2022/9/27
TGSW 2022

Critical point in heavy-quark QCD
at finite temperature
Kazuyuki Kanaya1) with Ryo Ashikawa2), Shinji Ejiri3),
Masakiyo Kitazawa2,4), Hiroshi Suzuki5), Naoki Wakabayashi3)(WHOT-QCD Collaboration)1) Univ. Tsukuba, 2) Osaka Univ., 3) Niigata Univ., 4) Kyoto Univ., 5) Kyushu Univ.

University of Tsukuba

QCD : the fundamental theory of quarks

Big Bang≈ +0.1 m sec (1012 K)
Phase transition:
hot quark matter (QGP)=> ordinary matter (protons etc.)

Important as the initialcondition of our Universe(atoms, molecules, stars, life, ...).

Nature of T > 0 QCD transition as function of mq's
The traditional picture given by this Columbia plot

The physical point locates in the crossover region.
Nature of the QCD transition off the phys. pt. is important<= properties at the phys. pt. affected by scaling of nearby critical points.

Nature of T > 0 QCD transition as function of mq's
The traditional picture given by this Columbia plot
SU(2)×SU(2)×U(1) σZ(3) Potts
Z(3) Potts + ext. field

SU(3)×SU(3)×U(1) σ

Universality arguments with effective chiral σ model around the chiral limits
Pisarski-Wilczek, Phys. Rev. D 29 (1984) 338
Wilczek, Int. J. Mod. Phys. A7 (1992) 3911
Rajagopal-Wilczek Nucl. Phys. B399 (1993) 395=> Nf = 2 : 2nd order in the O(4) univ. class // Nf ≥ 3 : 1st orderNature of T > 0 QCD transition as function of mq's
The traditional picture given by this Columbia plot is still under many discussions.

If U(1)A [broken by anomaly at all T's] is effectively restored around Tc=> Nf = 2 chiral trans. is either 2nd or 1st=> Nf = 3 1st order region may be smaller [anomaly was a source of ϕ 3pot.]

Nature of T > 0 QCD transition as function of mq's
The traditional picture given by this Columbia plot is still under many discussions...
2nd (?)
Z(2)

CD
Q

2nd
Z(2)

N

(?)

Phys.
point

1st

3

ms

crossoverF =*SU(3) YM

NF = 1 QCD

8Is the 2-flavor
chiral limit 2ndorder?

NF = 2 QCD

2nd
Z(2)

NF = 2+1 QCD

D'Elia+, PRD 72 (2005);
Cossu+, Lattice2008;
Philipsen+Pinke, PRD 93(2016)

See also

1st

mud

8

0
0

1st order for Nf = 2
on coarse lattices
with unimproved
lattice quarks

S.Aoki+, PRD 86 (2012)

If U(1)A [broken by anomaly at all T's] is effectively restored around Tc=> Nf = 2 chiral trans. is either 2nd or 1st=> Nf = 3 1st order region may be smaller [anomaly was a source of ϕ 3pot.]

Nature of T > 0 QCD transition as function of mq's
The traditional picture given by this Columbia plot is still under many discussions...
Is the 2-flavor
chiral limit 2ndorder?

Can the 3-flavor
1st order region
survive in thecontinuum limit?

Light-quark 1st order region for Nf ≈ 3 shrinks with a → 0 (Nt → ∞)Kuramashi+, PRD 101 (2020): improved Wilson, Nt=12
Dini+, PRD 105 (2022): improved KS, Nt=8

Nature of T > 0 QCD transition as function of mq's
The traditional picture given by this Columbia plot is still under many discussions...
Is the 2-flavor
chiral limit 2ndorder?

*Can the 3-flavor
1st order region
survive in thecontinuum limit?

Light-quark 1st order region for Nf ≈ 3 shrinks with a → 0 (Nt → ∞)Kuramashi+, PRD 101 (2020): improved Wilson, Nt=12
Dini+, PRD 105 (2022): improved KS, Nt=8

Nature of T > 0 QCD transition as function of mq's
The traditional picture given by this Columbia plot is still under many discussions...
Is the 2-flavor
chiral limit 2ndorder?

*

Where is the Critical Pointin the thermodynamic +continuum limit?

