Gluon Propagators in QC2D at High Baryon Density
Round 1
Reviewer 1 Report
In this paper, the transverse and longitudinal gluon propagators in the Landau-gauge lattice QCD with gauge group SU(2) at nonzero quark chemical potential and zero temperature are studied. Some useful results are obtained. This paper is interesting for readers. I recommend its publication.
However, the English needed to be paid more attention during revision.
Author Response
Rewiever 1:
However, the English needed to be paid more attention during revision.
Our answer:
Our corrections are indicated in green in the .pdf file of the revised manuscript.
We also checked spelling.
Reviewer 2 Report
This paper studies the behaviour of the Landau gauge lattice gluon
propagator at large quark chemical potential. As such, it is of
significant potential interest to researchers studying the QCD phase
diagram using functional methods such as Dyson-Schwinger equations and
the functional renormalization group. It could also complement and
provide an important corrective to other lattice studies using a
different action and significantly coarser lattices.
The research is well explained and presented and the quality of the
analysis is good. However, it suffers from one serious flaw which
means I cannot recommend acceptance of this manuscript.
The study is carried out on a single volume of (1.4fm)^4. The
temporal extent can be translated to an effective temperature of about
140 MeV. It is well known that the gluon propagator is very sensitive
to the volume, and that considerably larger volumes are required to
extract reliable results for infrared behavior. The gluon dressing
function has a maximum around p=500MeV, which is well below the
smallest nonzero momentum in this study, so their data cannot capture
this qualitative feature.
Author Response
Reviewer 2: "It is well known that the gluon propagator is very sensitive
to the volume"
Our answer:
We agree that many studies in both $SU(2)$ and $SU(3)$ gluodynamics
suggest significant dependence of the gluon propagator at zero and
small momenta on the lattice size. Still, we believe that our results
are correct at least qualitatively and deserve to be published.
Our arguments are as follows.
1) We choose lattice of relatively small size for the following reasons.
To reach large quark densities without lattice artifacts one needs sufficiently small
lattice spacing $a\mu_q \ll 1$. Therewith, to study the gluon propagators
in the infrared region one needs large physical volume.
As a compromise between these two requirements we choose $L = 3.4/\sqrt{\sigma} = 1.4$~fm,
having regard to a potential problem of finite volume effects at small momenta.
2) The volume dependence of the propagators studied in gluodynamics at zero
and nonzero temperature decreases fast with increasing momentum
and this also should be true for our results at nonzero momenta (p > 0.87 GeV).
3) It was shown for T=0 SU(2) gluodynamics in a series of papers
Phys.Rev. D81 (2010) 054503
Phys.Rev. D79 (2009) 074504
Phys.Rev. D77 (2008) 014504
that Z_2 symmetry of the action makes possible Z_2 nonperiodic gauge transformations (called Z_2 flips). When this is taken into account in the improved gauge fixing procedure the finite volume effects in the gluon propagator are substantially reduced. It was shown in particular that the finite volume effects practically disappear already at the minimal nonzero momentum and reduce substantially at zero momentum. Then it is natural to expect that in a theory with fermions the volume dependence of the
gluon propagator is similar to that of gluodynamics with improved gauge fixing because Z_2 symmetry is explicitly broken by fermions. Unfortunately, our expectations were not checked so far: volume dependence of the gluon propagator in theories with fermions have received only little attention,
we know only one work on this problem (Int.J.Mod.Phys. A27 (2012) 1250050), where close vicinity of pseudocritical temperature was investigated and thus at the moment there is no clear understanding of the volume dependence of the gluon propagators in the case under consideration ($T=0, \mu_q$ varies).
4) For the longitudinal propagator $D_L(p)$ there is a special argument. At sufficiently high density the chromo-electric screening length determined as the inverse of the chromo-electric mass can be evaluated in perturbation theory, $l_E = {1 \over m_E} \sim {1 \over g(\mu_q) \mu_q}$. Our results are in agreement with this prediction. Thus we expect that with increasing $\mu_q$ the finite volume effects for this propagator shouls decrease.
5) At the same time the screening length associated with the transverse propagator $D_T(p)$ is defined as the inverse of the chromo-magnetic screening mass $m_M$. Perturbation theory predicts zero value of the magnetic screening mass at high chemical potentials [hep-ph/9812287];
for this reason we should use nonperturbative estimates of $m_M$. On these grounds we expect that at sufficiently high $\mu_q$ (where perturbation theory works) $m_M$ goes down, the respective screening length becomes large, and to study the infrared behavior of $D_T(p)$ large lattices are needed.6) Thus, results on the $\mu_q$ dependence of the propagators at a fixed momentum $p\sim 1$~GeV should not be dramatically changed by the finite-volume effects and, therefore, we included such data into the updated version of our manuscript, see Fig.3 in the updated version of the manuscript and the respective text on lines 135-143 and 119 as well as Ref.38. Observation of the dependence of the propagators on $\mu_q$ is a new result as it is seen from, say, the study [arXiv:1812.08517]. It is a major issue of our study.
