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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
This is not the latest submitted version.
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Submission summary
| Authors (as registered SciPost users): | Wei-jie Fu |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2107.06482v2 (pdf) |
| Date submitted: | 2021-10-05 05:16 |
| Submitted by: | Fu, Wei-jie |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| 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 $z\simeq 2.023$.
Author comments upon resubmission
List of changes
1) We have enlarged the labels in Figs. 2, 3, 4.
2) 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.
4) 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 ...''
5) We have added a discussion in the right column on page 8: '' Note that the effective potential is broken in the ultraviolet, and the curvature of potential...''
6) We have added a paragraph below Fig. 13: ''As we have demonstrated above, when the temperature is above and not far away from the critical temperature...''
7) We have added a note below Eq.(75).
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) 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... ''
10) We have added a paragraph below Eq. (21) to discuss the regulators.
11) We have added several sentences at the end of page 12: ''Furthermore, we have inspected the process $\phi \rightarrow 3\phi$ in the spectral function...''
Current status:
Reports on this Submission
Report #1 by Anonymous (Referee 2) on 2021-10-18 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2107.06482v2, delivered 2021-10-18, doi: 10.21468/SciPost.Report.3697
Report
I thank the authors for taking many of my suggestions into account.
Concerning point 9, the authors now state in the manuscript that
"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 ones [77]."
I believe that dropping the last part of this statement reflects the message of [77] better. I therefore recommend to simply write
"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 [77]."
Concerning point 10, I believe that the statement
"For the qq-component, regularization is not necessary"
needs further clarification. In particular, in the paper by Duclut and Delamotte (arXiv: 1611.07301), it is stated that a regularization of the imaginary part of the 2-point function may indeed be necessary. In order to clarify this in the manuscript, I recommend to explicitly state that the regulator is chosen to be real and that therefore the imaginary part of the 2-point function is not regulated. Also a reference to the above mentioned paper should be made in this context.
In the same paper by Duclut and Delamotte, a dynamical critical exponent z close to 2 is found for the O(3) model. The authors may want to point this out when discussing different results on critical exponents and cite the paper also there.
Author: Wei-jie Fu on 2021-10-20 [id 1868]
(in reply to Report 1 on 2021-10-18)We thank the referee for his suggestions in the second report, which we have taken into account in the revised version.
1) Concerning point 9, thanks for the suggestion. We have made relevant modification for the text in the revised version.
2) Concerning point 10, related discussions have been modified below Eq.(21): “Note that in Equation (20) we do not include any regulator for the qq-component, since only the real parts of two-point functions are regulated in this work. This is adequate for cases in thermal equilibrium, where the Keldysh propagator is related to the retarded and advanced propagators by the fluctuation-dissipation relation as shown in Equation (36) in the following. But if the regulator in Equation (17) is extended to the one having a finite imaginary part, as done in some nonequilibrium calculations, e.g. [75], a nonvanishing qq-component of regulators is necessary. ”
Furthermore, we have added the reference [75] (arXiv: 1611.07301), which is also cited in the discussion about the dynamical critical exponent on page 14.
Thanks once more!