Net-Baryon Probability Distributions from Lattice Simulations

Round 1

Reviewer 1 Report

The Authors take advanced of lattice-based simulations of the net-baryon number distributions obtained at imaginary baryon chemical potential in the two-flavor limit to build the equation of state of dense and hot strong-interacting matter in a wider temperature range. 

They further discuss the consequences of the asymptotic behavior of these probability distributions on the reconstruction of the net-baryon density.

The manuscript contains original and sufficiently novel results.

The research design is appropriate and the methodology employed is adequately described. 

The manuscript is also suitably formatted for publication.

I recommend the manuscript for publication in its current version MPDI Particles.

Author Response

Thank you very much for your report!

Reviewer 2 Report

Please find the comments in the report.

Comments for author File: Comments.pdf

No special comments

Author Response

We thank Referee for indication of inaccuracies and helpful comments
We hope that the changes of the text made according to the advises of the Referee will improve presentation.

Point 1: In the abstract, line-5,“number as well as the respective cumulants and momenta”, none of the momenta appear in any equations of the paper, the thermodynamic statistic quantities are static, so is it a typo for momenta? “momenta” − > “moments”.

Response 1: Yes, it is typo, this has been corrected

Point 2: In the Eq.(14, 15) for the self-consistence of the paper, the authors should provide an explanation for why choose these two specific choices.

Response 2: The following text has been added to the manuscript: "Formula (14) is the lowest-degree polynomial that fits the data at $T=1.35T_c$ well. It is interesting to notice that it turns into the net-baryon density formula for free massless quarks when $a_1={N_f\over 3 N_c}$ and $a_3={N_f\over 3\pi^2 N_c^3}$. At $T=1.2T_c$, two-parameter polynomial fit formulas do not provide p-value greater than 0.05, whereas the three-parameter fit formulas (15) and $a_1\theta_I - a_3\theta_I^3  + a_5\theta_I^5$ give rather good fit quality ($p>0.5$). Therewith, the latter formula leads to unphysical behavior at $\theta_I=0, \theta_R\to \infty$ ($\hat\rho<0$ and $\hat p<0$) and thus we choose the fit formula (15)."

Point 3:  In the caption of Fig.(2), The authors should add one more line to explain what does the dashed line mean? The dashed line is a crossover which can change to the first order phase transition for large or small quark masses.

Response 3: The following text has been added to the figure caption:
"The dashed line separates phases with a finite ($|P|>0$) and an infninte ($|P|=0$) free energy of an isolated quark. At physical quark masses it furnishes a crossover transition line."

Point 4: In the Eq.(18, 19), the authors should give an explanation why choose the choice T = 0.93 T_c and T = 0.99 T_c .

Response 4: We have added the required explanation between formulas  (17) and (18):
"The coefficients $f_n$ can be evaluated in some models. It was shown \cite{Bzdak:2018zdg,Begun:2021nbf,Bornyakov:2016wld} that over the range $T_c< T< T_{RW}$ the Cluster Expansion Model (CEM) agrees with lattice data well and one of the parameters of this model, corresponding to the ratio $f_{n+1}/f_{n}$ at $n\to\infty$  tends to zero as the temperature becomes lower than $T_c$. We consider temperatures $T = 0.99 T_c$ and  $T = 0.93 T_c$  in the confinement domain in order to study temperature dependence of the probabilities $\mathbf{P}_n$ where the СЕМ reaches its  applicability limit."

In formula (19) the typo also has been corrected: "f_2=0.0053(7)" changed to "f_2= - 0.0053(7)"

Point 5a: In line 83, there is a typo “T = 0/99 Tc ” − > “T = 0.99 Tc ”. 

Response 5a: This typo, as well as the number of the formula preceding this equation, have been corrected

Point 5b: In the same paragraph, instead of using the words “upper solide curve” or “upper dashed curve”, it would be more clear if authors could add the corresponding color. 

Response 5b: The figure has been modified so that each curve 
has its own unique color. In the text each word "upper" or "lower"
has been replaced with an indication of the respective color.

Point 5c: The authors need to explain why the f2 < 0 curve in the right panel of Fig.3 deflects upwards? What is physics reason for these deflection? 

Response 5c: The following text has been added: 
"It deflects upwards from the $f_2=0$ case, which is a consequence of the EoS (\ref{eq:EoS_2param}) that follows from the expansion (\ref{eq:fit_trig_series}). The physical reason of such deflection can be explained as follows. In a wide class of statistical models such as the Excluded-Volume Hadron-Resonance Gas model (EV-HRG model) or the CEM, the sign of the coefficients $f_n$ alternates, which is a consequence of the excluded volume as was shown in \cite{Taradiy:2019taz}. The excluded-volume effect can be formulated in terms of the EoS as follows: the pressure of the gas of finite-volume particles increases with an increase of the density more rapidly than in the case of pointlike particles. This is precisely what one sees in the case of the Van-der-Waals gas as well as in Fig.\ref{fig:EoS}."

Point 6: In the line 138, “In the case of alternating 139 coefficients fn the formula (43) is not justified, however, we guess that..”, scientific facts cannot be relied on guess, the authors should make a more sound argument to explain this case. 

Response 6: We have replaced the phrase "we guess that" by the text: 
"in the case of alternating coefficients $f_n$ the probability mass function  $\mathbf{P}_n$ declineds even more rapidly than in the case when $f_n>0~\forall n$.

To justify this statement, one can consider the CEM, the EV-HRG model and the like, where the net-baryon number density at imaginary chemical potentials is described by formula (\ref{eq:fit_trig_series}) with alternating sign of $f_n$ and $|f_n|$ decrease as a geometric progression or faster. The probabilities $\mathbf{P_n}$ corresponding to such quark densities can be evaluated numerically using the algebraic procedures described in \cite{Begun:2021nbf}
both in the case $f_n=|f_n|$ and in the case of alternating sign of $f_n$. Such estimates show that the probabilities $\mathbf{P_n}$ in the case of alternating sign of $f_n$ decrease with $n$ more rapidly than those in the case of 
constatnt sign of $f_n$.
 
It should also be noticed that the alternating sign of $f_n$ was obtained in lattice simulations in QC$_2$D at $T_c< T< T_{RW}$. In Ref.\cite{Begun:2021nbf}
it was shown that fitting the CEM-parametrized fit-function to the lattice data on 
the imaginary part of the net-baryon number density over the segment  $-\pi<\theta_I<\pi$  gives alternating coeffcients $f_n$. "

We also have removed the text fragment: 
"Lee-Yang zeros in this model lie on the negative real semiaxis in the fugacity plane~\cite{Taradiy:2019taz}, and the expression for the grand canonical partition function at physical values of $\mu_B$  can be obtained by analytical continuation~\cite{Begun:2021nbf}. Analytical structure of the partition function
gives a hope that the convergence properties of the fugacity expansion in this case are similar to those in the free theory. A more rigorous check of our guess 
will be performed in a separate study."

For your convenience, we attach the .pdf file illustrating the decrease of the probabilities both in the case of alternating sign of $f_n$ and in the case when $f_n=|f_n|$. We take as an example the case of EV-HRG model, in which the coefficients $f_n$ are given by the formula

$f_n=c*(-q)^(n-1)*n^n/n!$ 

with the parameters $c=400.0;q=0.05$.

At different mentioned models and parameters the behavior of the probabilities is quite similar.

However, we haven't include this fugure in the manuscript because we are going to investigate the rate of decrease of the probabilities in the mentioned models and their modification in more detail.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

The authors have answered all my questions and corrected typos. I would recommend the publication of the present form.