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

### Submission summary

 As Contributors: Wei-jie Fu Arxiv Link: https://arxiv.org/abs/2107.06482v3 (pdf) Date submitted: 2021-10-20 03:59 Submitted by: Fu, Wei-jie Submitted to: SciPost Physics Academic field: Physics Specialties: High-Energy Physics - Phenomenology Approach: Theoretical

### Abstract

In this paper, the real-time dynamics of the $O(4)$ scalar theory is studied within the functional renormalization group formulated on the Schwinger-Keldysh closed time path. The flow equations for the effective action and its $n$-point correlation functions are derived in terms of the "classical" and "quantum" fields, and a concise diagrammatic representation is presented. An analytic expression for the flow of the four-point vertex is obtained. Spectral functions with different values of temperature and momentum are obtained. Moreover, we calculate the dynamical critical exponent for the phase transition near the critical temperature in the $O(4)$ scalar theory in $3+1$ dimensions, and the value is found to be $z\simeq 2.023$.

###### Current status:
Editor-in-charge assigned

We thank the referee for his suggestions in the second report, which we have taken into account in the revised version.

### List of changes

1) Minor modification for the text on page 14 is made.

2) We have modified the discussion in the paragraph below Eq.(21), and a new reference (arXiv: 1611.07301) is added.

### Submission & Refereeing History

Resubmission 2107.06482v3 on 20 October 2021
Resubmission 2107.06482v2 on 5 October 2021
Submission 2107.06482v1 on 23 July 2021

## Reports on this Submission

### Report

I thank the authors for considering my suggestions.

I am not convinced that the formulation concerning "thermal equilibrium" after Eq. (21) is completely correct, but this issue will probably have to be resolved by additional studies in the future. However, it is now clearly formulated what was done in the manuscript.

I therefore recommend publication without further changes.

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -

### Author:  Wei-jie Fu  on 2021-10-21  [id 1871]

(in reply to Report 1 on 2021-10-21)

We thank the referee for the recommendation of publication.

We think that the referee still has the question whether it is adequate to regulate only the real part of the 2-point functions in thermal equilibrium. Although this is not quite relevant to the main subject in this manuscript, we would like to address it more clearly.

We think that the answer is "yes". The reason is due to the fact that in thermal equilibrium, the real and imaginary parts are not independent, while they are related to each other through a principal value integral, as shown in e.g., Eq.(B7) and Eq.(B8) in Appendix B. This is also reflected by the fluctuation-dissipation relation.

We hope that this issue is addressed clearly now. Thanks.