# Finite temperature properties of QCD with two flavors and three, four, and
five colors

Thomas DeGrand  Department of Physics, University of Colorado, Boulder, CO
80309 USA<EMAIL_ADDRESS>
###### Abstract

I present a numerical study of the crossover between the low temperature
chirally broken phase and the high temperature chirally restored phase in
$SU(N_{c})$ gauge theory with $N_{c}=3-5$ colors and $N_{f}=2$ degenerate
fermion flavors. Fermion masses span a range of intermediate values
corresponding to pseudoscalar to vector meson masses $(m_{PS}/m_{V})^{2}\sim
0.25$ to 0.63. Observables include the temperature dependent chiral condensate
and screening masses. At each fermion mass these quantities show nearly
identical temperature dependence across $N_{c}$.

##  I Introduction and motivation

The limit of QCD when the number of colors $N_{c}$ is taken large has a long
history as a source of insight about $N_{c}=3$ QCD itself tHooft:1973alw ;
tHooft:1974pnl ; Witten:1979kh  . Many of the applications of large $N_{c}$
ideas to phenomenology actually involve nonperturbative quantities (masses or
decay constants), although the predictions are often based on counting the
color weight of Feynman diagrams. To what extent are these predictions true?
Checking them requires a lattice simulation, and there is a small literature
of lattice calculations away from $N_{c}=3$ to provide such tests. (See Refs.
Lucini:2012gg ; GarciaPerez:2020gnf ; Hernandez:2020tbc  for a selection of
reviews.)

At a qualitative level, lattice calculations confirm large $N_{c}$ intuition
rather nicely: meson masses, baryon masses, and decay constants scale as
$N_{c}^{p}f(m_{q})$ where $p$ is a characteristic power and $m_{q}$ is the
fermion mass. The interplay of small dynamical fermion mass and large $N_{c}$
is less well explored, but simple matrix elements (decay constants, the kaon
weak matrix element calculations of Refs Donini:2016lwz ; Donini:2020qfu )
mostly scale as expected. So do some chiral observables which are governed by
the pseudoscalar decay constant (scaling as $m_{PS}^{2}/f_{PS}^{2}\propto
m_{PS}^{2}/N_{c}$: compare to Ref. Hernandez:2019qed ) or by the condensate
(as in the topological susceptibility $\chi_{T}\propto m_{q}\Sigma\propto
m_{q}N_{c}$ DeGrand:2020utq ).

The subject of this paper is the crossover temperature for the transition from
the low temperature confining and chirally broken phase to the high
temperature deconfined and chirally restored phase, for QCD with $N_{c}=3$, 4,
and 5 colors and $N_{f}=2$ flavors of degenerate mass fermions in the
fundamental representation. The reason why this project might be interesting
is that there is not a single large $N_{c}$ story for what happens, there are
at least three possibilities.

The first possibility comes from the naive large $N_{c}$ expectation that
gluonic degrees of freedom dominate fermionic ones as
$N_{c}\rightarrow\infty$. There should be a finite temperature confinement
deconfinement transition which converges to the pure glue one in the large
$N_{c}$ limit. This is a first order transition with $T_{c}\sim 320$ MeV. In
$N_{c}=3$ the pure gauge transition is first order and as the fermion mass
falls from infinity the transition becomes a crossover. In this scenario the
large $N_{c}$ transition temperature would remain roughly constant across
$N_{c}$ as the fermion mass fell from infinity, and the critical point where
the first order region ends would move to smaller fermion mass as $N_{c}$
rises.

The second scenario assumes naive chiral symmetry breaking dominance. Even QCD
at large $N_{c}$ has an $SU(N_{f})\times SU(N_{f})$ symmetry which (modulo
issues with the eta-prime Kaiser:2000gs ) undergoes spontaneous symmetry
breaking to vectorial $SU(N_{f})$. The Pisarski - Wilczek analysis
Pisarski:1983ms  approximates the Goldstone sector as a linear sigma model.
For $N_{f}=2$ the system is expected to have a second order transition at zero
fermion mass, with $O(4)$ critical exponents. Second order transitions are
unstable under perturbation, so the transition becomes a crossover away from
$m_{q}=0$. All QCDs with any $N_{c}$ should then share a common behavior at
small fermion mass.

This paper doesn’t have data at the tiny (or zero) fermion masses needed to
say anything about the properties of any transition at zero fermion mass, but
it can ask a question which is related to the Pisarski - Wilczek analysis: how
does the crossover temperature scale with $N_{c}$? Linear sigma models contain
one dimensionful parameter, the vacuum expectation value of the scalar field,
and all derived dimensionful quantities (the pseudoscalar decay constant, and
the crossover temperature $T_{c}$ itself) are proportional to it. It is well
known from previous large $N_{c}$ spectroscopy comparisons that
$f_{PS}\propto\sqrt{N_{c}}$. Thus the naive prediction of the second scenario
is $T_{c}\propto\sqrt{N_{c}}$ Pisarski . Is it so?

The final scenario predates QCD. Confining theories are expected to show an
exponentially growing spectrum of resonances with mass, forming a Hagedorn
spectrum Hagedorn:1968zz . The tower of resonances implies a limiting
temperature $T_{0}$ and (as first stated explicitly by Cabibbo and Parisi
Cabibbo:1975ig , as far as I can tell) this implies a crossover temperature
$T_{c}\sim T_{0}$. In QCD, the Hagedorn temperature is about 160 MeV. The
extension of the story for large $N_{c}$ and nonzero $N_{f}$ is that the
spectrum of resonances is basically identical across $N_{c}$. Meson states
dominate baryon ones up to a few GeV. If two theories have the same spectrum,
then they will have the same critical properties. Notice that large $N_{c}$
with nonzero $N_{f}$ is different from quenched QCD: the latter case has a
much sparser spectrum below 2 GeV. There are only glueballs in contrast to all
the excited states labeled (for example) by quark model counting. The
prediction of the third scenario is that any $N_{c}\neq 3$ with $N_{f}=2$ will
qualitatively resemble $N_{c}=3$, $N_{f}=2$. This argument has been used to
justify the observation that the deconfinement transition in pure gauge
$SU(N_{c})$ is nearly independent of $N_{c}$ Lucini:2005vg ; Lucini:2012wq ,
since the glueball spectrum is also nearly $N_{c}$ independent.

The first scenario was already unlikely given that the deconfinement critical
point for $N_{c}=3$ is already at a very high mass. Ref Cuteri:2020yke
observes it at a pseudoscalar mass of about 4 GeV. And indeed, there is no
evidence in any of the simulations reported here for anything other than a
smooth crossover. (This makes the identification of a particular crossover
temperature problematic.) As to the other scenarios, my results indicate that
the temperature dependence of observables computed at common values of the
fermion mass show essentially identical behavior, including inflection points
at finite temperature, which is nearly independent of $N_{c}$.

To illustrate this statement, the temperature dependent and zero temperature
subtracted condensate, defined below in Eq. 12, is shown in Fig. 1. The naive
$N_{c}$ scaling of the condensate is divided out, and data is presented for
three values of the ratio $(m_{PS}/m_{V})^{2}$. This quantity is close to zero
at low temperature and becomes negative at high temperature as (speaking
loosely) chiral symmetry is restored and the finite temperature condensate
falls to zero. The different plotting symbols label different numbers of
colors. This picture illustrates the smooth crossover from broken to restored
chiral symmetry with nearly identical temperature variation across $N_{c}$.

Figure 1: The temperature dependent condensate, rescaled by $3/N_{c}$, as a
function of temperature, in appropriate units of $t_{0}$. Squares, octagons,
and diamonds label $N_{c}=3$, 4, and 5. (a) $(m_{PS}/m_{V})^{2}\sim 0.63$; (b)
$(m_{PS}/m_{V})^{2}\sim 0.5$; (c) $(m_{PS}/m_{V})^{2}\sim 0.25$.

A natural question to ask is, can one identify a feature in the data which
serves as a marker for a crossover temperature? The short answer is no. The
quantities I have studied have a smooth temperature dependence with no sharp
features. I was able to identify two quantities which could serve as markers:
there is a peak in the derivative $d\Sigma(T)/dT$, and there is a transition
region where screening masses cross over from near independence with
temperature to strong temperature dependence. Both features are broad.

The outline of the rest of the paper is as follows: Technical aspects of the
calculation (lattice action, data sets, a bit about data analysis methodology)
are described in Sec. II. This is all completely conventional. Then the
various observables are described and results are presented for them: the
temperature dependent condensate in Sec. III, screening masses in Sec. IV. I
mention the the Polyakov line susceptibility in Sec. V. Conclusions are
summarized in Sec. VI.

