Borel Representation of $τ$ Hadronic Spectral Function Moments in Contour-Improved Perturbation Theory

Abstract

We show that the Borel representations of τ hadronic spectral function moments based on contour-improved perturbation theory (CIPT) in general differ from those obtained within fixed-order perturbation theory (FOPT). We demonstrate that the Borel sums obtained from both types of Borel representations in general differ as well and that the apparently conflicting asymptotic behavior of the FOPT and CIPT series, which has been subject to many studies in the past literature, can be understood quantitatively from these results. The discrepancy between the CIPT and FOPT Borel sums, which we call the 'asymptotic separation', can be computed analytically and is related to inverse exponential terms in the strong coupling. The asymptotic separation arises from the singular and non-analytic infrared renormalon structures in the Borel function of the underlying Adler function where the leading dimension four gluon condensate renormalon has the highest weight. The size of the asymptotic difference is in general larger than that of the FOPT Borel sum ambiguity, but it can be modulated in a predictable way by choosing specific spectral function moments. Even though moments can be designed where the asymptotic difference is smaller than the FOPT Borel sum ambiguity, the asymptotic separation can as a matter of principle not be avoided entirely. The asymptotic separation has important implications for the standard operator product expansion approach used for spectral function moment predictions.

Using improved Operator Product Expansion in Borel-Laplace Sum Rules with ALEPH $τ$ decay data, and determination of pQCD coupling

“…These arguments, presented in a very general way in[1] (especially Appendix A there), are also compatible with somewhat related arguments presented in[26][27][28]. A construction of Borel transforms of CI sum rules was proposed and investigated in[52] (cf. also[53]).…”

Determination of perturbative QCD coupling from ALEPH $τ$ decay data using pinched Borel-Laplace and Finite Energy Sum Rules

“…In the Table the results are presented to three Somewhat related arguments for the preference of the FOPT methods over the CIPT methods, in FESRs of semihadronic τ -decays, were presented in Refs. [23][24][25]75]. We recall that the truncation index Nt was determined in each method in such a way that a relative stability of the full moments a (2,0) (σm) and a (2,1) (σm) is achieved under the variation of Nt.…”

Semileptonic tau decays beyond the Standard Model

“…On the theoretical side, one of the main limitations that may be overcome in the future are the uncertainties coming from higher-order and non-perturbative corrections [128,[144][145][146][147]. This would decrease some of the dominating SM uncertainties in our bounds, and it would allow us to use additional moments.…”

“…Some recent works aimed to understand higher-order corrections, which eventually may lead to a significant reduction on the perturbative uncertainty, can be found in the literature[128,[144][145][146][147].…”