Can the 3-flavor
1st order region
survive in thecontinuum limit?

ou
in r w
th o
is rk
ta
lk

Recent studies on the location of CP in heavy-quark QCD
Saito+ (WQHOT-QCD), PRD (2011/2014): HPE LO, Nt=4, Ns/Nt=6
Ejiri+ (WHOT-QCD), PRD (2020): HPE eff-NLO, Nt=6, Ns/Nt=4–6; Nt=8, Ns/Nt=3
Cuteri+, PRD (2021): Nf=2, fullQCD, Nt=6,8,10, Ns/Nt=4–7(10)=> We still have strong cutoff & spatial volume dependences.

Motivations
Binder cumulant analysis based on the Z(2) FSS expected around CP
So far, however, identification of
the Z(2) FSS is not a simple task--- removal of many high-T datarequired / correction terms to the
FSS introduced.

Nt=6
removedexcluded

These make the analyses slightly ambiguous& call careful systematic error estimations.

Z(2) FSS fit

Cuteri+, PRD (2021)=> Simulations with larger spatial volumes & high statistics
to identify the FSS more clearly.
=> Multi-point reweighting to vary coupling parameters continuously.
This talk is based on
Kiyohara+ (WQHOT-QCD), Phys.Rrev.D (2021) [DOI: 10.1103/PhysRevD.104.114509]
Wakabayashi+ (WHOT-QCD), Prog.Theor.Exp.Phys. (2022) [DOI: 10.1093/ptep/ptac019]
Ashikawa+ (WHOT-QCD), ongoing

Lattice setup
1
T =
Nt a

Nt a
a
Ns a = L

Nt : lattice size in the euclidian
time direction
a : lattice spacing

For a given T,
continuum limit: Nt → ∞Spatial system size L = Ns a <= we control it by the "aspect ratio" Ns / Nt = L T.
previous studies : L T ≈ 4 – 7 (10)our sudy :L T up to 12 or 15
To suppress compt. costs on large spatial volumes, we first revisited Nt=4 [Kiyohara+].
We are now extending the study to Nt=6 [Ashikawa+, ongoing].

Lattice setup
Our lattice action: plaquette gauge + standard Wilson quarks
Wilson quark kernel:
hopping term

Quark contribution to the effective action:
closed loops of B with κ [loop length]

Hopping Parameter Expansion to reduce simulation costs for large spatial volumes
LO:

NLO:

bent Polyakov loops Ωi

Polyakov loop Ω

B

B

B

B

κ4

κ Nt

κ6

κ Nt+2HPE ≈ 1/(amq) expansion
HPE worsens with a → 0 (Nt → ∞)=> higher order terms required with Nt → ∞.

Simulation incorporating LO + NLO meas.'s
Kiyohara, Kitazawa, Ejiri, KK (WHOT-QCD), PRD 104 (2021)

LO incorporated in the configuration generation

β → β* = β + 48Nf κ 4
λ∑x

Ω(x) term in the effective action (λ = 48Nf Nt κ 4 for Nt=4)
can be incorporated in PHB+OR parallel simulation efficiently
by keeping all temporal sites within a node=> Simulation cost ≤ 1/100 of full-QCD simulations : large spatial volume enabled
NLO incorporated in the measurements through multi-point reweighting
NLÔ⟨O(U)⟩
β,λ =

̂
⟨O(U)
e −δSLO−SNLO(β,λ)⟩LOβ̃,λ˜⟨e −δSg+LO−SNLO(β,λ)⟩LOβ̃,λ˜˜δSg+LO = Sg+LO(β, λ) − Sg+LO( β̃, λ)˜ => measure at (β*, λ)Simulations at several ( β̃*, λ)Overlap problem resolved by the inclusion of LO in
configuration generations <= essential on spatiallylarge lattices in this study

403x4

Study on Nt = 4 lattices
Kiyohara, Kitazawa, Ejiri, KK (WHOT-QCD), PRD 104 (2021)