##  II Technical aspects of the calculation

###  II.1 Simulation methodology, lattice actions, data sets

The simulations involved two degenerate flavors of Wilson–clover fermions. The
gauge action is the usual Wilson plaquette action. The fermion action uses
gauge connections defined as normalized hypercubic smeared links
Hasenfratz:2001hp ; Hasenfratz:2007rf ; DeGrand:2012qa ; DeGrand:2016pur . The
bare gauge coupling $g_{0}$ is set by the simulation parameter
$\beta=2N_{c}/g_{0}^{2}$. The bare quark mass $m_{0}^{q}$ is introduced via
the hopping parameter $\kappa=(2m_{0}^{q}a+8)^{-1}$. The clover coefficient is
fixed to its tree level value, $c_{\text{SW}}=1$. Gauge-field updates used the
Hybrid Monte Carlo (HMC) algorithm Duane:1986iw ; Duane:1985hz ;
Gottlieb:1987mq  with a multi-level Omelyan integrator Takaishi:2005tz  and
multiple integration time steps Urbach:2005ji  with one level of mass
preconditioning for the fermions Hasenbusch:2001ne . Lattices used for
analysis are spaced a minimum of 10 HMC time units apart. All data sets are
based on a single stream for each set of bare parameters.

The only new feature to report in these simulations is a first order bulk
transition in strong coupling for $SU(5)$. Its location depends, of course, on
the particular form of the bare action. These transitions are well known
features of the pure gauge systems and the one I found has a small $\kappa$
limit which appears to approach the value of gauge coupling where Ref.
Lucini:2005vg  observed a pure gauge transition. I only mention this in
passing; data sets used to do physics were selected to avoid this transition.

The choice of Wilson-clover fermions in a finite temperature study is not
optimal, since the best signals for finite temperature behavior with dynamical
fermions are ones which are sensitive to chiral symmetry. The reason I chose
this discretization is that I already had a large collection of zero
temperature simulations which could be used to find lines of constant physics
in bare parameter space, and because I was only looking for gross finite
temperature features.

###  II.2 Fixing the lattice spacing

The lattice spacing is set by the Wilson flow parameter $t_{0}$ other ;
Luscher:2010iy , and quantities will be presented as dimensionless ones by a
rescaling by an appropriate power of $t_{0}$.

The determination of $t_{0}$ is done on zero temperature lattices, in the
standard way, from the observable $E(t_{0})$ extracted from the field strength
tensor,

$t_{0}^{2}\left\langle E(t_{0})\right\rangle=C(N_{c}).$ (1)

$C(N_{c})$ is chosen to match what most other large $N_{c}$ simulations take,

$C(N_{c})=C\left(\frac{3}{8}\frac{N_{c}^{2}-1}{N_{c}}\right),$ (2)

with $C=0.3$ the usual value used in $SU(3)$. The extraction of $t_{0}$ from
the data is identical to what was done in Ref. DeGrand:2020utq  and details
may be found there.

I use values of $t_{0}$ computed at each bare parameter value, that is, there
is no extrapolation or interpolation in fermion mass. Let’s recall a few
useful numbers which will place results in context. The critical temperature
for pure gauge systems was published in Refs. Lucini:2005vg ; Lucini:2012wq .
The authors of these papers quote $T_{c}/\sqrt{\sigma}$ where $\sigma$ is the
string tension. I converted their numbers to $\sqrt{t_{0}}T_{c}$ using the
Sommer parameter $r_{0}\sim 0.49$ fm Sommer:1993ce ,
$r_{0}\sqrt{\sigma}=1.175$ for quenched $SU(3)$ and $SU(5)$ from Ref.
DeGrand:2012hd , and the quenched $N_{c}=3$ $\sqrt{t_{0}}=0.1638$ fm from
Refs. Lottini:2013rfa ; Bruno:2013gha  (as quoted in Ref. Sommer:2014mea ) to
give the pure gauge transition at $\sqrt{t_{0}}T_{c}=0.265$, 0.261 and 0.278
for $N_{c}=3$, 4, 5.

The nominal “physical point” value (with $2+1$ flavors) of fermions with
$T_{c}=150$ MeV, $\sqrt{t_{0}}=0.147$ fm of Ref. Soltz:2015ula  from chiral
observables, is $\sqrt{t_{0}}T_{c}=0.11$.

###  II.3 Data sets

At each $N_{c}$ the bare parameter space is (at least) three dimensional,
involving a bare gauge coupling, a bare fermion mass expressed through
$\kappa$ in units of the lattice spacing as $am_{q}$, and the temperature,
$T=1/(aN_{t})$ where $N_{t}$ is the length of the lattice in the Euclidean
temporal direction. The issue for a study like this is that no single
simulation (at any one set of bare parameters) is interesting in itself: all
features are broad. It is hard to avoid generating data sets at many
parameters and easy to get lost wading through them.

The literature suggests at least two ways to proceed.

The first approach is to map out a crossover line (or phase boundary, if it
exists) as a function of bare parameters at fixed $N_{t}$ values. Then zero
temperature simulations are done along the crossover lines to determine
physical quantities such as the lattice spacing and $(m_{PS}/m_{V})^{2}$.
(Examples of this approach I found useful were Refs. AliKhan:2001ek ;
Ejiri:2009hq ; Bornyakov:2009qh .) The issue with this approach is that since
there is no true phase transition, the crossover region is broad, and to
present a result such as $\sqrt{t_{0}}T_{c}$ versus $(m_{PS}/m_{V})^{2}$
involves collecting a large number of zero temperature data sets followed by
interpolation of their output onto a surface in bare coupling constant space.
The surface’s own bare parameters are poorly defined because the “transition”
is just a crossover. This issue is compounded of course by the need to work at
several values of $N_{c}$. The preliminary version of this project, Ref.
DeGrand:2018tzn  used this approach, but I found it to be unwieldy.

Instead, I took an approach inspired by Refs. Borsanyi:2012uq ;
Borsanyi:2015waa . I started with zero temperature data at selected bare
parameters and then varied the temperature by varying $N_{t}$ holding all
other bare parameters (and hence, the lattice spacing, $t_{0}$ and
$(m_{PS}/m_{V})^{2}$) fixed. A disadvantage of this approach is that it is
unlikely that one will obtain data precisely at a crossover temperature. This
can be compensated for, to some extent, by combining data sets from several
values of the lattice spacing (but the same $(m_{PS}/m_{V})^{2}$ values) to
fill in the curves. This of course introduces another disadvantage: the data
sets from different values of the lattice spacing have different lattice
artifacts. The naive approach of just plotting them all together ignores
lattice artifacts: they will appear in the scatter between points on a graph.
An advantage of the approach is that it allows one to use a “natural”
observable – the temperature -dependent condensate – to compare crossover
behavior across $N_{c}$. At the end, I have eight bare parameter sets per
$N_{c}$ [divided into three different $(m_{PS}/m_{V})^{2}$ values], five
values of $N_{t}$ per bare parameter, – 40 sets per $N_{c}$, 120 datasets in
all.

For zero temperature data sets I took $16^{3}\times 32$ volumes. Results for
many of the parameter sets have been published before (see
Refs.DeGrand:2016pur ; DeGrand:2020utq ) and additional sets were generated to
give several lattice spacings at three matching points in the fermion mass.
The squared ratio of the pseudoscalar to vector masses $(m_{PS}/m_{V})^{2}$
will be taken as a proxy for a fermion mass. and data were collected at
$r=(m_{PS}/m_{V})^{2}\sim 0.63$, 0.5 and about 0.25 (in the latter case,
ranging from 0.22-0.27). Useful results from the zero temperature data sets
are summarized in Table 1.

The calculation of a temperature dependent condensate described below in Sec.
III requires data sets at fixed spatial volume (for each lattice spacing),
again $16^{3}$ sites. I varied $N_{t}$ from 4 to 12 at $r=0.63$ and $r=0.5$,
and 6 to 16 at $r=0.25$ since the crossover temperature appeared to fall with
decreasing $r$. Almost all of these finite temperature data sets are 1600
trajectories long after equilibration, again saving lattices spaced 10
trajectories apart for further analysis. The results for screening masses and
related quantities are based on subsets of these data sets since these results
were not used for any precise calculations.

###  II.4 Data analysis

Most results are global observables, a single quantity averaged over the
simulation run. For most observables, there are not really enough data for a
reliable determination of an integrated autocorrelation time. Instead,
autocorrelation times are estimated and errors are assigned from a jackknife
analysis dropping successive measurements from the data stream. For an
observable $Q$ I compute an average $\left\langle Q\right\rangle$ and a
susceptibility $\chi_{Q}=\left\langle Q^{2}\right\rangle-\left\langle
Q\right\rangle^{2}$; the uncertainty of each comes from a jackknife. I varied
the size of the jackknife (dropping $n_{J}$ successive measurements).
Typically $\Delta Q$ rises as $n_{J}$ rises, reflecting the effect of time
autocorrelations, until it plateaus as $n_{J}$ approaches or exceeds the
autocorrelation time. At the same time the uncertainty in $\Delta Q$ has a
fractional error from loss of statistics,

$\frac{\Delta(\Delta Q)}{\Delta Q}=\sqrt{\frac{2}{n}}$ (3)

where $n=N/n_{J}$ for $N$ measurements. The increasing uncertainty in
$\left\langle\Delta Q\right\rangle$ eventually exceeds the variation of
$\Delta Q$ with $n_{J}$. The uncertainties quoted in the paper come from
jackknife averages taking $n_{J}$ where the quantity $\Delta(\Delta Q)$ from
Eq. 3 is larger than the apparent rise in the jackknife estimated $\Delta Q$.

