Lee–Yang Zeroes in the Baryon Fugacity Plane: The Role of High Densities
, Roman Rogalyov 2, Anatolii Korneev 1, Alexander Molochkov 1 and Atsushi Nakamura 1,3
Round 1
Reviewer 1 Report
The authors have presented a calculation of the canonical partition sums in QCD which are obtained through an integration of the imaginary baryon number density at imaginary chemical potential. The latter can be obtained by means of lattice QCD calculations since at imaginary chemical potential the lattice QCD approach does not suffer from a sign problem. As such, the imaginary chemical potential approach is widely used in the lattice community. However, the strategy to interpolate the lattice data with a polynomial of degree seven, and the deformation of the integration contour through the saddle point are a new development based on the preceding works of the authors. The resulting canonical partition sums and its zeros are very interesting and promising. The manuscript is well written. Prior to publication there are a few points that need to be clarified:
1.a The discussion of the order of the interpolating polynomial is not quite clear. The authors state that a negative 5th order coefficient in the polynomial would lead to a negative baryon number density at large real chemical potential if no further terms of higher order are considered. While this statement is certainly correct, the numerical coefficient that is presented as a result from the fit is positive. I thus conclude that the 5th order polynomial is also a valid interpolation of the data. Is there a sign missing in the numerical coefficient?
1.b In general, it would be good to show the original data and the interpolating polynomial in a figure. In particular the difference of a 5th and 7th order polynomial.
2. In Eq. (8) there is a sign missing in the lower integration boundary. Currently lower and upper integration boundaries are identical.
3. The discussion on neglecting the first and third terms in Eq. (10) is based on the numerical results. I would think that a symmetry argument based on the 2\pi-periodicity of the baryon number density would suffice here to prove that the first and second third terms cancel exactly.
4. After presenting the expression for the coefficients b2,b4,b6,b8,alpha the authors should elaborate a little more on how the expressions have been obtained. And what the strategy was used to obtain the expression for the density on the line B-C.
5. The results of the Lee-Yang zeros are very interesting. The authors present the scaling of the zeros with rho_max for two different temperatures. They use a power law and an exponential decay. Why is this ansatz used? I believe that at T=Tc the scaling is well defined in terms of a power low with a well defined critical exponent (gamma/nu). Can this limiting case be seen to emerge from the data?
Author Response
Dear Reviewer,
We carefully read your feedback on our work and tried to take into account all your requirements. You can find our answers in the attached PDF-file.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
In their ongoing efforts [10-12], the authors developed methods to compute the canonical partition functions ZC(n, T, V) using the lattice data for imaginary baryon density at both low and high temperatures. Earlier, in Ref. [12] the authors suggested the solution of the problem of computation of ZC(n, T, V) for temperatures above the Roberge-Weiss transition when the imaginary baryon density has discontinuity and applied it to the case of T = 1.35Tc when the baryon density is described by the polynomial of degree three. The authors improve on their method and apply it to the case of T = 31 1.20 Tc > TRW using higher degree polynomial to describe lattice data for imaginary baryon density. The authors in this article compute the canonical partition functions and the Lee-Yang zeros in Nf = 2 lattice QCD at temperature T = 1.20 Tc lying above the Roberge-Weiss phase transition temperature TRW. The phase transition is characterized by the discontinuities in the baryon number density at specific values of imaginary baryon chemical potential. Results are interesting to be considered for publication. Before this I suggest considering the following comments.
1. For the interest of large audience, improve the Introduction by expanding further the background of the main theme and research problem, research gap, and major novelty with regard to latest development in the field and advancement of methodology and or improvement of results over the earlier studies. In particular, as authors clarify that the current numerical results are slightly different from those presented earlier in Ref. [10] due to increased statistics, it is important to justify the major novelty of the current work in respect of the aforementioned points.
2. Expand caption of each figure for independent understanding of main results plots, better indicate the expression of equations plotted.
3. The authors talk about the discontinuities in the baryon number density at specific values of imaginary baryon chemical potential, this point needs to be further discussed with regard to the origin of this discontinuity and possible resolutions.
4. Does free energy play any role in the phase diagram of QCD while studying the Roberge-Weiss transition? Please expand discussion on this if so and provide the related numerical plots for better understanding its role in the phase diagram.
5. Clearly validate the reliability and novelty of the present results with available studies by due comparison.
Good
Author Response
Dear Reviewer,
We carefully read your feedback on our work and tried to take into account all your requirements. You can find our answers in the attached PDF file.
Author Response File: Author Response.pdf
Reviewer 3 Report
Please find the comments in the attached report.
Comments for author File: Comments.pdf
The English language is fine. No big issues detected
Author Response
Dear Reviewer,
We carefully read your feedback on our work and tried to take into account all your requirements. You can find our answers in the attached PDF file.
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
The authors have addressed all raised issues and have revised the paper coresspondingly. It is now in very good shape for publication.
Author Response
Thank you so much for your review.
Reviewer 2 Report
Ref: particles-2508453-peer-review-v2
Report 2
Though the authors have revised the manuscript considerably by addressing the comments of the referee; before acceptance, I suggest addressing the following issues.
1. The conclusion should be precise and concise and should be written in a way to be clearly understood regarding key findings and prospects without too much mentioning of equations and figures.
2. There is a considerable overlapping of text (7%) with only one reference (V. G. Bornyakov, N. V. Gerasimeniuk, V. A. Goy, A. A. Korneev, A. V. Molochkov, A. Nakamura, R. N. Rogalyov. "Numerical study of the Roberge-Weiss transition", Physical Review D, 2023). This should be reduced to 1 %.
Good
Author Response
Dear Reviewer 2,
Thank you for your feedback. We have tried to answer your requests.
1.The conclusion should be precise and concise and should be written in a way to be clearly understood regarding key findings and prospects without too much mentioning of equations and figures.
Answer 1: We modified the Conclusions section to reduce the number of quoted equations and figures.
2.There is a considerable overlapping of text (7%) with only one reference (V. G. Bornyakov, N. V. Gerasimeniuk, V. A. Goy, A. A. Korneev, A. V. Molochkov, A. Nakamura, R. N. Rogalyov. "Numerical study of the Roberge-Weiss transition", Physical Review D, 2023). This should be reduced to 1 %.
Answer 2: We modified the text to reduce the overlap with our previous publication on same subject.