˜ with ~106 meas.]
Simulations: Nt=4, Ns/Nt = LT = 6, 8, 9, 10, 12, each 3-6 x [( β̃*, λ)L = spatial lattice size, λ = 48Nf Nt κ 4 for Nt=4

around the transition line

History of ΩR = ReΩ

403x4

483x4

Distribution of ΩR on the transition line

4
⟨Ω
R⟩c
Ω
+ 3 along the transition line in the (β, κ) plane.
=> Binder cumulant B4 =
2
2
⟨ΩR⟩c

Study on Nt = 4 lattices
Kiyohara, Kitazawa, Ejiri, KK (WHOT-QCD), PRD 104 (2021)

Results at Nt=4 with HPE up to NLO
Z(2) FSS fits

Precision much improved over previous studies
Ns/Nt = LT ≥ 9 required for Z(2) FSS

B4Ω = 1.630(24)(2) using Ns/Nt ≥ 9, consistent with Z(2) value 1.604 within ≈ 1σ
λc = 0.00503(14)(2) [κc = 0.0603(4)] for Nt=4, Nf=2(cf.) Ejiri+ PRD(2020): κc = 0.0640(10) with eff. NLO

Scope and convergence of HPE
Wakabayashi, Ejiri, KK, Kitazawa (WHOT-QCD), PTEP (2022)

Are the effects of further higher orders of HPE really negligible?
Quark contribution to the effective action:ln det M(κ) = Nsite∑nDnκ n,

Wilson-type loops
Dn = W(n) +∑mloops of length nnoise average
−1
−1
n
Dn =
Tr[B ] ≈⟨⟨η †B nη⟩⟩noises
NsitenNsiten
Bxy =

†
(1
−
γ
)U
δ
+
(1
+
γ
)U
δy,x−μ]̂
̂
μ
x,μ
y,x+
μ
μ
y,μ
[
∑

W(4) = 96NcP,̂

W(6) = 256Nc(3Ŵ rec + 6Ŵ chair + 2Ŵ crown)μLm(Nt, n) = W(n) + L(Nt, n)

Polyakov-type loops
with m-windings

4Nc2Nt
L1(Nt, Nt) =ReΩ̂Nt
L1(Nt, Nt + 2) = 12Nc2Nt∑kReΩ̂ kWe developed a method to separately evaluate W(n) and Lm(Nt, n) from Dn
by combing the results with various twisted boundary conditions.

Scope and convergence of HPE
Wakabayashi, Ejiri, KK, Kitazawa (WHOT-QCD), PTEP (2022)Ŵ i, Pĵ in W(n) and Lm(Nt, n) take their maximum value 1 when we set Ux,μ = 1
In this case, we can calculate W(n) and Lm(Nt, n) analytically up to high orders.

=> Worst convergent case of HPE can be studied by combining them.
Convergence radius (lower bound for the Ux,μ ≠ 1 case)
L(Nt, n) =

W(n)

∑
m

Lm

Cauchy-Hadamard's
Nt = 26

Cauchy-Hadamard's
d'Alembert's

1/8

Nt = 4W(n)Convergence radius ⟶ 1/8, i.e. convergent up to the chiral limit.
n→∞<= free Wilson quarks when Ux,μ = 1=> HPE reliable at any mq, when sufficiently high orders are taken.

Scope and convergence of HPE
Wakabayashi, Ejiri, KK, Kitazawa (WHOT-QCD), PTEP (2022)

To which order we need to incorporate? <= depends on the value of κ
Deviation due to truncation (in the worst convergent case):

L(Nt, n)

W(n)

up to
NLO

up to NLO

up to LO

up to LOFor Nt=4: κc = 0.0603(4) [Kiyohara+ ('21)]=> LO may have at worst ≈10% error, NLO good enough
For Nt=6: κc = 0.0877(9) [Cuteri+ ('22)], 0.1286(40) [Ejiri+ ('20) using eff. pot.]=> NLO is ≥93% accurate. remaining error can be removed by NNLO or higher
For Nt=8: κc = 0.1135(8) [Cuteri+ ('22)]=> NNLO needed for ≥95% accuracy

Effective method to incorporate high orders
Wakabayashi, Ejiri, KK, Kitazawa (WHOT-QCD), PTEP (2022)

Calcuration of high order term becomes quickly difficult with increasing n.
We extend the idea of the effective NLO method [Ejiri+ ('20)] to high orders.
Basic observation: strong correlation of Wilson/Polyakov-type loops among different n.

