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

by Yang-yang Tan, Yong-rui Chen, Wei-jie Fu

Submission summary

As Contributors: Wei-jie Fu
Arxiv Link: (pdf)
Date submitted: 2021-07-23 04:40
Submitted by: Fu, Wei-jie
Submitted to: SciPost Physics
Academic field: Physics
  • High-Energy Physics - Phenomenology
Approach: Theoretical


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 $z\simeq 2.023$.

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Submission 2107.06482v1 on 23 July 2021

Reports on this Submission

Anonymous Report 1 on 2021-8-19 (Invited Report)


This work presents first results on the real-time dynamics of the O(4) model in 3+1 dimensions at finite temperature using the FRG approach formulated on the Schwinger-Keldysh contour. In particular, the authors compute the (degenerate) spectral functions in the restored phase, i.e., at temperatures beyond the critical temperature, and the dynamical critical exponent. This work represents an important further development of the real-time FRG approach on the Schwinger-Keldysh contour and a direct extension of recent work in 0+1 dimensions. However, the following remarks and questions need to be addressed before I can recommend publication in SciPost.

Requested changes

1) In Figs. 2, 3 and 4 the labels are very small and barely readable. The authors may want to increase the font size.

2) In Eq. (64) the self-energy is introduced. It may be useful to add an equation that expresses the 2-point function in terms of the self-energy.

3) In section V, the authors state that the flow of $m_\pi^2$ is extracted from (68). It may be useful to add an equation that makes this extraction explicit.

4) The critical temperature in the momentum-dependent calculation (T=20.4 MeV) is considerably smaller than in the momentum-independent case (T=145 MeV). Is there a deeper explanation for this? Is it possible that the “kink” in Fig. 6 is due to some numerical effect?

5) Fig. 5 shows that the squared pion mass changes its sign during the flow in the momentum-dependent case. Does this represent unphysical behavior? A more detailed discussion of this effect would be interesting.

6) The various negative values in the 4-point coupling, the 2-point function, and the spectral function are traced back to the term describing Landau damping. Clearly these negative values are a problem and need to be clearly understood. A more detailed discussion including potential ways to improve the situation would be interesting.

7) Fig. 7 shows values up to k=700 MeV. It may be interesting to add the initial value at the UV cutoff.

8) The left-hand side (lhs) plot in Fig. 12 is a log-log plot while the rhs one is a log-linear plot. It may be better to use the same axes for both plots for better comparability. Also, the rhs plot shows a "0" as well as negative values on the log-axis. This needs to be corrected or discussed since a "0" cannot appear here. Moreover, it would be nice to see the quasi-particle peak also in the rhs plot.

9) Concerning the critical exponent z, I recommend to also cite the recent work by Schweitzer et al. (arXiv: 2007.03374). Also a reference to Schlichting et al. (arXiv: 1908.00912) should be made. In the former work, the dynamic critical behavior of relativistic $Z_2$ theories is studied and classified in terms of the Hohenberg/Halperin scheme. To which of these `model’ classes does the O(4) model studied in the present work belong? Does the extracted value for the dynamical critical exponent agree with this expectation? A more detailed discussion on this point would be useful.

10) The regulator used in the present work, Eq. (20) has no entries in the cc as well as the qq component. Why is the qq-component not regularized? Is the cc-component always zero during the flow, as it should be?

11) Is the location of the threshold of the process phi -> 3 phi consistent with the location of the quasi-particle peak, i.e. is it located at three times the mass (see e.g. Fig. 12)?

  • validity: high
  • significance: top
  • originality: high
  • clarity: high
  • formatting: excellent
  • grammar: good

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