Generally, I observed that for observables related to the temperature
dependent condensate the errors taken from jackknifes dropping one or two
successive lattices (spaced 10 molecular dynamics time steps apart) were
indistinguishable within the uncertainty of Eq. 3. Volume averaged Polyakov
lines were measured every trajectory in the normal course of data collection.
There, Eq. 3 indicated that typically jackknife errors saturated with
$n_{J}=5-20$ (the higher number coming closer to the crossover temperature).
The errors quotes in tables and shown in graphs were taken from this
procedure.

##  III The temperature dependent condensate and related quantities

###  III.1 Defining the condensate

With Wilson fermions the chiral condensate is a bit awkward to measure
Borsanyi:2012uq ; Giusti:1998wy . The bare condensate for Wilson fermions at
bare mass $m_{1}$ has an expansion

$\left\langle\bar{\psi}_{0}\psi_{0}\right\rangle=c_{0}+c_{1}(m_{1}-m_{0})+c_{2}(m_{1}-m_{0})^{2}+\dots$
(4)

where $m_{0}$ is the bare mass at which the axial Ward identity fermion mass
(defined below) vanishes, $m_{1}$ is the bare mass at the simulation point,
and the $c_{i}$’s contain cutoff (divergent) behavior, $c_{0}\sim 1/a^{3}$,
$c_{1}\sim 1/a^{2}$, $c_{2}\sim 1/a$. The divergent pieces are independent of
temperature, and so the difference

$\left\langle\bar{\psi}\psi\right\rangle_{sub}=\left\langle\bar{\psi}\psi\right\rangle_{T}-\left\langle\bar{\psi}\psi\right\rangle_{T=0}$
(5)

is finite and sensible. Inspired by the Gell-Mann, Oakes, Renner relation, a
potential definition for $\left\langle\bar{\psi}\psi\right\rangle_{sub}$ is

$\left\langle\bar{\psi}\psi\right\rangle_{sub}\propto m_{AWI}\left[\int
d^{4}x\left\langle 0|P(x,t)P(0,0)|0\right\rangle_{T}-\int d^{4}x\left\langle
0|P(x,t)P(0,0)|0\right\rangle_{T=0}\right]$ (6)

where $P(x,t)=\bar{\psi}(x,t)\gamma_{5}\psi(x,t)$ is the pseudoscalar current.
The first term on the right hand side of Eq. 6 is evaluated on an
$N_{s}^{3}\times N_{t}$ lattice where $T=1/(aN_{T})$ and the second term is
evaluated on an $N_{s}^{3}\times N_{t}$ lattice where $N_{t}\gg N_{s}$.

With the “130 MeV” definition of $f_{PS}$ the Gell-Mann, Oakes, Renner
relation between the condensate $\Sigma$ and other observables is

$\Sigma=\frac{m_{PS}^{2}f_{PS}^{2}}{4m_{q}}.$ (7)

In this convention, with the axial current
$A_{\mu}^{a}=\bar{\psi}\gamma_{\mu}\gamma_{5}(\tau^{a}/2)\psi$, and the
pseudoscalar density $P^{a}=\bar{\psi}\gamma_{5}(\tau^{a}/2)\psi$, the vacuum
to pseudoscalar matrix elements are $\left\langle
0|A_{0}|PS\right\rangle=m_{PS}f_{PS}$ and $\left\langle
0|P|PS\right\rangle=m_{PS}^{2}f_{PS}/(2m_{q})$ . The partial conservation of
axial current relation is

$\partial_{\mu}A_{\mu}(x,t)=2m_{q}P(x,t).$ (8)

Matrix elements of this relation define $m_{q}$ to be the Axial Ward Identity
(AWI) fermion mass, through the two-point functions

$\partial_{t}\sum_{\bf x}\left\langle A_{0}^{a}({\bf x},t){\cal
O}^{a}\right\rangle=2m_{q}\sum_{\bf x}\left\langle P^{a}({\bf x},t){\cal
O}^{a}\right\rangle,$ (9)

where ${\cal O}^{a}$ can be any convenient source.

The massive correlator in finite volume with periodic temporal boundary
conditions in temporal length $N_{t}$, saturated by a single state of mass
$m_{PS}$, is

$C(t)=\sum_{x}\left\langle P(x,t)P(0,0)\right\rangle=|\left\langle
0|P|PS\right\rangle|^{2}\frac{\cosh(m_{PS}(N_{t}/2-t))}{2m_{PS}\sinh(m_{PS}N_{t}/2)}.$
(10)

Thus its integral over the simulation volume is

$\int_{0}^{N_{t}}dtC(t)=\frac{|\left\langle
0|P|PS\right\rangle|^{2}}{m_{PS}^{2}}=\left(\frac{m_{PS}^{2}f_{PS}}{2m_{q}}\right)^{2}\frac{1}{m_{PS}^{2}}=\frac{\Sigma}{m_{q}}.$
(11)

In the passage from lattice to continuum regularization there is a factor of
$Z_{A}^{2}$ on the right hand side of Eq. 11 where
$Z_{A}=(1-(3\kappa)/(4\kappa_{c}))z_{A}$ is the tadpole improved $Z-$ factor;
$z_{A}=1+c\alpha/(4\pi)\sim 1$. $z_{A}$ is close to unity for the lattice
action used here and so we omit it from further discussion.

Borsanyi et al Borsanyi:2012uq ; Borsanyi:2015waa  wrote in the days before
the use of $t_{0}$ and so they presented plots of
$(m_{q}\left\langle\bar{\psi}\psi\right\rangle_{sub})/m_{PS}^{4}$ as a
dimensionless observable. I instead will look at the quantity

$\frac{3}{N_{c}}t_{0}^{3/2}\Sigma(T)=\frac{3}{N_{c}}t_{0}^{3/2}\times
m_{q}(\Delta_{PP}(T)-\Delta_{PP}(T=0))$ (12)

where (explicitly showing the conversion from the lattice quantity computed
with clover fermions to a continuum one)

$\Delta_{PP}(T)=\hat{\Delta}_{PP}(N_{t})(1-\frac{3\kappa}{4\kappa_{c}})^{2}.$
(13)

and the lattice quantity measured with the usual convention for the definition
of lattice field variables is

$\hat{\Delta}_{PP}(N_{t})=\sum_{t=0}^{N_{t}}\sum_{x}\left\langle
P(x,t)P(0,0)\right\rangle.$ (14)

The factor of $t_{0}^{3/2}$ in Eq. 12 renders the observable dimensionless and
the overall factor of $3/N_{c}$ is included so that plots can show collapse to
a common curve across $N_{c}$ when the condensate scales proportional to
$N_{c}$ as expected by large $N_{c}$ counting.

Data for the integrated pseudoscalar correlator is recorded in Tables 2, 3 and
4. Figure 1 shows the rescaled temperature dependent condensate from Eq. 12 at
three values of $r=(m_{PS}/m_{V})^{2}=0.63$, 0.5 and about 0.25. These plots
are sufficient to show that the finite temperature behavior of the systems is
reasonably independent of $N_{c}$. However, the curves are too featureless to
identify an inflection point as a signal for a crossover temperature.

###  III.2 Checking for effects of finite volume and nonzero lattice spacing

Two potential issues with the calculation should be discussed before
proceeding: the first is whether the simulation volume could affect the
results. The second is a check of the lattice spacing dependence of the data
presented in Fig. 1.

