arXiv subjects

B. A. Magradze

Publications and source records attributed to B. A. Magradze.

6 recordsLinked to original sources

arXiv2012

Strong Coupling Constant from $τ$ Decay within a Dispersive Approach to Perturbative QCD

We present a new dispersive framework for the extraction of the strong coupling constant $α_s$ from $τ$-lepton decays. A new feature of our procedure is the use of the quark-hadron duality on the limited region $s_{\rm d}<s<m_τ^{2}$. The duality point $s_{\rm d}$ and the $\bar{\rm MS}$ strong coupling constant $α_{s}(m_τ^{2})$ are self-consistently extracted from the $τ$ data for the non-strange vector spectral function. We use 2005 ALEPH and 1998 OPAL experimental data on the vector spectral function. We compare the new framework with the contour improved perturbation theory up to order $α_s^{5}$. The new procedure yields systematically lower values for $α_s$. From the 2005 ALEPH data, we obtain $α_{s}(m_τ^{2})=0.308\pm 0.014_{\rm exp}\pm 0.005_{\rm th}$ which corresponds to $α_{s}(M_{\rm_{z}}^{2})=0.1170\pm 0.0018_{\rm exp}\pm 0.0007_{\rm th}\pm 0.0005_{\rm ev}$. The extracted value for the duality point $s_{\rm d}$ is found surprisingly stable against perturbation theory corrections $s_{\rm d}= 1.71\pm 0.05_{\rm exp}\pm 0.00_{\rm th}\,\, {\rm GeV^{2}}$. From the 1998 OPAL data, we obtain $α_{s}(m_τ^{2})=0.290\pm 0.023_{\rm exp}$ and $s_{\rm d}=1.68\pm 0.10_{\rm exp}\,\, {\rm GeV^{2}}$.

hep-ph↗

arXiv2012

Testing the Concept of Quark-Hadron Duality with the ALEPH $τ$ Decay Data

We propose a modified procedure for extracting the numerical value for the strong coupling constant $α_s$ from the $τ$ lepton hadronic decay rate into non-strange particles in the vector channel. We employ the concept of the quark-hadron duality specifically, introducing a boundary energy squared $s_{\rm p}>0$, the onset of the perturbative QCD continuum in Minkowski space \cite{BLR,Rafa,PPR}. To approximate the hadronic spectral function in the region $s>s_{\rm p}$, we use Analytic Perturbation Theory (APT) up to the fifth order. A new feature of our procedure is that it enables us to extract from the data simultaneously the QCD scale parameter $Λ_{\bar{\rm MS}}$ and the boundary energy squared $s_{\rm p}$. We carefully determine the experimental errors on these parameters which come from the errors on the invariant mass squared distribution. For the $\bar{\rm MS}$ scheme coupling constant, we obtain $α_s(m^{2}_τ)=0.308\pm 0.014_{\rm exp.}$. We show that our numerical analysis is more stable against higher-order corrections than the standard one. The extracted value for the duality point $s_{\rm p}$ is found surprisingly stable against perturbation theory corrections $s_{\rm d}= 1.71\pm 0.05_{\rm exp}\pm 0.00_{\rm th}\,\, {\rm GeV^{2}}$.Additionally, we recalculate the "experimental" Adler function in the infrared region using final ALEPH results. The uncertainty on this function is also determined.