Distribution of L(Nt, n) vs. the Polyakov loop Ω
qQCD simulation on 323x(6, 8), blue/red slightly below/above ßtransnormalized by the Uxµ=1 result L0
0.06

0

0.02

0.04

0.02

-0.02
-0.02

0.06

Re Ω

0.02
0

0

0.02

0.04

L/L0

0.04

0.02
0

-0.02
-0.02

0.06

0

0.02

0.04

-0.02
-0.02

0.06

0.04

0.04

n=14

0.03

0.03

0.02

0.02

0.02

-0.01
-0.01

0

0.01 0.02 0.03 0.04

Re Ω

0.01

0.01

0

0

-0.01
-0.01

0

0.01 0.02 0.03 0.04

Re Ω

0.02

0.04

0.06

Re Ω

0.03

0

0

Re Ω

n=10

0.01

n=18

0.04

Re Ω

Nt=8

0.06

0.04

0
0

n=14

L/L0

0.02

-0.02
-0.02

0.06

0.04

L/L0

L/L0

0.04

n=10

L/L0

n=6

L/L0

0.06

L/L0

Nt=6

-0.01
-0.01

n=18

0

0.01 0.02 0.03 0.04

Re Ω

Effective method to incorporate high orders

Fig. 9. As Fig. 8 but on a 323 × 8 lattice at β = 6.0320 (blue) and 6.0660 (red). The top lef…, bottom right panels show the results of n = 10, 12, …, 20, respectively.

Wakabayashi, Ejiri, KK, Kitazawa (WHOT-QCD), PTEP (2022)
Table 3. Coefficients cn of Eq. (51) for n = Nt –20 on Nt = 6
and 8 lattices.

This linear correlation suggests us to approximate

L(Nt, n) ≈ L 0(Nt, n) cn ReΩ̂
known from Uxµ=1

Nt = 6

Nt = 8c6
10.8112(20)(7)1c8
0.6280(15)(3)0.8327(114)(95)c100.4736(29)(15)0.6408(36)(27)c12
0.3609(26)(11)0.4841(22)(10)c14
PTEP 2022, 033B05 c16
N. Wakabayashi et al.
0.3106(25)(10)0.3616(21)(6)1.0159(90)(33)
0.2679(16)(3)c18
Table 5. Coefficients dcn20and f n of Eq. (64)obtained on Nt = 6 and 8 lattices.
The numbers in the first− 0.02771(57)(13)0.2020(13)(2)

slope
measured bysimulation
parentheses are the statistical errors by the jackknife method, and those in the second parentheses are
the errors of the noise method.

W(n) ≈ W (n) (dnP̂ + fn)
0

0.15

n=10

10
1.3835(156)(10)− 0.6590(91)(6)1.3433(117)(8)− 0.6367(70)(5)2⟨L(N1.1752(130)(7)⟩ 0.6025(78)(4)1
121.2140(178)(9)− 0.6235(103)(5)t , n )Re #̂/$L −
=
c
.
n
14
1.0256(196)(9)− 0.5533(114)(5)
0.9825(141)(7)0 (N , n )2 /$L2 ⟩ − 0.5303(85)(4)
L
⟨(Re
#̂)
t
16
0.8607(219)(9)− 0.4811(127)(5)0.8052(153)(8)− 0.4512(92)(5)0.7481(258)(10)0.4296(150)(6)− 0.3870(103)(5)The error18propagationfrom $L−to
cn is given by0.6698(173)(9)200.7290(337)(12)− 0.4275(196)(7)0.6071(219)(12)"# − 0.3606(131)(7)

W/W

0

0.148
0.146

though the correlation
weaker than L(Nt,n)

0.144
0.142
0.14
0.578

! f n (Nt = 6)nd n (Nt = 6)
d n (Nt = 8)
f n (Nt = 8)n and Nt . Minimizing χ 2 = i [(L(i) − L0 (Nt , n ) cn Re #̂(i))/$L(i)]2 , with L(i) t
4
1
0
1
0
ith configuration
and $L(i) theerror of L(i)−due
to the noise m
L(Nt , n ) on
6 the1.3625(73)(12)− 0.4070(42)(7)1.3366(66)(8)0.3922(39)(5)− 0.6089(72)(6)1.4256(96)(8)− 0.5869(57)(5)
is given by
best value8 of cn 1.4644(123)(11)

