Erratum to: The branching ratio $\omega \rightarrow \pi ^+\pi ^-$ revisited

1 Erratum to: Eur. Phys. J. C (2017) 77:98 https://doi.org/10.1140/epjc/s10052-017-4651-x

In the published paper [1], a data set for the pion vector form factor from the BaBar experiment [2] was used that was not consistent with the ones employed for the other experiments, since it did not include the contributions from the vacuum polarization, while it contained those of the final-state radiation. The use of the official form factor data changes some of the results, as we report in this erratum.

Fig. 1

Comparison of the values for the branching ratio for $\omega \rightarrow \pi ^+\pi ^-$ extracted from the various fits to the different data sets, where circles refer to Fit 1, squares to Fit 2, diamonds to Fit I, the triangles-up to Fit II, triangles-left to Fit 1-$\rho$, (red) triangles-down to Fit 2-$\rho$, triangles-right to Fit 1-$\phi$, and crosses to Fit 2-$\phi$. The red thick solid line denotes the average of the values, the gray band the corresponding uncertainty found from our preferred analysis – Fit 2-$\rho$ – including all data sets. Note that the values extracted from $\eta '\rightarrow \pi ^+\pi ^-\gamma$ refer to pseudo-data generated according to preliminary results

In Table 1 we report the results of the various fits to the corrected data set as well as the quantities derived from the fit parameters like the $\omega \rightarrow \pi \pi$ branching fraction and the square of the pion radius. The results for the branching fractions extracted from the different fits are compared to those extracted from the other data sets in Fig. 1.

It is obvious from Fig. 1 that with the corrected data set for the BaBar experiment, the fluctuations of the results for the $\omega \rightarrow \pi \pi$ branching ratio derived from the different fit strategies are reduced significantly to a level compatible with those of the other experiments. Furthermore, the branching ratio extracted is now much more in line with the other experiments than before. Note that the difference between the branching ratio $(1.74\pm 0.10)\%$, quoted in Table 1, and the branching ratio $(1.44\pm 0.11)\%$ given in Ref. [3] can be partially traced back to our using the $\omega$ width as an input (fixed from other experiments, in particular $e^+e^-\rightarrow 3\pi$), while it was varied in the analyses of Refs. [2, 3].

Throughout this erratum we stick to the strategy already chosen in Ref. [1], namely that the final results are derived from weighted averages of the findings of Fit 2-$\rho$.

We then find for the $\omega \rightarrow \pi \pi$ branching ratio with the corrected BaBar data included

$$\begin{aligned} \mathscr {B}(\omega \rightarrow \pi ^+\pi ^-) = (1.52\pm 0.08) \times 10^{-2} , \end{aligned}$$

(1)

where, according to the prescription of the Particle Data Group (cf. the introduction of Ref. [4]), a scaling factor of 1.8 had to be applied to the uncertainty.

Table 1 Fit results for the pion vector form factor using the corrected BaBar data [2, 5]

This is to be compared to the value extracted with the BaBar data omitted

$$\begin{aligned} \mathscr {B}(\omega \rightarrow \pi ^+\pi ^-) = (1.46\pm 0.08) \times 10^{-2} , \end{aligned}$$

(2)

where a scale factor of 1.5 was applied. The latter value was already reported in Ref. [1]. Clearly, once the correct BaBar data set is employed in the analysis, the $\omega \rightarrow \pi \pi$ branching ratio comes out statistically consistent with the other values. Moreover, the mass parameters of both the $\omega$ and the $\rho$ are found to be reasonably consistent with those of the other extractions, and the fit does not call for a complex phase of the coupling $g_{\omega \pi \pi }$, in line with unitarity. From this point of view there is no reason anymore to prefer the value given in Eq. (2) to the one of Eq. (1).

The situation is somewhat different for the pion charge radius. The central value for the radius derived from the BaBar data now even more strongly deviates from the values derived from the other experiments: our preferred fit (Fit 2-$\rho$) results in (cf. Table 1)

$$\begin{aligned} \langle r_V^2\rangle = (0.4416 \pm 0.0004 \pm 0.0005) \,\text {fm}^2 , \end{aligned}$$

(3)

where the first error denotes the statistical uncertainty given by the fit and the second one the uncertainty by possible residual radiative corrections estimated in Ref. [1]. This is to be compared to the value extracted from an average over the other experiments with the BaBar data omitted [1]

$$\begin{aligned} \langle r_V^2\rangle = (0.4361\pm 0.0007\pm 0.0005) \,\text {fm}^2 , \end{aligned}$$

(4)

where a scale factor of 1.5 was included in the uncertainty as detailed in Ref. [1].¹ We emphasize that the first uncertainties quoted in Eqs. (3) and (4) are based solely on the statistical errors derived from the fits performed with a given fixed set of parameters for the $\pi \pi$ phase shifts [7]. In particular the ranges for those parameters also quoted in Ref. [7] were not considered. Preliminary studies trying to include these uncertainties in the analysis using Bayesian statistics indicate that the full uncertainty might be dominated by the systematics resulting from this procedure. We therefore neither perform a combined fit for the radii nor quote a final result at this stage. The aforementioned more advanced studies indicate that the uncertainty for the $\omega \rightarrow \pi \pi$ branching fraction changes only marginally.

Note that the central value for the square of the pion radius derived from the BaBar data using the parametrization of Ref. [5] reads

$$\begin{aligned} \langle r_V^2\rangle = (0.433 + i \ 0.004) \,\text {fm}^2 . \end{aligned}$$

(5)

Ref. [8] quotes the value $(0.4319\pm 0.0016)\,\text {fm}^2$ for the real part of $\langle r_V^2\rangle$. The difference in the central value may be understood from different rounding procedures. In this reference no number is given for the imaginary part, but we expect its uncertainty to be of comparable size. Thus the real part of the pion radius derived from the parameterization employed in Ref. [5] is even lower than the number reported in Eq. (4), however, it comes with a nonvanishing imaginary part in conflict with general principles, which demonstrates that this parametrization is inapt to derive a reliable value of the charge radius – while a single Gounaris–Sakurai term is consistent with unitarity and analyticity, a sum of those terms with complex coefficients as used in Ref. [5] violates both.