Abstract
In this addendum to Ref. [1] we show that the mismatch between the \(\rho \)–\(\omega \) mixing parameter \(\epsilon _{\rho \omega }\) as extracted from \(\eta '\rightarrow \pi ^+\pi ^-\gamma \) and \(e^+e^-\rightarrow \pi ^+\pi ^-\) can be resolved by including higher orders in the expansion in \(e^2\) in the description of the \(\eta '\rightarrow \pi ^+\pi ^-\gamma \) decay. We repeat the analysis in this extended framework and update the numerical results accordingly.
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Addendum to: Eur. Phys. J. C https://doi.org/10.1140/epjc/s10052-022-10247-7
1 Extended formalism
Following the notation from Ref. [1] throughout, the spectrum for \(P\rightarrow \pi ^+\pi ^-\gamma \) can be expressed as
generalizing Eq. (D.14) in Ref. [1] by the next order in the expansion in \(e^2\) (the sign convention is such that \(g_{P\omega \gamma }<0\)). The most important change, numerically, concerns \(\epsilon _{\rho \omega }\rightarrow \epsilon _{\rho \omega }-e^2g_{\omega \gamma }^2\) in the numerator of the \(\omega \) propagator, corresponding to the photon contribution in \(\epsilon _{\rho \omega }\) as defined in resonance chiral perturbation theory [2,3,4]. In our formalism, \(\epsilon _{\rho \omega }\), determined from a fit to the bare cross section for \(e^+e^-\rightarrow \pi ^+\pi ^-\), does not include this VP effect, in line with the definition in Ref. [5] (numerically, it evaluates to \(e^2g_{\omega \gamma }^2=0.34(1)\times 10^{-3}\)). This shift removes the tension observed between \(\eta '\rightarrow \pi ^+\pi ^-\gamma \) and \(e^+e^-\rightarrow \pi ^+\pi ^-\) in Ref. [1].
The coefficients appearing in Eq. (3.9) of Ref. [1] are generalized according to Eq. (1.1):
In the following, we provide the updated numerical results when including the additional \(e^2\) effects as given in Eq. (1.1), implemented in the fit via Eq. (1.2).
2 Numerical results
The updated fit parameters are collected in Table 1, Fig. 1, and Table 2. The main difference to the results presented in Ref. [1] is that the shift \(\epsilon _{\rho \omega }\rightarrow \epsilon _{\rho \omega }-e^2g_{\omega \gamma }^2\) removes the tension between \(e^+e^-\rightarrow \pi ^+\pi ^-\) and the \(\eta '\rightarrow \pi ^+\pi ^-\gamma \) spectrum, markedly improving the quality of the combined fit.
The updated results for the TFF are shown in Fig. 2 and Table 3. In particular, the prediction for the slope parameter
is reduced by about \(1\sigma \), which traces back not to the change in \(\epsilon _{\rho \omega }\) (which is marginal given the fact that the fit is dominated by \(e^+e^-\rightarrow \pi ^+\pi ^-\)), but to a stronger curvature in the polynomial P(s) (the coefficient \(\gamma \) of the quadratic term increases by a factor 3).
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Acknowledgements
We thank Pablo Sánchez-Puertas for pointing out the issue of one-photon-reducible contributions to \(\epsilon _{\rho \omega }\), which ultimately explains the tension observed in Ref. [1]. Financial support by the SNSF (Project Nos. 200020_200553 and PCEFP2_181117), the DFG through the funds provided to the Sino-German Collaborative Research Center TRR110 “Symmetries and the Emergence of Structure in QCD” (DFG Project-ID 196253076 – TRR 110), and the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 824093 is gratefully acknowledged.
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Holz, S., Hanhart, C., Hoferichter, M. et al. Addendum: A dispersive analysis of \(\varvec{\eta '\rightarrow \pi ^+\pi ^-\gamma }\) and \(\varvec{\eta '\rightarrow \ell ^+\ell ^-\gamma }\). Eur. Phys. J. C 82, 1159 (2022). https://doi.org/10.1140/epjc/s10052-022-11094-2
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DOI: https://doi.org/10.1140/epjc/s10052-022-11094-2