Erratum to: Eur. Phys. J. C (2023) 83:111 https://doi.org/10.1140/epjc/s10052-023-11244-0

The present Erratum corrects two calculation errors for the helicity relaxation time \(\tau _H\), presented in Appendix B. Correcting these errors leads to a larger value of \(\tau _H\), namely \(4.80\ {\textrm{fm}}/c\) instead of \(2.54\ {\textrm{fm}}/c\), in the context of Eq. (110). Specifically:

1. The integration with respect to the momentum magnitude \(p'\) implied by \(dP'\) in Eq. (160) yields a factor of 1/2 due to the structure of the energy-conservation delta function in Eq. (161), \(\delta (E_{cm} - p'{}^0 - k'{}^0) \rightarrow \delta (E_{cm} - 2p'{}^0)\) – this factor was missed. Thus, Eqs. (167) and (170) must be divided by 2, while (174) must be multiplied by 2, as follows:

$$\begin{aligned} I_\Omega&= \frac{1}{2(2\pi )^5} \int _{-1}^1 dx\, \tilde{f}_{0{\textbf{p}}'} \tilde{f}_{0{\textbf{k}}'} (\tilde{f}_{0{\textbf{p}}} + f_{0{\textbf{p}}'}) \nonumber \\&\quad \times \int _0^{2\pi } \frac{d\varphi _{p'}}{2\pi } (1 - \cos \theta _{cm})^2\,; \end{aligned}$$
(167)
$$\begin{aligned} \tau _H^{-1}&= \frac{g \alpha _{\textrm{QCD}}^2 \beta ^3}{3 \pi ^3} \int _{-1}^1 d\cos \gamma \int _0^\infty dp\, p \nonumber \\&\quad \times \int _0^\infty dk\, k \int _{-1}^1 dx\, f_{0{\textbf{p}}} f_{0{\textbf{k}}} \tilde{f}_{0{\textbf{p}}'} \tilde{f}_{0{\textbf{k}}'} (\tilde{f}_{0{\textbf{p}}} + f_{0{\textbf{p}}'}) \nonumber \\&\quad \times \left[ \left( 1- \frac{p_{||}}{p_{cm}}x \right) ^2 + \frac{1 - x^2}{2} \frac{p_\perp ^2}{p_{cm}^2}\right] ; \end{aligned}$$
(170)
$$\begin{aligned} \tau _H&= \frac{6\pi ^3 \beta }{g\alpha _{\textrm{QCD}}^2 \mathcal {I}}\,. \end{aligned}$$
(174)

2. On the right hand side of Eq. (168), the factor \(k^0\) should read \(k^0 / 2\), i.e.

$$\begin{aligned} p_{||} = \frac{{\textbf{p}}_{cm} \cdot ({\textbf{p}}+ {\textbf{k}})}{|{\textbf{p}}+ {\textbf{k}}|} = \frac{E_{cm}(p^0 - k^0)}{2|{\textbf{p}}+ {\textbf{k}}|}\,. \end{aligned}$$
(168)

This error propagates to Eqs. (175) and (176), which are modified to:

$$\begin{aligned} \mathcal {I}&= \int _0^\infty dz\, z^3 \int _{-1}^1 dx \int _{-1}^1 d\delta \int _{|\delta |}^1 d\xi \nonumber \\&\quad \times \left( \frac{3 - x^2}{2} \xi - 2x\delta + \frac{3x^2 - 1}{2\xi } \delta ^2\right) \nonumber \\&\quad \times [e^{\frac{z}{2}(1+\delta )}+1]^{-1} [e^{\frac{z}{2}(1-\delta )}+1]^{-1} \nonumber \\&\quad \times [e^{-\frac{z}{2}(1+x \xi )}+1]^{-1} [e^{-\frac{z}{2}(1 - x\xi )}+1]^{-1} \nonumber \\&\quad \times \{ [e^{-\frac{z}{2}(1+\delta )}+1]^{-1} + [e^{\frac{z}{2}(1+x\xi )}+1]^{-1}\}\,; \end{aligned}$$
(175)
$$\begin{aligned} \mathcal {I}&\simeq 5.09434\,. \end{aligned}$$
(176)

The corrections described above lead to a value for \(\mathcal {I}\) larger by about \(6\%\) compared to the one reported in the original paper (\(\mathcal {I}_{\textrm{old}} = 4.81255\)).

3. The errors corrected by points 1 and 2 above propagate to the main text, changing Eqs. (109) and (110) as follows:

$$\begin{aligned} \tau _H&= 0.392 \times \frac{\pi ^3 \beta }{N_f \alpha _{\textrm{QCD}}^2} \nonumber \\&\simeq \left( \frac{250\ {\textrm{MeV}}}{k_B T}\right) \left( \frac{1}{\alpha _{\textrm{QCD}}}\right) ^2 \left( \frac{2}{N_f}\right) \nonumber \\&\quad \times 4.80 \ {\textrm{fm}}/c\,; \end{aligned}$$
(109)
$$\begin{aligned} \tau _H&\simeq 4.80\ {\textrm{fm}}/c\,. \end{aligned}$$
(110)

The corrections summarized above lead only to quantitative modifications of our results. The paper can still be read in its original form. The text and all our conclusions retain full validity, even more so as a larger value for the helicity relaxation time strengthens the scope of our analysis of quantum fluids at finite helical chemical potential since the longer the helicity relaxation time, the stronger the effects of the helical imbalance.