Finite volume effects were studied in an earlier paper, Ref. DeGrand:2016pur ,
involving some of the data sets used here. Here is a recapitulation of that
analysis, which basically follows the treatment of Sharpe Sharpe:1992ft .
Simulation volume effects typically arise from tadpole contributions due to
pseudoscalar meson emission and absorption, where the meson returns not to its
original emission point but to an image point. The pseudoscalar correlator for
a particle of mass $m$ in a box of length $L_{\mu}$ in direction $\mu$ can be
written as

$\Delta(m,x)\rightarrow\sum_{n_{\mu}}\Delta(m,x+n_{\mu}L_{\mu}).$ (15)

The infinite volume propagator, call it $\bar{\Delta}(m,x)$, is the $n=0$ term
in the sum. The finite volume tadpole is

$\Delta(m,0)=\bar{\Delta}(m,0)+\bar{I}_{1}(m,L)$ (16)

where $\bar{I}_{1}(m,L)$ is the sum over images. If a typical infinite volume
observable has a chiral expansion

$O(L=\infty)=O_{0}[1+C_{0}\frac{1}{f_{PS}^{2}}\bar{\Delta}(m,0)]$ (17)

then the finite volume correction is

$O(L)-O(L=\infty)=O_{0}[C_{0}\frac{1}{f_{PS}^{2}}\bar{I}_{1}(m,L)].$ (18)

Sharpe Sharpe:1992ft  has shown that nearest image contribution gives a useful
lower bound on the finite volume correction. It is

$I_{1}(m,L)\sim
6\left(\frac{m^{2}}{16\pi^{2}}\right)\left(\frac{8\pi}{(mL)^{3}}\right)^{1/2}\exp(-mL).$
(19)

The factor of 6 counts the closest neighboring points at positive and negative
offsets.

We can use Eq. 19, plus our tables of lattice masses and decay constants, to
check to see which of our data sets might be compromised by volume. The
result, $2I_{1}(m,L)/f_{PS}^{2}$ (the 2 is needed to convert our 130 MeV
definition of the decay constant to the standard chiral literature’s 93 MeV)
is shown in Fig. 2 for the low mass end of our data set, mostly the sets
labelled $(m_{PS}/m_{V})^{2}\sim 0.25$.

Typical values of $C_{0}$ for chiral observables in Eq. 17 are order unity
numbers and so this indicates that finite volume does not seem to be an issue
with this data set as long as one is not looking for precision at the few per
cent level. Note that the usual shorthand taking $m_{PS}L$ to be greater than
some minimum value is not really applicable to $N_{c}>3$ because of the
$1/f_{PS}^{2}$ in Eq. 18.

Figure 2: Expected finite size effect from Eq. 19, from our tabulated data.
Symbols are squares for $SU(3)$, octagons for $SU(4)$, diamonds for $SU(5)$.
Only the small fermion mass part of the data collection is shown.

Cutoff dependence can be shown by breaking out the data in Fig. 1 (and related
quantities) into plots where the $x$ axis is a measure of the lattice spacing.
I will take this measure to be $a^{2}/t_{0}$, the inverse of the lattice flow
parameter. An issue with such tests is that the bare parameters are not
carefully matched. A better way to proceed will be to do a combined fit of
observables related to chiral symmetry breaking to a version of chiral
perturbation theory which includes lattice artifacts. To do this well requires
many more zero temperature data sets than the ones used here, and is a topic
for future work.

A first indirect test is to look at the zero temperature condensate. This is
useful for a check related to Fig. 1: One expectation would be that the curves
in Fig. 1 would show a sigmoid behavior with $\Sigma(T)$ zero at low
temperature since (loosely speaking) the condensate is unchanged as the
temperature rises, then a fall as the finite temperature condensate goes to
zero, followed by a plateau at a constant value, basically the negative of the
zero temperature condensate. One could identify a crossover temperature as the
midpoint on the sigmoid.

The issue with doing this is that $T=1/(aN_{t})$ so going to high temperature
at fixed $a$ means going to smaller $N_{t}$, and at some point $N_{t}$ is so
small that lattice artifacts must appear. An alternative is to do a direct
measurement of the condensate at $T=0$ and compare it to Fig. 1. There are
modern ways based on Ref. Giusti:1998wy , but maybe a quicker (though dirtier)
way is to use the Gell Mann, Oakes, Renner relation, Eq. 7, using a single
elimination jackknife from separate fits to the AWI quark mass, the decay
constant, and the pseudoscalar mass. This will allow for a couple of checks:
first, what are the lattice spacing effects in the data? and second, will it
let us bracket the crossover region in temperature?

Fig. 3 shows $\Sigma(m)$ plotted as a function of $a^{2}/t_{0}$ for the three
collections of bare parameters used in the $(m_{PS}/m_{V})^{2}=0.63$, 0.5, and
0.25 sets of finite temperature. The different plotting symbols correspond to
different $N_{c}$ values. Lattice artifacts are clearly present [though the
reader is cautioned again, the data sets at an individual $N_{c}$ are not
matched very well in quark mass and some of the variation may be due to
intrinsic fermion mass dependence in $\Sigma(m)$]. With $\sqrt{t_{0}}$ a
nominal 0.15 fm, the deviation of the data at strong coupling from its value
at weak coupling is not surprising.

Figure 3: Condensate from the Gell-Mann - Oakes - Renner relation versus
$a^{2}/t_{0}$. Squares, octagons, and diamonds label $N_{c}=3$, 4, and 5. (a)
$(m_{PS}/m_{V})^{2}\sim 0.63$; (b) $(m_{PS}/m_{V})^{2}\sim 0.5$; (c)
$(m_{PS}/m_{V})^{2}\sim 0.25$.

Comparing Figs. 1 and 3, the zero temperature condensate appears (rather
noisily) to be $\frac{3}{N_{c}}t_{0}^{3/2}\Sigma\sim 0.03-0.04$ at $r=0.63$,
0.02 at $r=0.5$ and 0.015 at $r=0.25$. The results in Fig. 1 seem to be
plausible, in the sense that they fall to roughly the negative of the zero
temperature condensate.

A further check before returning to physics should involve the finite
temperature data sets themselves. I will break up the data for the
temperature-dependent condensate into bins in the temperature $\sqrt{t_{0}}T$
and plot it versus $a^{2}/t_{0}$. This is shown in Fig. 4. The picture is a
bit awkward to present; the data show temperature dependence within each bin
in addition to scale dependence. I have color coded the various bins and used
two sets of plotting symbols for each $N_{c}$ value to separate the different
$\sqrt{t_{0}}T$ values.

The highest temperature data sets are the ones with the largest lattice
spacing dependence. These are uniformly taken with $N_{t}=4$. Their collection
was an attempt to get high above the crossover temperature to try to see a
flattening in $\Sigma(T)$ at high temperature, while still keeping to a large
physical value of the simulation volume. Their only use is for the largest
temperature bins in a calculation of $d\Sigma(T)/dT$ in Sec. III.3. They are
not used in the pictures of screening masses in Sec. IV.

Figure 4: Temperature dependent condensate versus $a^{2}/t_{0}$. Two sets of
plotting symbols, shown in panel (a), label $N_{c}=3$, 4, and 5. The panels
are (a) $(m_{PS}/m_{V})^{2}\sim 0.63$; (b) $(m_{PS}/m_{V})^{2}\sim 0.5$; (c)
$(m_{PS}/m_{V})^{2}\sim 0.25$. In panel (a) the data proceeding down from the
top are (1) $\sqrt{t_{0}}T<0.11$ in black with the first set of symbols, (2)
$\sqrt{t_{0}}T\sim 0.18$ in purple with the second set of symbols, (3)
$\sqrt{t_{0}}T\sim 0.21$ in red with the first set of symbols, (4)
$\sqrt{t_{0}}T\sim 0.25$ in blue with the second set of symbols, (5)
$\sqrt{t_{0}}T>0.35$ in black with the first set of symbols. In panel (b),
proceeding down from the top are (1) $\sqrt{t_{0}}T<0.11$ in black with the
first set of symbols, (2) $\sqrt{t_{0}}T\sim 0.15$ in purple with the second
set of symbols, (3) $\sqrt{t_{0}}T\sim 0.23$ in red with the first set of
symbols, (4) $\sqrt{t_{0}}T>0.35$ in blue with the second set of symbols. In
panel (c) again from the top down are (1) $\sqrt{t_{0}}T\sim 0.10$ in black
with the first set of symbols, (2) $\sqrt{t_{0}}T\sim 0.13$ in purple with the
second set of symbols, (3) $\sqrt{t_{0}}T\sim 0.20$ in red with the first set
of symbols, (4) $\sqrt{t_{0}}T>0.25$ in blue with the second set of symbols.

###  III.3 Looking for a peak

It would be better to have an observable with a peak, so I looked at two more
quantities related to the condensate. One of them produced a signal. It is the
temperature derivative of $\Sigma(T)$, just taken from the finite difference

$\frac{\Delta\Sigma(T_{m})}{\Delta
T}=\frac{\Sigma(T_{1})-\Sigma(T_{2})}{T_{1}-T_{2}}$ (20)

where $T_{m}=(T_{1}+T_{2})/2$. The difference in Eq. 20 can just be taken from
the integrated pseudoscalar correlator at each value of $N_{t}$, without doing
the $T=0$ subtraction. Of course, only data sets at the same bare parameters
can be used. Fig. 5 shows this (rescaled by $N_{c}$ and the appropriate power
of $t_{0}$). The differences are (for example for $r=0.25$) $N_{t}=6-8$, 8-12,
12-16, and 16-32.