0.58

0.582

0.584

P

(Re #̂)2
1$cn = √
$L2
Nconf |L0 (Nt , n#)|

−1/2

,

"
=> Higher order effects can be effectively incorporated
in+theLO simulation
byRe ".ˆ
L (N , n)e
L (N , n)eκ = L (N , N ) κ

nmax

nmax !
∞
!

+
m

t

+mµ/T

−
m

t

−mµ/T

n
c

0

t

t

n
c,LO

(67)

is tthenumber of configurations.
where Nconfn=N
m=1
nmax
TheCorresponding
results of the
fits
are
shown
by
the
green
lines
Figs.
9. We
summ
to the effective theory discussed in Sect.
4.3in
based
on 8theandstrongcorrelation
0nn
let theus assume
Table 3, inloops,
which
numbers
sultsamongfor cthen inPolyakov-typef thatt in the first parenthesestn are the stati

1
β → β* = β + Nf
W 0(n) dn κ
6 ∑n=4

λ → λ* = N N∑

L (N , n) c κ

1 0
ˆ
L+
n) cn,m Re ",(68)m (Nt , n) ≈ Lm (Nt ,n=N
2
t
19/27
where cn, m is a constant to be determined by a Monte
Carlo simulation. Then, Eq. (67) becomes

Extension to non-degenerate cases (Nf=2+1 etc.)
straightforward.
$
1 !!
nmax

∞

%
n
L0m (Nt , n) cn,m e+mµ/T + c∗n,m e−mµ/T κcn = L0 (Nt , Nt ) κc,LO.

(69)

Study on Nt = 6 lattices
Ashikawa+ (WHOT-QCD), ongoing

Nt=6, Ns/Nt = LT = 6, (7,) 8, 9, 10, 12, 15 ongoing
Status of B4Ω with NLO:

λ = 128Nf Nt κ 6 for Nt=6, Nf=2, NLO

Preliminary:B4Ω ∼ 1.61 − 1.62 with Ns/Nt = 12–15(cf.) Z(2) value = 1.604
λc ∼ 0.00097 − 0.00101 => κc ∼ 0.093 NLO => κc ∼ 0.0905 up to 20th order(cf.) κc = 0.0877(9) by a full QCD simulation [Cuteri+ ('22)]

Nt = 4 vs. Nt = 6
Ashikawa+ (WHOT-QCD), ongoing

Violation of FSS larger on finer lattice => larger Ns/Nt = LT required

Origin of the violation:
Contamination of remnant Z(3) distribution
from mq=∞ ?
Mixing with energy-like op. in the scaling ??

Conclusion & outlook
HPE provides us with a reliable and powerful way to study QCD
with heavy quarks
Convergent up to chiral limit + enable large Ns/Nt simul.'s + analytic in Nf
up to κc of Nt=4, Nf=2 : LO: ≥90% / NLO: ≥99% accurate
around κc of Nt=6, Nf=2 : NLO: ≥93% accurate
Higher orders needed to remove remaining truncation error and for Nt≥8.

Effective method to incorporate high orders developed=> reduce simulation cost largely
At Nt=4, Ns/Nt≥9 needed for Z(2) FSS.
=> NLO study of B4Ω : κc = 0.0603(4) for Nf=2
At Nt=6, larger violation of FSS, require larger Ns/Nt
Preliminary with Ns/Nt≥12, κc ∼ 0.090 including high orders.
Nt=6 ongoing: more statistics.
HPE powerful also at finite densities: in progressWe miss our best friend+collaborator

Yusuke Taniguchi
who passed away on July 22, 2022.
1968–2022

backup slides

Study on Nt = 4 lattices
Kiyohara, Kitazawa, Ejiri, KK (WHOT-QCD), PRD 104 (2021)

