Real-time dynamics of the $O(4)$ scalar theory within the fRG approach

We are grateful to the referee for his comments and suggestions, which we have taken into account in the revised version. In the following, we will respond to the remarks and questions raised by the referee one by one.

1) Thanks for the suggestion. We have enlarged the labels in the relevant figures in the revised version.

2) In the revised version, equation (68) was modified a bit, such that the flow of 2-point function is related to the self-energy.

3) A new equation (74) has been added to make this extraction explicit in the revised version.

4) In order to address this issue more explicitly, we have added a new paragraph at the end of page 8 in the revised version: '' It is interesting to explore underlying reasons accounting for the difference between the momentum independent and dependent results. When the momentum dependence is included, the flow of the effective four-point coupling in Equation (62) is suppressed at finite external momenta. Consequently, the coupling in the flow of the inverse retarded propagator in Equation (70), which contributes mostly around $ |\vec{q}|\sim k$ due to the 3-$d$ momentum integral, is relatively larger than that for the case without momentum dependence. The larger coupling leads to an increased flow of the two-point function as well as a larger meson mass square $m_{\pi,k}^2$ in the infrared, as shown by the blue solid line in the left panel of Figure 5. Hence, lower temperature is required to decrease $m_{\pi,k}^2$ at $k \rightarrow 0$ in order to realize the phase transition. Furthermore, it is found that the kink-like structure of the blue line around about $T=100$ MeV in Figure 6 arises from the fact that when the temperature is below $\sim 100$ MeV, the meson mass square in the region of low $k$ behaves as $m_{\pi,k}^2\sim -k^2$, which results in a small energy factor $E_{\pi,k}(k)$ in Equation (70), cf. also Equation (72), and eventually increases the flow of the inverse retarded propagator even further. That is the reason why the critical temperature in the case with momentum dependence is significantly lower than that without momentum dependence.''

5) No, it doesn't represent unphysical behavior. In the revised version, we have added a discussion in the right column on page 8, trying to make it more clearly: '' Note that the effective potential is broken in the ultraviolet, and the curvature of potential at $\phi_c=0$, i.e., the squared meson mass, cf. Equation (9), is negative when the temperature is below $T_c$. When the temperature is increased above $T_c$, the squared meson mass evolves from the negative to a positive value with the decrease of the RG scale $k$. Therefore, the critical temperature just corresponds to the case that the meson mass square is vanishing at $k \rightarrow 0$.''

6) In the revised version, we have added a paragraph below Fig. 13 to discuss this issue: ''As we have demonstrated above, when the temperature is above and not far away from the critical temperature, the spectral function is positive. However, it is found that when the temperature is quite larger than the critical one, Landau damping contributes a negative value to the spectral function in the regime of small $p_0$. Whether it is an artifact in our computation certainly needs more sophisticated investigations. For instance, the truncation for the flow equation of the four-point vertex as shown in Figure 4 might have to be improved in the region of high temperature, where the momentum dependence should also be encoded for the four-point vertices on the r.h.s. of the flow equation in Figure 4. We hope to report the relevant studies in future work.''

7) The initial value for the imaginary part of the averaged effective vertex in Eq.(75) is vanishing at the UV cutoff. We have added a note below Eq.(75) to address it.

8) We have updated Fig. 12 and used the same axes for both plots. The quasi-particle peak is now more visible in the right plot. Furthermore, in the right panel a symmetric log scale is applied for the $y$-axis in order to take into account both positive and negative values of the spectral function, where the log scale is implicitly translated into a linear one upon crossing the zero point. A sentence has been added in the caption of Fig. 12 to address this issue.

9) Thanks for the suggestions. In the revised version, we have added three references (two mentioned above and also arXiv:hep-ph/9210253). A new paragraph below Eq. (86) was added to discuss this issue: ''Interestingly, this value of the dynamical critical exponent obtained in this work is compatible with a very recent result for Model A in three spatial dimensions, $z=1.92(11)$, obtained from real-time classical-statistical lattice simulations [75]. Here we have used the standard classification for the universality of critical dynamics [14]. However, the critical dynamics of the relativistic $O(4)$ scalar theory should be more closely related to Model G, based on the analysis by Rajagopal and Wilczek [76], see also [77]. The dynamical critical exponent of Model G is $z=3/2$ in three dimensions. Though direct calculation of the dynamical critical exponent for the $O(4)$ model from classical-statistical lattice simulations has not yet arrived at a conclusive result because of errors, it indicates that $z$ is in favor of 2 for small lattice size and $3/2$ for large one [77]. Furthermore, it is also found that the dynamical critical exponent in a relativistic $O(N)$ vector model is close to 2 [63]. In summary, whether the critical dynamics of the relativistic $O(4)$ scalar theory falls into Model A or Model G is still an open question, and more insightful studies are required. ''

10) Vanishing of the cc-component of regulator is due to the fact that there is no such term as $\sim \phi_c^2$ in the effective action in Eq. (1), because the effective action in the Keldysh formalism can be expressed as the difference between those on the forward and backward branches, as shown in Eq. (A1). For the qq-component, regularization is not necessary. The reasons are listed as follows. On the one hand, the Keldysh propagator is related to the retarded and advanced propagators in thermal equilibrium by the fluctuation-dissipation relation as shown in Eq. (36). On the other hand, the dynamics of a system is governed by the retarded and advanced propagators on the level of two-point correlation functions, while the Keldysh propagator represents its statistical properties [18]. It is, therefore, adequate to suppress dynamic quantum fluctuations of different RG scales by encoding regulators only in the retarded and advanced propagators. In the revised version, we have added a paragraph below Eq. (21) to discuss it.

11) Thanks for the question. Following this question, we have inspected the process $\phi \rightarrow 3\phi$ in the spectral function in Fig. 12. This process can be traced back to the imaginary part of the internal momentum averaged effective vertex in Fig. 8, where in the left panel one can see that the sudden rise of the threshold function $I_{1,k}(p-q)$ just corresponds to $p_0=3E_{\pi,k}$. However, its contribution to the spectral function is almost hidden by processes related to, e.g., $I_{1,k}(p+q)$ and $I_{2,k}(p-q)$ as shown in Fig. 9, and hence the process $\phi \rightarrow 3\phi$ is hard to be observed from the spectral function in Fig. 12. In the revised version, we have added several sentences at the end of page 12 to discuss it: ''Furthermore, we have inspected the process $\phi \rightarrow 3\phi$ in the spectral function...''