Determination of perturbative QCD coupling from ALEPH $$\tau $$ decay data using pinched Borel–Laplace and Finite Energy Sum Rules

Abstract

We present a determination of the perturbative QCD (pQCD) coupling using the V+A channel ALEPH $$\tau $$ τ -decay data. The determination involves the double-pinched Borel–Laplace Sum Rules and Finite Energy Sum Rules. The theoretical basis is the Operator Product Expansion (OPE) of the V+A channel Adler function in which the higher order terms of the leading-twist part originate from a model based on the known structure of the leading renormalons of this quantity. The applied evaluation methods are contour-improved perturbation theory (CIPT), fixed-order perturbation theory (FOPT), and Principal Value of the Borel resummation (PV). All the methods involve truncations in the order of the coupling. In contrast to the truncated CIPT method, the truncated FOPT and PV methods account correctly for the suppression of various renormalon contributions of the Adler function in the mentioned sum rules. The extracted value of the $${\overline{\mathrm{MS}}}$$ MS ¯ coupling is $$\alpha _s(m_{\tau }^2) = 0.3116 \pm 0.0073$$ α s ( m τ 2 ) = 0.3116 ± 0.0073 [$$\alpha _s(M_Z^2)=0.1176 \pm 0.0010$$ α s ( M Z 2 ) = 0.1176 ± 0.0010 ] for the average of the FOPT and PV methods, which we regard as our main result. On the other hand, if we include in the average also the CIPT method, the resulting values are significantly higher, $$\alpha _s(m_{\tau }^2) = 0.3194 \pm 0.0167$$ α s ( m τ 2 ) = 0.3194 ± 0.0167 [$$\alpha _s(M_Z^2)=0.1186 \pm 0.0021$$ α s ( M Z 2 ) = 0.1186 ± 0.0021 ].

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…”

“…If we took, instead of the two mentioned D = 6 terms in the OPE, the simple D = 6 and D = 8 OPE terms [∼ 1/(Q 2 ) 3 and ∼ 1/(Q 2 ) 4 ], the central value would decrease by about δα s (m 2 τ ) ≈ −0.008. In our previous work [13] we used the OPE with simple D = 6, 8 terms, and took for d 4 higher values d 4 = 338 ± 63 than here Eq. ( 7).…”

Reconciling the Contour-Improved and Fixed-Order Approaches for $τ$ Hadronic Spectral Moments I: Renormalon-Free Gluon Condensate Scheme

“…(2.12), where the gluon condensate (GC) operator product expansion (OPE) correction and the asymptotic character of the series related to it are suppressed, 1 the CIPT approach leads to systematically smaller values for the truncated perturbative series. As a result α s determinations based on CIPT yield systematically larger values than being based on FOPT [5][6][7][8][9][10].…”

“…In our scheme it is convenient to treat the function c0 (R 2 ), which is conceptually part of the GC OPE correction, like a tree-level contribution that is not supposed to be expanded any more in powers of the strong coupling. 10 Parametrizing the subtraction series generated by Eq. (3.1) back into a Borel function, the perturbation series for the Adler function DRF (s)…”

On the use of the Operator Product Expansion in finite-energy sum rules for light-quark correlators

“…One might argue that the tests of Eqs. (7,8) are possibly somewhat inconclusive. However, the R-ratio data allow us to subject the tOPE strategy to a more stringent test, by considering the s 0 dependence of tOPE-based fits.…”