Comparison with LO analysis=> effects of NLO corrections

LO ≈ NLO with Ns/Nt=LT ≥ 9
Shift due to NLO is small ( ≈ 2.6%), suggesting LO dominance around κc for Nt=4=> previous Nt=4 LO results seems OKRcÞ as ΩR at the two local minima
the function
valuesṼðxof
0.00498(14)(2)0.00503(14)(2)VðΩR ; λc ; LTÞ. No further adjustments3−y are made in
0.00505(14)(2)VðΩR ; λc ; LTÞ ¼ ṼððΩR − hΩR iÞðLTÞ h Þ; ð45Þofthefigure. The error bands do not include the uncertainty of theadditive
lower
panel
an enlargementof theis subtracted
from
up toconstant.
an additive The
constant,wherehΩR i isþ dðLTÞy −y Þ: region
ΩRindicatedto adjust theby
centerthe distribution.
the ofdottedrectangle in the upper panel.
To seeif thebehaviorofEq.
(45) is results
satisfied,N … ð43Þ
PHYS.
REV.
Dthe
104,
114509(2021)
From Fig.
12,
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for LT ¼
Kiyohara,
PRD114509
104 (2021)(2021)KIYOHARA,KITAZAWA,
EJIRI,KANAYA Kitazawa, Ejiri, KK (WHOT-QCD),
PHYS. REV. D 104,
we show in Fig. 12 the effective potential VðΩR ; λc ;andLTÞ
8–12at agree
errors with
the CP almost
obtained atcompletely
five values of within
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ixing, we try fits3−yð1Þthe scaling
Eq.
(45).
This
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, ofwhere
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0.518.
termVðΩwe; λcset; LTÞthat VðΩ; λc ; LTÞþRs of BΩ
4 at three λVðΩR ; λ; LTÞ ¼ − ln
pðΩ
Þ
;
ð44Þ
ð2Þadjust R
ð1Þ
ð1Þ
For inthethefigure,wearbitrary
constant; λc ; LTÞthe
¼ λ;LT0,where
and(> Ωthe) are
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shows figure.
a cleardeviation
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for
ima. The model
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lowerdependence
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From
Eq.
(14),
nst
the data. As
a [27,29].
largervolumes
at additive
Ω
≪
Ω
and
Ω
≫
Ω
while
it
R
R
region indicated by the dotted rectangle in, theupper panel.
with
Eq. (43)byð1Þtheð2Þ for LT ¼
FrominFig.
12,
we findΩthatnumerical
results
VðΩ
;
λ;
LTÞ
at
the
CP
will
be
described
byasingleagrees
well
with
them
the
range
≲
Ω
≲
Ω
. ThisR
2
Rfind that χ still
8–12 agree almost completely within the errors with FIG. 13. Positions of peaks of the distribution functio
suggests
from the Zð2Þ FSS by lattices
nctionÞ as that the deviation
becomes Ṽðxlargerthe scaling relation Eq. (45). This result nicely supportss also found thatwith small LT, discussed
intheSec.
B, is due
thatCP.inFrom
the the measured on the transition line.
the FSS inZð2ÞIV
universality
classtoat then the error for all
upper panelofÞFig.
12,smallwe noteLT.
that the effective potential
tails
of
the
distribution
pðΩ
for
R
3−y
h
suggests that thetbackup slideshha clear deviationfromthe results for− ¼hΩ6 RshowsiÞðLTÞ
Þ;
ð45ÞVðΩR ; λc ; LTÞ ¼ ṼððΩforR LT
ð1Þð2ÞΩR is negligiblen.

