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

Abstract

Abstract We present a determination of the perturbative QCD (pQCD) coupling using the V+A channel ALEPH $$\tau $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> </mml:math> -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 $${\overline{\mathrm{MS}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>MS</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> coupling is $$\alpha _s(m_{\tau }^2) = 0.3116 \pm 0.0073$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msubsup> <mml:mi>m</mml:mi> <mml:mrow> <mml:mi>τ</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>0.3116</mml:mn> <mml:mo>±</mml:mo> <mml:mn>0.0073</mml:mn> </mml:mrow> </mml:math> [ $$\alpha _s(M_Z^2)=0.1176 \pm 0.0010$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msubsup> <mml:mi>M</mml:mi> <mml:mi>Z</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>0.1176</mml:mn> <mml:mo>±</mml:mo> <mml:mn>0.0010</mml:mn> </mml:mrow> </mml:math> ] 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, $$\alpha _s(m_{\tau }^2) = 0.3194 \pm 0.0167$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msubsup> <mml:mi>m</mml:mi> <mml:mrow> <mml:mi>τ</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>0.3194</mml:mn> <mml:mo>±</mml:mo> <mml:mn>0.0167</mml:mn> </mml:mrow> </mml:math> [ $$\alpha _s(M_Z^2)=0.1186 \pm 0.0021$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msubsup> <mml:mi>M</mml:mi> <mml:mi>Z</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>0.1186</mml:mn> <mml:mo>±</mml:mo> <mml:mn>0.0021</mml:mn> </mml:mrow> </mml:math> ].

Topics & Concepts