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1 Erratum to: Eur. Phys. J. C (2020) 80:1190 https://doi.org/10.1140/epjc/s10052-020-08748-4
First, the sign of the regulator in Eq. (6) is incorrect. It should read \(-i{\varepsilon }\), i.e.,
Next, an additional footnote has be added at the end of the first paragraph in Section III which reads “It is worth mentioning that comparison between the numerical results for \(\Omega \) and \(\Sigma \) in Eq. (8) using the approach presented in this work and Passarino–Veltman reduction techniques implemented in software packages such as, e.g., LoopTools [1] was carried out for different points of phase space to assess the performance of our method. It was found that our results were in agreement with those from the above package for points far from the edge of phase space, which provides a level of confidence in our approach, but in sharp disagreement for points near the edge of phase space. This is, however, a well-known drawback of the Passarino–Veltman reduction and variants due to the appearance of Gram determinants in the denominator, which spoils the numerical stability when they become small or even zero giving rise to spurious singularities (see, e.g., Refs. [2,3,4]). For processes with up to four external particles, this usually happens near the edge of phase space [2], which is consistent with our findings.”
The paragraph in page 5 starting with “Given the very wide decay width...” is replaced by “Given the very wide decay width of the \(\rho ^0\) resonance, which, in turn, is associated to its very short lifetime, the use of the usual Breit–Wigner approximation for the \(\rho ^0\) propagator is not justified. Instead, an energy-dependent width for the vector propagator ought to be considered, which may be written for a generic \({\hat{q}}^2\) as follows
where \(\theta (x)\) is the Heaviside step function. Strictly, one would now need to plug Eq. (14) into Eq. (2) and perform the loop integral, which represents a computation challenge in its own right and is outside of the scope of the present work.Footnote 1
With this in mind, and for the sake of simplicity, we resolve to stick with the Breit–Wigner approximation for the \(\rho ^0\) propagator despite being a potential source of error. The energy-dependent propagator is not needed, though, for the \(\omega \) and \(\phi \) resonances, as their associated decay widths are narrow and, therefore, use of the usual Breit–Wigner approximation suffices.”
Finally, in Eq. (15) there is a missing \(|e|\) factor. It should read
The above corrections lead to a new set of results and dilepton energy spectra which are summarised in Table 1 and Fig. 1 of this erratum, respectively.
Notes
One could write, for example, the \(\rho ^0\) energy-dependent propagator \(f(s)=\frac{m_{\rho }^2}{m_{\rho }^2-s-i m_{\rho } \varGamma _{\rho }(s)}\) as a once-subtracted dispersion relation, \(f(s)=f(s_0)+\frac{s-s_0}{\pi }\int _{s_{\text {th}}}^\infty \frac{\text {Im}f(s^{\prime })\ \text {d}s^{\prime }}{(s^{\prime }-s_0)(s^{\prime }-s-i\epsilon )}\), where \(s_{\text {th}}\) is the particle production threshold, in the case at hand \(s_{\text {th}}=4m_{\pi }^2\), and \(s_0\) is the subtraction point such that \(s_0<s_{\text {th}}\), e.g. \(s_0=0\). One would then perform the loop integral in the usual way, leaving the dispersion integral to the end of the computation.
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Acknowledgements
We would like to thank Marvin Zanke for pointing out two errors in the original paper.
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Escribano, R., Royo, E. Erratum to: A theoretical analysis of the semileptonic decays \(\eta ^{(\prime )}\rightarrow \pi ^0l^+l^-\) and \(\eta ^\prime \rightarrow \eta l^+l^-\). Eur. Phys. J. C 82, 743 (2022). https://doi.org/10.1140/epjc/s10052-022-10717-y
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DOI: https://doi.org/10.1140/epjc/s10052-022-10717-y