We use a new compilation of the hadronic $R$-ratio from available data for the process $e^+e^-\to$ hadrons below the charm mass to determine the strong coupling $\alpha_s$, using finite-energy sum rules. Quoting our results at the $\tau$ mass to facilitate comparison to the results obtained from similar analyses of hadronic $\tau$-decay data, we find $\alpha_s(m_\tau^2)=0.298\pm 0.016\pm 0.006$ in fixed-order perturbation theory, and $\alpha_s(m_\tau^2)=0.304\pm 0.018\pm 0.006$ in contour-improved perturbation theory, where the first error is statistical, and the second error combines various systematic effects. These values are in good agreement with a recent determination from the OPAL and ALEPH data for hadronic $\tau$ decays. We briefly compare the $R(s)$-based analysis with the $\tau$-based analysis.
Here are our answers to the referee's comments:
1. Add captions to Tables and enumerate them.
Response: We did this.
2. In section 2 the authors infer that more data in the region s<4 GeV2 should imply a more precise determination of αs as compared with the higher s region. However, I do not see why this should be true in general, since theoretical uncertainties play a crucial role in the extraction of αs.
Response: This is discussed in much more detail in Ref. , to which we refer through footnote 1.
3. Explain briefly how the errors quoted in the last column of would-be Table 1 are obtained.
Response: This is very simple: by propagation of the data errors. We added a note stating this.
4. The error on αs displayed in Figure 4 is not the total error. This is stated by the authors in footnote 5, but it should also be stated in the caption of Figure 4. The total error should possibly be indicated in the Figure, or reported in the text.
Response: We added a sentence to the caption indicating that the individual error bars on the points reflect the fit errors. The purpose of this plot is explained in detail in the text, and it's goal is not to show the final values of the error bars on αs, which can be found in Eq. (6).
5. Data for w4 should also be included in Figure 4, according to the content of would-be Table 2.
Response: We did not do this to avoid clutter in the plot. We added a sentence to this extent in the caption of this figure.
6. What is the effect of neglected higher dimensional condensates on the data in Figure 4, and how do they compare with the duality violations (black circles) especially in the low smin0 region?
Response: No higher dimensional condensates have been neglected, as explained in the text. The condensates we included follow directly from Cauchy's theorem.
7. The authors should address more clearly the stability of their fitted results reported in Figure 4 when varying smin0 and smax0. An analogous observation applies to section 5, see point 9.
Response: This is discussed in detail in the paper this writeup summarizes, see Ref. , to which we refer extensively for all details.
8. What is the channel displayed in Figure 5 left panel, V, A, V+A? And what are the uncertainties in the fitted curves?
Response: We chose the V channel, as this makes most sense in comparison with the e+e-based plots. We added a clarification to the caption.
9. Section 5 concludes that the determination of αs from e+e− data and the one from τ data are consistent. However, there is hardly enough information that can be extracted from Figure 5 and the surrounding text. Importantly, what happens to the final value of αs and to the fitted parameters of the duality-violation model when one varies smin0 in the τ-data fits?
This is a relevant point. I think that an accurate study of this dependence is needed in order to assess the stability and consistency of the results and to draw conclusions. If a complete analysis cannot be worked out in a reasonably short time, the authors can at least acknowledge this point in their contribution and formulate their final remarks accordingly.
Response: This writeup is not concerned with the τ-based analysis, for which we refer to Refs. [11,18], and references therein.