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.
In this article we review the results of our recent work on the difference between the Borel representations of $$\tau $$ τ hadronic spectral function moments obtained with the CIPT and FOPT methods. For the presentation of the theoretical results we focus on the large-$$\beta _0$$ β 0 approximation, where all expressions can be written down in closed form, and we comment on the generalization to full QCD. The results may explain the discrepancy in the behavior of the FOPT and CIPT series that has been the topic of intense discussions in previous literature and which represents a major part of the theoretical uncertainties in current strong coupling determinations from hadronic $$\tau $$ τ decays. The findings also imply that the OPE corrections for FOPT and CIPT differ and that the OPE corrections for CIPT do not have standard form.
In this article we review the results of our recent work on the difference between the Borel representations of τ hadronic spectral function moments obtained with the CIPT and FOPT methods. For the presentation of the theoretical results we focus on the large-β0 approximation, where all expressions can be written down in closed form, and we comment on the generalization to full QCD. The results may explain the discrepancy in the behavior of the FOPT and CIPT series that has been the topic of intense discussions in previous literature and which represents a major part of the theoretical uncertainties in current strong coupling determinations from hadronic τ decays. The findings also imply that the OPE corrections for FOPT and CIPT differ and that the OPE corrections for CIPT do not have standard form.
In a recent work by some of us it was shown that the long-standing discrepancy between the QCD perturbation series for the inclusive hadronic tau decay rate computed in the CIPT and FOPT expansion approaches can be understood from the fact that CIPT has an infrared (IR) sensitivity that it not compatible with the standard form of the operator production expansion (OPE). For concrete IR renormalon models for the QCD Adler function the resulting CIPT-FOPT discrepancy, the asymptotic separation, can be calculated analytically from the Borel representation of the CIPT series expansion. If the known perturbative corrections for the QCD Adler function at the 5-loop level already have a sizeable contribution from the asymptotic behavior related to the gluon condensate (GC) renormalon, the asymptotic separation is dominated by that renormalon. This implies that the CIPT expansion can be reconciled with FOPT, when a renormalon-free scheme for the GC is adopted. In this talk we discuss such a renormalon-free scheme for the GC, which involves perturbative subtractions in analogy to using short-distance quark mass schemes instead of the pole mass. Using a concrete realistic high-order Borel model for the Adler function consistent with the known corrections up to 5 loops and containing a sizeable GC renormalon contribution, we show that the CIPT-FOPT discrepancy can be avoided when switching to the renormalonfree GC scheme. At the same time, the perturbative convergence of τ hadronic spectral funtion moments strongly sensitive to the GC OPE corrections is considerably improved. We show that these improvements may lead to higher precision for strong coupling determinations.
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