larger volumes at ΩR ≪ Ω and ΩR ≫ Ω , while itwell withthemin theminima
range Ωð1Þ ≲ ΩR ≲ Ωð2Þ . This
B. Gapagrees
between
the
two
(1)
(2)
suggestshΩthat thedeviationfrom the Zð2ÞFSS by latticesiissubtracted
fromto an additive constant, whereR
withargument
small LT, discussed
Sec. IVis dueto that in the
Using Eq. (13), theof Sec.inVA
onB,theeffective
to adjustthe center of thetailsdistribution.
of the distribution pðΩR Þ for small LT.
Rl results
for theΩΩpotential can be extended away from the CP along theweshowifthethe scaling behavior of Eq. (45) is satisfied,Toalsoseetransition line. In this subsection,
study
betweenB. GapwebetweenthethetwogapminimaN f dependence ofehatshowin theFig.two12local
the minima
effectiveVðΩRof; λSec.
LTÞc ; VAof potentialVðΩ
λc ;argument
LTÞ,
Using
Eq. (13),on the effective
N f enters theR ; thepotential can
extended
from the CP along thethethecombination
CP obtained at five values
ofbeLT,
as awaya function
transition line.
In this subsection,
we study the gap between
β % . Therefore, λc
ð2Þ
ð1Þ
3−y h
−Ω
:
ΔΩ
¼
Ω
ðΩ
−
hΩ
iÞðLTÞ
,
where
we
set
3
−
y
¼
0.518. ð46Þthe two local minima of VðΩhR ; λc ; LTÞ,R to obtainR
ows us2ore of κthefigure,we adjust the arbitrary
constant
c at N f ¼
ð2Þ
ð1Þ
−Ω
:
ΔΩ
¼
Ω
onger applicable
According to Eq. (15), this quantity should behave
aroundð46Þ
rical analyses theat CP asAccording to Eq. (15), this quantity should behave aroundt the results of λc
the CP ass that the NLO
yh −3 ΔΩ̃ððλ −λ ÞðLTÞ1=ν Þ;ΔΩðλ; LTÞ ¼ ðLTÞΔΩðλ; LTÞ
¼ ðLTÞy −c3 ΔΩ̃ððλ −λc ÞðLTÞ1=ν Þ;ð47Þð47Þ
h

N OF ΩR

that pðΩR Þ obeys the FSS.
Þ Inobeys
the FSS.
provided that pðΩRprovided
Fig. 13 we show the λ dependence of Ωð1Þ and Ωð2Þ . As

behavior of the

ð1Þ

ð2Þ

FIG. 13. Positions of peaks of the distribution function pðΩR Þ
measured on the transition line.

backup slides

PTEP 2022, 033B05

N. Wakabayashi et al.

Wakabayashi,(WHOT-QCD), PTEP (2022)
Table 1. Wilson loop terms
W0 (n) for theEjiri,case KK,
Ux,µ Kitazawa= 1.
W0 (4)
288 W0 (20)1.54422361 × 1014
W0 (36)−5.58410362 × 1027
W0 (6)
8448 W0 (22)2.83682900 × 1015
W0 (38)−2.91018925 × 1029
W0 (8)
245952 W0 (24)
−2.40028584 × 1016
W0 (40)−1.50223497 × 1031
W0 (10)7372800 W0 (26)
−6.88836562 × 1018
W0 (42)−7.71380102 × 1032
W0 (12)225232896 W0 (28)
−5.41133954 × 1020
W0 (44)−3.95168998 × 1034
W0 (14)6906175488 W0 (30)
−3.39122203 × 1022
W0 (46)−2.02386871 × 1036
PTEP
2022, 033B05
N. Wakabayashi et al.
W0 (16)208431502848 W0 (32)−1.93668514 × 1024
W0 (48)−1.03783044 × 1038
W0 (18)6.00259179 × 1012 W0 (34)−1.05424635 × 1026
W0 (50)−5.33468075 × 1039
Table 2. Polyakov-type loop expansion terms L0m (Nt , n) for the case Ux,µ = 1 .

0

+

3

Downloaded from https://academic.oup.com/ptep/article/2022/3/033B05/6
Downloaded from https://academic.oup.