Figure 5: $\Delta\Sigma(T)/\Delta T$, rescaled by $3/N_{c}$, as a function of
temperature, in appropriate units of $t_{0}$. Squares, octagons, and diamonds
label $N_{c}=3$, 4, and 5. (a) $(m_{PS}/m_{V})^{2}\sim 0.63$; (b)
$(m_{PS}/m_{V})^{2}\sim 0.5$; (c) $(m_{PS}/m_{V})^{2}\sim 0.25$.

Fig. 5 shows a broad obvious feature at $\sqrt{t_{0}}T\sim 0.18$, pretty much
independent of $(m_{PS}/m_{V})^{2}$. This corresponds to a temperature of
about 225 MeV, an intermediate value between the quenched and physical
transition points. The figure certainly shows no difference between $N_{c}=3$,
4 and 5.

Attempts to refine this statement came to nothing. They were mostly based on
doing fits to an arbitrary peaked function, a Gaussian,

$y(x)=C_{1}\exp\left(-\frac{1}{2}\frac{(x-x_{0})^{2}}{\sigma^{2}}\right)$ (21)

with $x=\sqrt{t_{0}}T$. Fits to all data sets with the same $N_{c}$ value (for
each choice of $(m_{PS}/m_{V})^{2}$) generally had poor chi-squared, probably
due to lattice artifacts: the data span a wide range of $t_{0}$ values. Fits
to a single set of bare parameters fared better, though the issue is that
there are only four values of $N_{t}$. Some of the data sets (especially the
ones at coarse lattice spacing) do not themselves include the peak and then of
course the fit fails immediately.

I also attempted to compute a susceptibility from the time histories of
$\Delta_{PP}(T)$. This was unsuccessful so I do not report on it.

##  IV Screening masses

Meson screening masses in the scalar, pseudoscalar, vector, and axial vector
channels are taken from two-point correlation functions extending in a spatial
lattice direction. The temperature dependent pseudoscalar decay constant is
also extracted from these spatial correlators. Propagators are constructed
with composite boundary conditions to double the effective length of the
lattice Blum:2001xb ; Aoki:2005ga ; DeGrand:2007tx .

In the low temperature, chirally broken phase, the spectroscopy of screening
masses should qualitatively resemble ordinary $T=0$ spectroscopy with a light
pion and no degeneracies in the spectrum. When chiral symmetry is restored
parity partners (the pion and the scalar mesons, the vector and axial vector
mesons) should become degenerate, and all four states should become degenerate
when $U(1)_{A}$ is restored. A naive expectation for a screening mass is that
it behaves something like

$m_{H}^{2}=4\left[\left(\frac{\pi}{N_{t}}\right)^{2}+m_{q}^{2}\right]$ (22)

where $\pi/N_{t}$ is the lowest nonzero Matsubara frequency associated with
antiperiodic boundary conditions in a lattice of temporal length $N_{t}$.
Since $1/N_{t}=aT$, this gives $m_{H}=2\pi T$ at high temperature.

Results for screening masses (scaled by $\sqrt{t_{0}}$) are displayed in Fig.
6 for the pseudoscalar and scalar states and Fig. 7 for the vector an axial
vector states, They show the expected behavior. (The diagonal lines are just
$y=2\pi T$, the small $m_{q}$ (or large $T$) limit of Eq. 22.) The very dirty
signals for the $a_{0}$ (scalar) and $a_{1}$ (axial vector) mesons in the
chirally broken phase are also expected. There, the pseudoscalar meson is
light and the noise to signal ratio $\sigma(t)/C(t)\sim\exp((m_{H}-m_{\pi}))t$
is exponentially bad Lepage:1989hd ; Parisi:1983ae ; in the chirally restored
phase all states are massive and the noise to signal ratio improves.

Figure 6: Pseudoscalar and scalar meson screening masses ($\sqrt{t_{0}}m_{PS}$
and $\sqrt{t_{0}}m_{S}$) versus $\sqrt{t_{0}}T$. (a) $m_{PS}/m_{V}=0.63$, (b)
$m_{PS}/m_{V}=0.5$ (c) $m_{PS}/m_{V}=0.25$. The panel shows the plotting
symbols with PS and S for pseudoscalar and scalar mesons, respectively. The
$SU(3)$ results are shown in black, $SU(4)$ in red, and $SU(5)$ in blue. The
line is just $m_{H}=2\pi T$.  Figure 7: Vector and axial vector meson
screening mass ($\sqrt{t_{0}}m_{V}$ and $\sqrt{t_{0}}m_{A}$) versus
$\sqrt{t_{0}}T$. (a) $m_{PS}/m_{V}=0.63$, (b) $m_{PS}/m_{V}=0.5$ (c)
$m_{PS}/m_{V}=0.25$. The panel shows the plotting symbols with V and A for
vector and axial vector meson masses respectively. The $SU(3)$ results are
shown in black, $SU(4)$ in red, and $SU(5)$ in blue. The line is just
$m_{H}=2\pi T$.

The pseudoscalar and vector screening masses make a clear transition from a
temperature independent value at low $T$ to linear behavior at high $T$. This
suggests that linear fits $O=c_{1}+c_{2}T$ should show better quality (smaller
chi-squared) when fits keeping only data in one phase are included, and
$c_{2}$ would be much larger in the higher $T$ phase. The fit would
deteriorate when points in both phases were included. This would determine a
crossover temperature.

The transition can be seen in individual data sets, and no fit is needed to
see it, just a table of the mass values. In most sets the change in slope
occurs in the middle of the data set, with (typically) two or three masses
with nearly the same value at low temperature and the remaining higher
temperature masses rising linearly with temperature. Data set by data set, one
can identify a $\sqrt{t_{0}}T_{low}$ and a $\sqrt{t_{0}}T_{high}$ where masses
remain unchanged for $T\leq T_{low}$ and rise linearly for
$\sqrt{t_{0}}T\geq\sqrt{t_{0}}T_{high}$. Generally, $\sqrt{t_{0}}T_{low}$ and
$\sqrt{t_{0}}T_{high}$ are consistent between the pseudoscalar and vector mass
sets. The individual data sets give a relatively wide interval between
$\sqrt{t_{0}}T_{low}$ and $\sqrt{t_{0}}T_{high}$ simply because $T=1/(aN_{t})$
and the values of $N_{t}$ are small.

Fits to all mass values at each $N_{c}$ and $(m_{PS}/m_{V})^{2}$ value also
show jumps in chi-squared when a fit including high temperature points extends
too low, or fits to low temperature points extend too high. The issue is that
even with data in one phase, the chi-squared tended to be unacceptably large
due to the $a$ dependent variation in the masses. The best one can say is that
the crossover region is in the range $\sqrt{t_{0}}T_{c}\sim 0.15-0.24$, which
is not inconsistent with the location of the broad peak in $d\Sigma/dT$.

The last screening quantity to present is the pseudoscalar decay constant.
Here, the following alternative seems to show a sharper result than plots of
$f_{PS}$ versus temperature at fixed $r$. In the chirally broken phase,
$f_{PS}$ is nonvanishing at zero fermion mass (due to spontaneous breaking of
chiral symmetry) and rises modestly with fermion mass (due to to explicit
breaking of chiral symmetry from the fermion mass). In the chirally restored
phase there is only explicit symmetry breaking, and $f_{PS}$ should be
proportional to $m_{q}$. A plot of $f_{PS}/m_{q}$ will show collapse to a
common value when that situation occurs. Fig. 8 shows that behavior. I have
scaled $f_{PS}$ by $\sqrt{3/N_{c}}$. Here the different plotting symbols in
each panel represent different values of $(m_{PS}/m_{V})^{2}$ with the
diamonds representing the lightest fermion mass.

Figure 8: Pseudoscalar decay constant divided by the AWI fermion mass
($f_{PS}/m_{q}$) versus $\sqrt{t_{0}}T$. (a) $SU(3)$, (b) $SU(4)$, (c)
$SU(5)$. The plotting symbols correspond to $(m_{PS}/m_{V})^{2}=0.63$ for
squares, 0.5 for octagons, and 0.25 for diamonds.

##  V Polyakov line and its susceptibility

It is well known that the Polyakov line is not a sensitive observable at small
fermion mass, but I present one picture for completeness. Following Ref.
Lucini:2012wq , I define the volume averaged Polyakov line as

$l_{P}=\frac{1}{N_{s}^{3}N_{c}}\sum_{x}{\rm
Tr}\,\prod_{t=0}^{N_{t}-1}U_{4}(x,t)$ (23)

and then the susceptibility is

$\chi_{P}=N_{s}^{3}(\left\langle|l_{P}|^{2}\right\rangle-\left\langle|l_{P}|\right\rangle^{2}).$
(24)

I measure these quantities from time histories of the average Polyakov line
taking a jackknife average over the simulation run. I have only measured the
original Polyakov line, not any smoothed one.