7) It should also be noted that our Section 4 is completely devoted to
$\mu_q$ dependence of the propagators at high momenta, where finite-volume effects are negligible.
Reviewer 2:
"The gluon dressing function has a maximum around p=500 MeV, which is well below the smallest nonzero momentum in this study, so their data cannot capture this qualitative feature."
Our answer:
We cannot agree with this statement. The gluon dressing function peaks at
a momentum about 1 GeV (see, e.g. hep-lat/0209040,arXiv:0912.4475)
which is greater than our minimal momentum. Our data do capture this qualitative feature, we will present the respective results in a forthcoming study.
Reviewer 3 Report
Dear Editor,
In the manuscript the authors find in Landau lattice SU(2) QCD a substantial dependency of the chromomagnetic and chromoelectric Linde screening mass on the quark chemical potential. This finding is in disagreement with another paper, Ref. [24].
The paper is well written and organized and the results are significant. After the authors have implemented the suggestions for improvement and addressed my concerns (see below) I believe the paper is suitable for publication in MDPI-Particles.
In the abstract the authors write “This observation contradicts to earlier findings by other groups.” Here I would replace “contradicts to” with “does not agree with”. The word “contradict” here seems too strong as it suggests an inconsistency. It is,however, not clear what the origin of the differences is and the authors actually state later that it might come from using a different lattice action and a different lattice spacing. If that would be the case, then the 2 approaches would not contradict each other as their ingredients are different. To make this point stronger, the authors might want to try to reproduce the results of Ref. [24] with the same lattice action and lattice spacing (a=0.138 fm), just to exclude other origins of the discrepancies. In the Introduction some References should be included, for instance: “e.g. they play a crucial role in the Dyson-Schwinger equations approach and similar approaches ([hep-ph/0605173, nucl-th/0301049, 1606.09602, 1811.01003].” A pion mass of 740 MeV seems very high to me. How sensitive are the results on the value of pion mass? If they are not very sensitive, this should be mentioned in the paper. Fig. 1 should appear on page 5.Author Response
Reviewer 3:
"In the abstract the authors write “This observation contradicts to earlier findings by other groups.” Here I would replace “contradicts to” with “does not agree with”"
Our answer:
We performed these changes: see lines 15 and 174
Reviewer 3:
"To make this point stronger, the authors might want to try to reproduce the results of Ref. [24] with the same lattice action and lattice spacing (a=0.138 fm), just to exclude other origins of the discrepancies."
Our answer:
We agree that to study the effects of the finite lattice spacing and of lattice discretization of the Dirac operator is an important subject. This is definitely a subject of a separate study.
Reviewer 3:
In the Introduction some References should be included, for instance: “e.g. they play a crucial role in the Dyson-Schwinger equations approach and similar approaches ([hep-ph/0605173, nucl-th/0301049, 1606.09602, 1811.01003].”
Our answer: We added these references, see lines 35 and 220-225
Reviewer 3:
A pion mass of 740 MeV seems very high to me. How sensitive are the results on the value of pion mass? If they are not very sensitive, this should be mentioned in the paper.
Our answer:
Comment in lines 69-72 is added: "Similar values for the pion mass were used in Ref. [24] as well as in earlier studies. The dependence of the gluon propagators on the pion mass in QC$_2$D was not investigated so far. This important issue will be a subject of our future studies."
Reviewer 3:
"Fig. 1 should appear on page 5."
Our answer:
We moved this figure as indicated
Round 2
Reviewer 2 Report
The authors have presented a series of arguments why their results
should be considered reliable despite the small lattice volume. While
I agree with a number of the points they make, in particular regarding
the gluon propagator at larger momenta, I remain unconvinced that the
results for screening masses can be considered even qualitatively
reliable without having explicitly assessed the volume dependence.
I appreciate that the results at larger momenta can be considered more
reliable and that their additional paragraph on pp6-7 and new figure 3
goes some way to addressing my concerns. However, the potential
limitations are still not made sufficiently clear.
Ideally, further simulations should be carried out with a larger
volume at zero and at least one non-zero mu. In the absence of this,
it should be clearly stated that these results are potentially subject
to finite volume effects and an extended discussion of this should be
included and referenced in the conclusion This may include some of the
points made in their response.
Regarding the specific point about the maximum of the dressing
function, I apologise for my mistake, although I note that this
maximum is only marginally present in their data, with only one
momentum value below 1 GeV.
Author Response
1. Possible changes of the zero-momentum value of the propagator
with an increase of the lattice volume are mentioned in the Concusions.
2. A detailed discussion of possible finite-volume effects
from our previous answer to the Referee is inserted in the text
between Figs. 2 and 3
Please see the attachment
Author Response File: 
Author Response.pdf
Round 3
Reviewer 2 Report
The authors have addressed my concerns to my satisfaction, and I therefore recommend that this paper can be published in its present form.
