Analytic continuation and perturbative expansions in QCD

Abstract

Starting from the divergence pattern of perturbative quantum chromodynamics, we propose a novel, non-power series replacing the standard expansion in powers of the renormalized coupling constant $a$. The coefficients of the new expansion are calculable at each finite order from the Feynman diagrams, while the expansion functions, denoted as $W_n(a)$, are defined by analytic continuation in the Borel complex plane. The infrared ambiguity of perturbation theory is manifest in the prescription dependence of the $W_n(a)$. We prove that the functions $W_n(a)$ have branch point and essential singularities at the origin $a=0$ of the complex $a$-plane and their perturbative expansions in powers of $a$ are divergent, while the expansion of the correlators in terms of the $W_n(a)$ set is convergent under quite loose conditionsComment: 18 pages, latex, 5 figures in EPS forma

Determination of the Strong Coupling From Hadronic Tau Decays Using Renormalization Group Summed Perturbation Theory

“…In this model, FOPT expansion approaches better the 'true value'. A method for taming the divergent behaviour of the QCD perturbative expansions was proposed in [27,28,29,30,31,32], using the series acceleration by the conformal mappings of the Borel plane and the implementation of the known nature of the leading singularities in this plane.…”

“…Consequently one can study the Borel transform of the QCD Adler function which has ultraviolet (UV) and infrared (IR) renormalon singularities in the Borel plane. The divergent behaviour can be considerably tamed by using techniques of series acceleration based on conformal mappings and 'singularity softening' [27,28,29,30,31,32]. The method is not applicable to the perturbative series in powers of α s since the expanded correlators are singular at α s = 0, but can be applied in the Borel plane.…”

Expansions ofτhadronic spectral function moments in a nonpower QCD perturbation theory with tamed large order behavior

“…In the present work, we investigate the moments considered in [29] also in the frame of RGSPT. More significantly, in the present paper we investigate the moments also in the frame of a novel formulation of QCD perturbation theory, defined some time ago in [31][32][33] starting from the divergent character of the standard series. The method uses the idea of series acceleration by means of a conformal mapping [34], applied to the Borel plane of QCD correlators.…”

Strong Coupling From the Tau Hadronic Width by Non-Power QCD Perturbation Theory

“…More detailed arguments are given in two lemmas formulated and proved in [8,10]. For the Adler function in QCD, the optimal mapping w(u) and the corresponding perturbative expansion were proposed and investigated in [23,24].…”

“…where the expansion functions W (14)- (17) have been discussed in [23,24,10]. A remarkable feature is that the expansion functions resemble the expanded function, being singular at a s = 0 and having divergent series in powers of a s .…”