivated ansatz incorporating the leading UV renormalon and the first two leading IR renormalons. The trunacted D = 0 part of the Sum Rules is evaluated by two variants of the fixed-order perturbation theory (FO), by Principal Value of the Borel resummation (PV), and by contour-improved perturbation theory (CI). For the experimental V+A channel spectral function we use the ALEPH τ -decay data. We point out that the truncated FO and PV evaluation methods account correctly for the renormalon structure of the Sum Rules, while this is not the case for the truncated CI evaluation. We extract the value of the MS coupling αs(m 2 τ ) = 0.3198 +0.0149 −0.0143[αs(M 2 Z ) = 0.1187 ± 0.0018] for the average of the two FO methods and the PV method, which we consider as our main result. If we included in the average also CI extraction, the value would be αs(m 2 τ ) = 0.3270 +0.0244 −0.0241 [αs(M 2 Z ) = 0.1195 ± 0.0030]. This work is an extension and improvement of our previous work [1] where we used for the truncated OPE a more naive (and widely used) form and where the extracted values for αs(M 2 Z ) were somewhat lower.

Borel-Laplace Sum Rules with ALEPH $τ$ decay data, using OPE with improved anomalous dimensions

“…We use for the full Adler function the Operator Product Expansion (OPE) with the terms of dimension D = 0, 4, 6. The D = 6 OPE contribution in this work, with two terms, is improved in comparison with our previous works [1,2], in the sense that it involves the recently known noninteger values γ (1) (O (j) 6 )/β0 of the effective leading-order anomalous dimensions. The higher order terms of the D = 0 part of the Adler function are generated in a renormalon-motivated approach, in such a way that the resulting renormalon ambiguities originating from the infrared renormalons at u = 2, 3 can be cancelled by the corresponding D = 4, 6 OPE contributions of the Adler function.…”

“…The OPE is believed to describe reasonably well the behaviour of the Adler function D(Q 2 ) in this frontier regime between the intermediate and low momenta, |Q 2 | ≈ 3 GeV 2 , and it contains relevant contributions of dimension D = 0, 4, 6 and possibly higher. 2 On the other hand, in the D = 0 contribution d(Q 2 ) D=0 of the Adler function, the terms D = 4, 6, . .…”

“…The Borel-Laplace sum rules are then used to extract, at each truncation index Nt in the considered methods (FO, FO, PV), the values of the coupling αs and of the condensates O4 , O(1) 6and O(2) 6. The optimal values of the index Nt, in each method, is then determined by requiring the local insensitivity of the resulting double-pinched momenta a (2,0) and a (2,1) . Averaging over the three methods, the extracted value of the MS coupling is αs(m 2 τ ) = 0.3179 +0.0051 −0.0087 , corresponding to αs(M 2 Z ) = 0.1184 +0.0007 −0.0011 .…”

“…. The optimal values of the index Nt, in each method, is then determined by requiring the local insensitivity of the resulting double-pinched momenta a (2,0) and a (2,1) . Averaging over the three methods, the extracted value of the MS coupling is αs(m 2 τ ) = 0.3179 +0.0051 −0.0087 , corresponding to αs(M 2 Z ) = 0.1184 +0.0007 −0.0011 .…”

Extraction of $α_s$ using Borel-Laplace sum rules for tau decay data

“…The resulting extracted values of the coupling a (i,j) 0.324 0.341 ± 0.008 -0.332 ± 0.012 Pich&R.Sánchez, 2016 [11] a (i,j) 0.320 ± 0.012 0.335 ± 0.013 -0.328 ± 0.013 Boito et al, 2014 [12] DV in a (i,j) 0.296 ± 0.010 0.310 ± 0.014 -0.303 ± 0.012 our prev. work, 2021 [13] BL (O6, O8) 0.308 ± 0.007 -0.316 +0.008 −0.006 0.312 ± 0.007 this work, 2022 (also [14]) BL (O…”