The Polyakov line susceptibility is shown in Fig. 9. The bare Polyakov line is
not a scaling quantity, so the results for different $N_{c}$ values should not
coincide. The only comment to make about these noisy figures is that the
magnitude of the susceptibility falls with the fermion mass, as seen for
$N_{c}=3$ Clarke:2019tzf .

At small fermion masses, where one is closer to the $m_{q}=0$ critical point,
the $N_{c}=3$ Polyakov line and its derivatives do show structure associated
with the critical point Clarke:2020htu . Since my data were collected at
heavier quark masses where everything is crossover, the Polyakov line was not
a particularly useful observable.

Note that the shoulder in the Polyakov line susceptibility appears at
$\sqrt{t_{0}}T\sim 0.15$ or so, about where the screening masses begin to take
their high temperature functional form.

Figure 9: Polyakov line susceptibility as a function of temperature, scaled by
$\sqrt{t_{0}}$. Squares, octagons, and diamonds label $N_{c}=3$, 4, and 5.
Lines connect simulation results from the same bare parameters. (a)
$(m_{PS}/m_{V})^{2}\sim 0.63$; (b) $(m_{PS}/m_{V})^{2}\sim 0.5$; (c)
$(m_{PS}/m_{V})^{2}\sim 0.25$.

##  VI Conclusions

The question this study was designed to address was whether the finite
temperature crossover behavior of $SU(N_{c})$ gauge theories with $N_{f}=2$
flavors of fermions showed different behavior as the number of colors was
varied. These simulations, with low statistics and carried out on small
volumes, studied the temperature dependence of the condensate (as determined
from the volume integral of the pseudoscalar correlator) and of screening
masses and the pseudoscalar decay constant. A smooth crossover from a low
temperature confining and chirally broken phase to a high temperature chirally
restored and deconfined phase is observed. This crossover behavior is of
course not surprising given our extensive knowledge of $N_{c}=3$: the one new
result is that the temperature dependence of these quantities (when compared
at fixed values of $m_{PS}/m_{V}$) shows no observable dependence on $N_{c}$.

It was not possible to determine a crossover temperature (to the extent that
such a quantity makes sense when the crossover is smooth) but, with some
plausible assumptions, it appears to be someplace between the known $SU(3)$
result with physical quark masses ($\sqrt{t_{0}}T\sim 0.11$) and the large
$N_{c}$ pure gauge result ($\sqrt{t_{0}}T_{c}\sim 0.26$).

Of course, one can also make comparisons with earlier work. Refs.
AliKhan:2001ek ; Ejiri:2009hq  present $SU(3)$ results for $T_{c}/m_{V}$ as a
function of $m_{PS}/m_{V}$. With a nominal crossover temperature
$\sqrt{t_{0}}T_{c}\sim 0.18$ and $m_{V}$ taken from Table 1, $T_{c}/m_{V}\sim
0.20$, 0.23, 0.25 at $r=0.63$, 0.5, 0.25, in reasonably good agreement with
Refs. AliKhan:2001ek ; Ejiri:2009hq . Ref. Bornyakov:2009qh  shows a plot of
$r_{0}T_{c}$ versus $r_{0}m_{PS}$ where $r_{0}$ is the Sommer parameter
Sommer:1993ce , $r_{0}/\sqrt{t_{0}}\sim 3$. This conversion gives
$r_{0}T_{c}\sim 0.54$ at $r_{0}m_{PS}=2.2$, 1.6 and 1.1 at $r=0.63$, 0.5,
0.25, to be compared with $r_{0}T_{c}=0.48-0.6$ from Ref. Bornyakov:2009qh
for the same range of $r_{0}m_{PS}$ values.

Of the three scenarios described in the Introduction, the first two seem to be
disfavored. No evidence was seen for any first order behavior at large mass at
any $N_{c}$ studied. Of course, it could be that $N_{c}=5$ is still not “large
$N_{c}$” from the point of view of thermodynamics. Scenario two suggests
crossover behavior, which is seen, but the scenario that $T_{c}\propto f_{PS}$
is disfavored (of course, over the range of fermion masses studied) because
the crossover temperature shows no obvious $N_{c}$ separation despite the
(known) variation of the pseudoscalar decay constant with respect to $N_{c}$.
As far as the third scenario, lattice simulations show that the lowest part of
the meson spectrum show little $N_{c}$ dependence. The third scenario assumes
that the two results I have presented – a common spectrum and a common
crossover behavior – are correlated. Of course, the scenario asks that the
correlation persists high in the meson spectrum (that the density of states is
as given by Hagedorn) and lattice simulations say nothing about that.

The results in this paper are rather poor quality, but at the mass values
studied, everything is smooth. The situation at very low mass and very high
mass remains open. I think that to do more work in this area, one should not
continue to use Wilson-clover fermions – at least, not in pilot tests. The
importance of chiral symmetry, the need to subtract results from finite and
zero temperature simulations, and the fierce scaling of the cost with
simulation volume of thermodynamical observables argue in favor of using some
modern version of staggered fermions for dedicated simulations.

The obvious next target in future studies of large $N_{c}$ systems at finite
temperature could be a more direct attack on thermodynamic observables – the
internal energy, pressure, speed of sound, and related quantities. The pure
gauge calculations of Panero Panero:2009tv  are an inspiration, and they have
been extensively cited in the phenomenological and gravitational duality
literature of QCD thermodynamics. But we know that the critical behavior of
$N_{c}=3$ with dynamical fermions is different from the behavior of pure gauge
systems. Does this matter, for the questions which interest researchers in
these areas? If so, numerical simulation which includes dynamical fermions
might be a worthwhile project.

###### Acknowledgements.

Daniel Hackett participated in the earliest stages of this project. I am
grateful to Rob Pisarski for a conversation about large $N_{c}$ expectations
for QCD thermodynamics. My computer code is based on the publicly available
package of the MILC collaboration MILC . The version I use was originally
developed by Y. Shamir and B. Svetitsky. This material is based upon work
supported by the U.S. Department of Energy, Office of Science, Office of High
Energy Physics under Award Number DE-SC-0010005. Some of the computations for
this work were also carried out with resources provided by the USQCD
Collaboration, which is funded by the Office of Science of the U.S. Department
of Energy using the resources of the Fermi National Accelerator Laboratory
(Fermilab), a U.S. Department of Energy, Office of Science, HEP User Facility.
Fermilab is managed by Fermi Research Alliance, LLC (FRA), acting under
Contract No. DE- AC02-07CH11359.