hep-ph↗

arXiv2006

A novel series solution to the renormalization group equation in QCD

Recently, the QCD renormalization group (RG) equation at higher orders in MS-like renormalization schemes has been solved for the running coupling as a series expansion in powers of the exact 2-loop order coupling. In this work, we prove that the power series converges to all orders in perturbation theory. Solving the RG equation at higher orders, we determine the running coupling as an implicit function of the 2-loop order running coupling. Then we analyze the singularity structure of the higher order coupling in the complex 2-loop coupling plane. This enables us to calculate the radii of convergence of the series solutions at the 3- and 4-loop orders as a function of the number of quark flavours $n_{\rm f}$. In parallel, we discuss in some detail the singularity structure of the ${\bar{\rm MS}}$ coupling at the 3- and 4-loops in the complex momentum squared plane for $ 0\leq n_{\rm f} \leq 16 $. The correspondence between the singularity structure of the running coupling in the complex momentum squared plane and the convergence radius of the series solution is established. For sufficiently large $n_{\rm f}$ values, we find that the series converges for all values of the momentum squared variable $Q^2=-q^2>0$. For lower values of $n_{\rm f}$, in the ${\bar{\rm MS}}$ scheme, we determine the minimal value of the momentum squared $Q_{\rm min}^2$ above which the series converges. We study properties of the non-power series corresponding to the presented power series solution in the QCD Analytic Perturbation Theory approach of Shirkov and Solovtsov. The Euclidean and Minkowskian versions of the non-power series are found to be uniformly convergent over whole ranges of the corresponding momentum squared variables.

hep-ph↗

arXiv2004

Practical techniques of analytic perturbation theory of QCD

The Lambert-W explicit solutions to the QCD renormalization group (RG) equation are considered up to fourth order in the ${\bar {MS}}$ scheme. We compare, systematically, these solutions with the conventional asymptotical (iterative) approximations and with the exact numerical solutions to the RG equation in the domain with three quark flavours. Applications of these solutions in analytic perturbation theory (APT) are discussed. Using these (Lambert-W, asymptotical and exact numerical) solutions we reconstruct the expansion functions for the non-power APT series in the space- and time-like regions. These expansion functions are examined in the infrared region. It is shown that the Lambert-W solutions provide the excellent accuracy.

hep-ph↗

arXiv2001

Explicit expressions for Euclidean and Minkowskian QCD observables in analytic perturbation theory

Technical aspects of the Shirkov-Solovtsov's analytic perturbation theory (APT) are considered. We construct explicitly two sets of specific functions, ${\mathfrak{A}_n(s)}$ and ${{\cal A}_n(Q^2)}$ that determine the nonpower as ymptotic expansions for Minkowskian and Euclidean QCD observables in APT. The results, up to third order, are written in terms of the Lambert W-functions. As an input we used the exact two loop and the three loop (corresponding to Padé transformed be ta-function) RG solutions for common invariant coupling $α_s$. In addition, the exact three-loop coupling is expanded in powers of the exact two-loop solution. The excellent accuracy is achieved with few terms of this series. We derive order by order elegant systems of equations for both sets of the functions. Then we construct the global versions of the APT functions with quark thresholds in the $\bar{MS}$ scheme and give numerical results.

hep-ph↗

arXiv2001

QCD coupling up to third order in standard and analytic perturbation theories

We analyze two sets of specific functions, that/which form the basis of the nonpower asymptotic expansions both in the timelike and spacelike regions for single scale dependent QCD observables in the Shirkov--Solovtsov's Analytic Perturbation Theory (APT) free of unphysical singularities. These functions are explicitly derived up to the third order in the closed form in terms of the Lambert-W function. As an input we used the exact two loop and the three loop (corresponding to Padé transformed beta-function) RG solutions for common invariant coupling α_s. The elegant recurrence formulas, helpful for numerical analysis, are obtained for the both sets of the APT functions. Then we construct the global versions of APT functions using the continuity conditions (at the quark thresholds) on the α_s in the \bar{MS} scheme and give numerical results. For first three of these functions \mathfrak{A}_n(s) and {\cal A}_n(Q^2); n=1,2,3 in the large interval of the momentum transfer and energy (1 GeV <Q,{\sqrt s}< 170 GeV), numerical tables are presented. From these we observe that, for the timelike arguments, the differences between functions \mathfrak{A}_n(s) and the corresponding powers of the standard iteratively approximated coupling α_s^n(s) are not negligible even for moderate energies in the five--flavor region.

hep-ph↗