Renormalons in integrated spectral function moments and αs extractions

Abstract

Precise extractions of α s from τ → (hadrons) + ν τ and from e + e − → (hadrons) below the charm threshold rely on finite energy sum rules (FESR) where the experimental side is given by integrated spectral function moments. Here we study the renormalons that appear in the Borel transform of polynomial moments in the large-β 0 limit and in full QCD. In large-β 0 , we establish a direct connection between the renormalons and the perturbative behaviour of moments often employed in the literature. The leading IR singularity is particularly prominent and is behind the fate of moments whose perturbative series are unstable, while those with good perturbative behaviour benefit from partial cancellations of renormalon singularities. The conclusions can be extended to QCD through a convenient scheme transformation to the C-scheme together with the use of a modified Borel transform which make the results particularly simple; the leading IR singularity becomes a simple pole, as in large-β 0 . Finally, for the moments that display good perturbative behaviour, we discuss an optimized truncation based on renormalisation scheme (or scale) variation. Our results allow for a deeper understanding of the perturbative behaviour of integrated spectral function moments and provide theoretical support for low-Q 2 α s determinations.

Conformal mapping of the Borel plane: Going beyond perturbative QCD

“…(3) iteratively along the circle, starting from a given aðs 0 Þ. The comparison between the standard FO and CI perturbative QCD expansions of the moments has been investigated in [39,52], where substantial differences between the two ways of renormalization-group summation have been noticed. Here we are interested in the method of conformal mapping of the Borel plane, which can be applied to improve the expansions of the quantities δ ð0Þ w i much like that of the Adler function itself.…”

“…For the FO version, one must follow the steps applied in Sec. II to the Adler function, using now as starting point the expansion (39). We define first the Borel transform…”

“…where we adopted the principal value anticipating the presence of singularities of the Borel transform B δ w i ðuÞ on the integration axis. The analytic properties of the Borel transform B δ w i ðuÞ defined in (41) in the complex u plane have been investigated some time ago in [53] and more recently in [33,39,40]. Inserting the Laplace-Borel representation (10) into the integral (37) and permuting the integrals we obtain…”

“…The integral can be calculated exactly for polynomial weights. In particular, for w i ðxÞ ¼ x n , one has [39]…”

“…In Sec. V we discuss the perturbative expansions of the moments of the spectral function, using recent results on their singularities in the Borel plane [33,39]. In Sec.…”