$N_{c}$ | $\beta$ | $\kappa$ | $\kappa_{c}$ | $am_{q}$ | $a\,m_{PS}$ | $a\,m_{V}$ | $(m_{PS}/m_{V})^{2}$ | $t_{0}/a^{2}$ | N
---|---|---|---|---|---|---|---|---|---
$r\sim 0.63$ |  |  |  |  |  |  |  |  |
3 | 5.3 | 0.1250 | 0.12923 | 0.153 | 0.757(1) | 0.925(5) | 0.670(7) | 0.871(3) | 30
3 | 5.4 | 0.1250 | 0.12838 | 0.107 | 0.563(1) | 0.707(1) | 0.634(3) | 1.657(3) | 400
3 | 5.45 | 0.1250 | 0.12795 | 0.093 | 0.497(2) | 0.636(5) | 0.611(11) | 2.284(21) | 30
4 | 10.1 | 0.1240 | 0.12794 | 0.150 | 0.716(2) | 0.881(3) | 0.661(6) | 1.189(7) | 30
4 | 10.2 | 0.1245 | 0.12792 | 0.108 | 0.556(1) | 0.701(1) | 0.629(3) | 1.966(3) | 190
4 | 10.3 | 0.1240 | 0.12759 | 0.112 | 0.537(2) | 0.657(3) | 0.668(8) | 2.727(18) | 50
5 | 16.3 | 0.1230 | 0.12795 | 0.162 | 0.726(1) | 0.875(3) | 0.688(5) | 1.467(7) | 30
5 | 16.4 | 0.1240 | 0.12785 | 0.119 | 0.582(1) | 0.725(1) | 0.644(3) | 2.030(2) | 190
5 | 16.6 | 0.1240 | 0.12740 | 0.106 | 0.514(2) | 0.638(2) | 0.649(6) | 2.942(14) | 50
$r\sim 0.5$ |  |  |  |  |  |  |  |  |
3 | 5.25 | 0.1280 | 0.12964 | 0.080 | 0.545(2) | 0.773(4) | 0.497(6) | 0.863(2) | 30
3 | 5.4 | 0.1265 | 0.12838 | 0.058 | 0.395(1) | 0.563(2) | 0.492(4) | 2.019(5) | 400
3 | 5.45 | 0.1265 | 0.12795 | 0.044 | 0.331(2) | 0.486(6) | 0.464(13) | 2.747(17) | 91
4 | 10.0 | 0.1270 | 0.12926 | 0.104 | 0.623(3) | 0.860(6) | 0.525(9) | 0.855(2) | 40
4 | 10.2 | 0.1262 | 0.12792 | 0.054 | 0.377(1) | 0.561(2) | 0.452(4) | 2.270(4) | 190
4 | 10.3 | 0.1260 | 0.12759 | 0.049 | 0.343(3) | 0.491(5) | 0.488(13) | 3.106(17) | 90
5 | 16.2 | 0.1260 | 0.12853 | 0.087 | 0.531(1) | 0.757(4) | 0.492(6) | 1.290(7) | 30
5 | 16.4 | 0.1258 | 0.12785 | 0.063 | 0.404(1) | 0.592(2) | 0.466(4) | 2.272(5) | 190
5 | 16.6 | 0.1252 | 0.12740 | 0.069 | 0.406(2) | 0.549(3) | 0.547(8) | 3.108(19) | 40
$r\sim 0.25$ |  |  |  |  |  |  |  |  |
3 | 5.4 | 0.1276 | 0.12838 | 0.021 | 0.234(2) | 0.444(8) | 0.278(11) | 2.412(7) | 400
3 | 5.45 | 0.1273 | 0.12795 | 0.018 | 0.228(4) | 0.437(6) | 0.272(12) | 3.164(23) | 90
4 | 10.2 | 0.1272 | 0.12792 | 0.022 | 0.238(2) | 0.472(8) | 0.254(10) | 2.520(20) | 101
4 | 10.3 | 0.1270 | 0.12759 | 0.017 | 0.201(3) | 0.432(6) | 0.216(9) | 3.375(21) | 90
5 | 16.4 | 0.1270 | 0.12785 | 0.025 | 0.248(1) | 0.493(3) | 0.253(4) | 2.483(6) | 210
5 | 16.6 | 0.1268 | 0.12740 | 0.019 | 0.208(2) | 0.435(14) | 0.229(16) | 3.565(23) | 50
Table 1:  Zero temperature data sets from $16^{3}\times 32$ volumes. (The $SU(5)$ $\beta=16.6$ $\kappa=0.1268$ set is $16^{3}\times 48$.). The last column gives the number of measurement lattices in the set. $\kappa_{c}$ is the value of the hopping parameter where the Axial Ward Identity fermion mass vanishes. It is needed to perform tadpole renormalization of condensate - related quantities.  $N_{c}$ | $\beta$ | $\kappa$ | $N_{t}=4$ | $N_{t}=6$ | $N_{t}=8$ | $N_{t}=12$ | $N_{t}=32$
---|---|---|---|---|---|---|---
3 | 5.3 | 0.1250 | 16.64(2) | 18.11(6) | 18.34(5) | 18.61(12) | 18.35(9)
3 | 5.4 | 0.1250 | 16.49(2) | 17.30(4) | 18.06(5) | 18.02(8) | 18.11(8)
3 | 5.45 | 0.1250 | 16.45(2) | 17.14(3) | 17.61(3) | 17.86(5) | 17.89(9)
4 | 10.1 | 0.1240 | 22.03(2) | 23.44(4) | 24.08(5) | 24.01(9) | 23.88(8)
4 | 10.2 | 0.1245 | 21.93(2) | 22.91(2) | 23.72(4) | 23.73(6) | 23.94(6)
4 | 10.3 | 0.1240 | 21.84(1) | 22.68(2) | 23.24(3) | 23.36(4) | 23.42(5)
5 | 16.3 | 0.1230 | 27.40(2) | 28.58(3) | 29.35(3) | 29.52(8) | 29.41(13)
5 | 16.4 | 0.1240 | 27.41(2) | 28.60(3) | 29.54(4) | 29.54(5) | 29.54(7)
5 | 16.6 | 0.1240 | 27.26(1) | 28.28(2) | 28.82(2) | 29.17(4) | 29.21(4)
Table 2:  The (bare, lattice regulated) quantity $\hat{\Delta}_{PP}(N_{t})$ as defined in Eq. 14 for $r\sim 0.63$. Lattice volumes are $16^{3}\times N_{t}$.  $N_{c}$ | $\beta$ | $\kappa$ | $N_{t}=4$ | $N_{t}=6$ | $N_{t}=8$ | $N_{t}=12$ | $N_{t}=32$
---|---|---|---|---|---|---|---
3 | 5.25 | 0.1280 | 16.90(9) | 18.94(9) | 20.12(16) | 19.94(12) | 19.96(18)
3 | 5.4 | 0.1265 | 16.58(2) | 17.43(8) | 17.96(5) | 18.78(11) | 18.72(16)
3 | 5.45 | 0.1265 | 16.49(2) | 17.26(2) | 17.65(3) | 18.36(8) | 18.45(8)
4 | 10.0 | 0.1270 | 22.44(9) | 25.88(10) | 26.22(9) | 26.16(14) | 26.00(9)
4 | 10.2 | 0.1262 | 22.10(2) | 23.08(2) | 23.96(5) | 25.12(12) | 24.68(8)
4 | 10.3 | 0.1260 | 21.99(1) | 22.92(2) | 23.41(3) | 24.35(7) | 24.34(8)
5 | 16.2 | 0.1260 | 27.79(2) | 29.87(7) | 31.51(7) | 31.51(10) | 31.77(23)
5 | 16.4 | 0.1258 | 27.61(2) | 28.82(2) | 30.52(5) | 30.54(6) | 30.66(4)
5 | 16.6 | 0.1252 | 27.40(1) | 28.50(2) | 29.01(2) | 29.95(6) | 29.79(5)
Table 3:  The (bare, lattice regulated) quantity $\hat{\Delta}_{PP}(N_{t})$ at $r\sim 0.5$. Lattice volumes are $16^{3}\times N_{t}$.  $N_{c}$ | $\beta$ | $\kappa$ | $N_{t}=6$ | $N_{t}=8$ | $N_{t}=12$ | $N_{t}=16$ | $N_{t}=32$
---|---|---|---|---|---|---|---
3 | 5.4 | 0.1276 | 17.43(3) | 18.07(6) | 19.87(15) | 20.05(18) | 20.80(27)
3 | 5.45 | 0.1273 | 17.32(2) | 17.69(3) | 19.08(11) | 19.69(21) | 19.48(15)
4 | 10.2 | 0.1272 | 23.25(2) | 23.88(3) | 26.69(14) | 26.55(17) | 27.04(24)
4 | 10.3 | 0.1270 | 23.00(2) | 23.79(11) | 25.60(13) | 25.33(13) | 25.72(11)
5 | 16.4 | 0.1270 | 29.00(3) | 30.21(7) | 33.10(13) | 33.05(17) | 32.95(17)
5 | 16.6 | 0.1268 | 28.66(2) | 29.28(3) | 31.59(9) | 31.97(23) | 32.25(16)
Table 4:  The (bare, lattice regulated) quantity $\hat{\Delta}_{PP}(N_{t})$ at
$r\sim 0.25$. Lattice volumes are $16^{3}\times N_{t}$.

## References

  * (1) G. ’t Hooft, Nucl. Phys. B 72, 461 (1974). doi:10.1016/0550-3213(74)90154-0
  * (2) G. ’t Hooft, Nucl. Phys. B 75, 461 (1974). doi:10.1016/0550-3213(74)90088-1
  * (3) E. Witten, Nucl. Phys. B 160, 57-115 (1979) doi:10.1016/0550-3213(79)90232-3
  * (4) B. Lucini and M. Panero, Phys. Rept. 526, 93 (2013) doi:10.1016/j.physrep.2013.01.001 [arXiv:1210.4997 [hep-th]].
  * (5) M. Garcia Perez, Proc. Sci., LATTICE2019 (2020) 276 arXiv:2001.10859 [hep-lat].
  * (6) P. Hernández and F. Romero-López, Eur. Phys. J. A 57, no.2, 52 (2021) doi:10.1140/epja/s10050-021-00374-2 [arXiv:2012.03331 [hep-lat]].
  * (7) A. Donini, P. Hernández, C. Pena and F. Romero-López, Phys. Rev. D 94, no.11, 114511 (2016) doi:10.1103/PhysRevD.94.114511 [arXiv:1607.03262 [hep-ph]].
  * (8) A. Donini, P. Hernández, C. Pena and F. Romero-López, Eur. Phys. J. C 80, no.7, 638 (2020) doi:10.1140/epjc/s10052-020-8192-3 [arXiv:2003.10293 [hep-lat]].
  * (9) P. Hernández, C. Pena and F. Romero-López, Eur. Phys. J. C 79, no.10, 865 (2019) doi:10.1140/epjc/s10052-019-7395-y [arXiv:1907.11511 [hep-lat]].
  * (10) T. DeGrand, Phys. Rev. D 101, no.11, 114509 (2020) doi:10.1103/PhysRevD.101.114509 [arXiv:2004.09649 [hep-lat]].
  * (11) R. Kaiser and H. Leutwyler, Eur. Phys. J. C 17, 623 (2000) doi:10.1007/s100520000499 [hep-ph/0007101].
  * (12) R. D. Pisarski and F. Wilczek, Phys. Rev. D 29, 338 (1984). doi:10.1103/PhysRevD.29.338
  * (13) Thanks to Rob Pisarski for discussions about this point.
  * (14) R. Hagedorn, Nuovo Cim. A 56, 1027 (1968). doi:10.1007/BF02751614
  * (15) N. Cabibbo and G. Parisi, Phys. Lett. 59B, 67 (1975). doi:10.1016/0370-2693(75)90158-6
  * (16) B. Lucini, M. Teper and U. Wenger, JHEP 0502, 033 (2005) doi:10.1088/1126-6708/2005/02/033 [hep-lat/0502003].
  * (17) B. Lucini, A. Rago and E. Rinaldi, Phys. Lett. B 712, 279 (2012) doi:10.1016/j.physletb.2012.04.070 [arXiv:1202.6684 [hep-lat]].
  * (18) F. Cuteri, O. Philipsen, A. Schön and A. Sciarra, Phys. Rev. D 103, no.1, 014513 (2021) doi:10.1103/PhysRevD.103.014513 [arXiv:2009.14033 [hep-lat]].
  * (19) A. Hasenfratz and F. Knechtli, Phys. Rev. D 64, 034504 (2001). doi:10.1103/PhysRevD.64.034504 [hep-lat/0103029].
  * (20) A. Hasenfratz, R. Hoffmann and S. Schaefer, JHEP 0705, 029 (2007). doi:10.1088/1126-6708/2007/05/029 [hep-lat/0702028].
  * (21) T. DeGrand, Y. Shamir and B. Svetitsky, Phys. Rev. D 85, 074506 (2012). doi:10.1103/PhysRevD.85.074506 [arXiv:1202.2675 [hep-lat]].
  * (22) T. DeGrand and Y. Liu, Phys. Rev. D 94, no. 3, 034506 (2016) Erratum: [Phys. Rev. D 95, no. 1, 019902 (2017)] doi:10.1103/PhysRevD.95.019902, 10.1103/PhysRevD.94.034506 [arXiv:1606.01277 [hep-lat]].
  * (23) S. Duane and J. B. Kogut, Nucl. Phys. B 275, 398 (1986). doi:10.1016/0550-3213(86)90606-1
  * (24) S. Duane and J. B. Kogut, Phys. Rev. Lett. 55, 2774 (1985). doi:10.1103/PhysRevLett.55.2774
  * (25) S. A. Gottlieb, W. Liu, D. Toussaint, R. L. Renken and R. L. Sugar, Phys. Rev. D 35, 2531 (1987). doi:10.1103/PhysRevD.35.2531
  * (26) T. Takaishi and P. de Forcrand, Phys. Rev. E 73, 036706 (2006). doi:10.1103/PhysRevE.73.036706 [hep-lat/0505020].
  * (27) C. Urbach, K. Jansen, A. Shindler and U. Wenger, Comput. Phys. Commun. 174, 87 (2006). doi:10.1016/j.cpc.2005.08.006 [hep-lat/0506011].
  * (28) M. Hasenbusch, Phys. Lett. B 519, 177 (2001). doi:10.1016/S0370-2693(01)01102-9 [hep-lat/0107019].
  * (29) R. Narayanan and H. Neuberger, JHEP 0603, 064 (2006) doi:10.1088/1126-6708/2006/03/064 [hep-th/0601210]. M. Luscher, Commun. Math. Phys. 293, 899 (2010) doi:10.1007/s00220-009-0953-7 [arXiv:0907.5491 [hep-lat]].
  * (30) M. Lüscher, JHEP 1008, 071 (2010) Erratum: [JHEP 1403, 092 (2014)] doi:10.1007/JHEP08(2010)071, 10.1007/JHEP03(2014)092 [arXiv:1006.4518 [hep-lat]].
  * (31) R. Sommer, Nucl. Phys. B 411, 839 (1994). doi:10.1016/0550-3213(94)90473-1 [hep-lat/9310022].
  * (32) T. DeGrand, Phys. Rev. D 86, 034508 (2012) doi:10.1103/PhysRevD.86.034508 [arXiv:1205.0235 [hep-lat]].
  * (33) S. Lottini [ALPHA Collaboration], PoS LATTICE 2013, 315 (2014) doi:10.22323/1.187.0315 [arXiv:1311.3081 [hep-lat]].
  * (34) M. Bruno et al. [ALPHA Collaboration], PoS LATTICE 2013, 321 (2014) doi:10.22323/1.187.0321 [arXiv:1311.5585 [hep-lat]].
  * (35) R. Sommer, PoS LATTICE 2013, 015 (2014) doi:10.22323/1.187.0015 [arXiv:1401.3270 [hep-lat]].
  * (36) R. A. Soltz, C. DeTar, F. Karsch, S. Mukherjee and P. Vranas, Ann. Rev. Nucl. Part. Sci. 65, 379 (2015) doi:10.1146/annurev-nucl-102014-022157 [arXiv:1502.02296 [hep-lat]].
  * (37) A. Ali Khan et al. [CP-PACS Collaboration], Phys. Rev. D 64, 074510 (2001) doi:10.1103/PhysRevD.64.074510 [hep-lat/0103028].
  * (38) S. Ejiri et al. [WHOT-QCD Collaboration], Phys. Rev. D 82, 014508 (2010) doi:10.1103/PhysRevD.82.014508 [arXiv:0909.2121 [hep-lat]].
  * (39) V. G. Bornyakov, R. Horsley, S. M. Morozov, Y. Nakamura, M. I. Polikarpov, P. E. L. Rakow, G. Schierholz and T. Suzuki, Phys. Rev. D 82, 014504 (2010) doi:10.1103/PhysRevD.82.014504 [arXiv:0910.2392 [hep-lat]].
  * (40) T. DeGrand, D. C. Hackett and E. T. Neil, PoS LATTICE2018, 175 (2018) doi:10.22323/1.334.0175 [arXiv:1809.00073 [hep-lat]].
  * (41) S. Borsanyi et al., JHEP 1208, 126 (2012) doi:10.1007/JHEP08(2012)126 [arXiv:1205.0440 [hep-lat]].
  * (42) S. Borsanyi et al., Phys. Rev. D 92, no. 1, 014505 (2015) doi:10.1103/PhysRevD.92.014505 [arXiv:1504.03676 [hep-lat]].
  * (43) L. Giusti, F. Rapuano, M. Talevi and A. Vladikas, Nucl. Phys. B 538, 249-277 (1999) doi:10.1016/S0550-3213(98)00659-2 [arXiv:hep-lat/9807014 [hep-lat]].
  * (44) S. R. Sharpe, Phys. Rev. D 46, 3146-3168 (1992) doi:10.1103/PhysRevD.46.3146 [arXiv:hep-lat/9205020 [hep-lat]].
  * (45) T. Blum et al. [RBC], Phys. Rev. D 68, 114506 (2003) doi:10.1103/PhysRevD.68.114506 [arXiv:hep-lat/0110075 [hep-lat]].
  * (46) Y. Aoki, T. Blum, N. H. Christ, C. Dawson, T. Izubuchi, R. D. Mawhinney, J. Noaki, S. Ohta, K. Orginos and A. Soni, et al. Phys. Rev. D 73, 094507 (2006) doi:10.1103/PhysRevD.73.094507 [arXiv:hep-lat/0508011 [hep-lat]].
  * (47) T. DeGrand and S. Schaefer, [arXiv:0712.2914 [hep-lat]].
  * (48) G. P. Lepage, “The Analysis Of Algorithms For Lattice Field Theory,” invited lectures at the 1989 TASI summer school, Boulder CO, June 4-30, 1989. CLNS-89-971.
  * (49) G. Parisi, Phys. Rept. 103, 203-211 (1984) doi:10.1016/0370-1573(84)90081-4
  * (50) D. A. Clarke, O. Kaczmarek, F. Karsch and A. Lahiri, PoS LATTICE2019, 194 (2020) doi:10.22323/1.363.0194 [arXiv:1911.07668 [hep-lat]].
  * (51) D. A. Clarke, O. Kaczmarek, F. Karsch, A. Lahiri and M. Sarkar, Phys. Rev. D 103, no.1, L011501 (2021) doi:10.1103/PhysRevD.103.L011501 [arXiv:2008.11678 [hep-lat]].
  * (52) M. Panero, Phys. Rev. Lett. 103, 232001 (2009) doi:10.1103/PhysRevLett.103.232001 [arXiv:0907.3719 [hep-lat]].
  * (53) https://github.com/milc-qcd/milc_qcd/