L01 (4, 4)
48 L01 (10, 10)1228.8L01 (18, 18)
174762.67
000L1 (4, 6)1728 L1 (10, 12)331776
L1 (18, 20)
160432128
00 Nt
0L1 (4, 8)4579212 ×
L12(10, 14)52862976
L1 (18, 22)75497472000
00Nt
L1 (Nt , N, Nt + 2) = 36 L
×01 (18,2 (Nt) =t − 1). 2.36626 × (33)
L01 (4, 10)645120L01 (10,,16) L1 (Nt6258180096
24)
1013
N
L01 (4, 12)−26224128 L01t (10, 18)
5.99330 × 1011
L01 (18, 26)
5.50232 × 10150L14)present case−3201067008
L01we(10,can20)also 4.87727
× 1013
L01 (18, 28)
1.01809 × 1018
Forof Ux,µ = 1,
show that
1 (4,theL01 (4, 16)−2.14087 × 1011 L01 (10, 22)
3.47446 × 1015
L01 (18, 30)
1.57315 × 1020
017L01 (4, 18)−1.19007 × 1013 L
2.20156
L01 (20, 20)
629145.6
0 1 (10, 24)
m−1 0× 10
L1 (mN19t , n ). 0(34)
014 Lm 0(Nt , n ) = (−1)
L1 (4, 20)−6.00757 × 10
L1 (10, 26)
1.24531 × 10
L1 (20, 22)717225984
L01 (4, 22)−2.84486 × 1016 L01 (10, 28)
6.20798 × 1020
L01 (20, 24)
4.11140 × 1011
00022 substituting
Using
this property,
we×canfor m >
Eq. (33).
L−1.281051018calculate
L01 (10,some30) Lm2.59861
×1
10by
L01 (20, 26)L1 from1.54445
× 1014
1 (4, 24)
0
16
= 1 ona lattice
Nt and Lsufficiently
large 4.24543
Ns . Sincethecalculate
Dn for U
L01We(4, 26)−5.50874
×x,µ1019
L01 (12,12) with various4096
× 10
1 (20, 28)
0
21
0
0
L
28)
−2.25576
× 10
14)9.17892 ×
1018
link
are uniform
in thisLcase,of 1622016
one diagonalLelementindex
1 (4,variables
1 (12,calculation
1 (20, 30) for the position
L01 (4, 30)−8.69402 × 1022 L01 (12, 16)
360603648
L01 (22, 22)2287802.18
is 0sufficient. For the color and spinor
indexes, we take their trace.0More concretely, we
prepare0
L1 (6, 6)
128 L1 (12, 18)57416810496
L1 (22, 24)
3170893824⃗ei having
a non-vanishingelement
a 26)
position, andat theith
aL0pseudo-fermion
field11520
12
L01 (12,20)
7.19497
× 1012 onlyL01at(22,2.17478
× 10
1 (6, 8)† n 14n0]ii = ⃗e×
B10⃗eicombinationof the colorand spinor
wecalculateelement [B9.64167
L716544
L01 (12,indexes
22)
7.51820
× 1014the diagonal
L01 (22, 28)i
1 (6, 10)
!4Nc16 n000LL1 (12,24) D6.80443
10 [B ]ii .L1 (22, 30)
3.09123 × 1017
forall12)
combinations35891712
of i. We thencalculate
1 (6,n from × i=1
0
18
0
L01To
(6, compute
14)1464910848
L
(12,
26)
5.46987×10L
8388608
1
lattices with Nt = 4, 6,20…, n max10 (24,
/2 +24)
1, the spatial latticesize
Dn up to n = n max on
00
L1 (6, 16)43817011200 L1 (12, 28)
3.96931
×
10
L
(24,
26)
13891534848
1+
N0s(6,= 18)
n max + 2 is3.17933
sufficient.
We
11 also0 compute Dn with the periodic
22
0boundary condition. For n <
L
×
10
L
(12,
30)
2.62442
×10
L
1.12307 × 1013
1
1
1 (24, 28)backup slides
Wakabayashi, Ejiri, KK, Kitazawa (WHOT-QCD), PTEP (2022)

PTEP 2022, 033B05

N. Wakabayashi et al.

Fig. 11. Effective critical point κ c, eff in two-flavor QCD for Nt = 6 as a function of nmax . The black circle
and red square symbols are for κ c, LO obtained on a 243 × 6 and a 323 × 6 lattice, respectively.
Table 4. Effective critical point κ c, eff in degenerate
Nf -flavor QCD on Nt = 6 lattices.

20

0.7290(337)(12)

backup!slides!"
nmax

PTEP 2022, 033B05∞− 0.4275(196)(7)0.6071(21

#

+mµ/T
−
−mµ/T
n
L+
(N
,
n)e
+
L
(N
,
n)e
κ
t
t
m
c =L
N.mWakabayashi et al.

Wakabayashi, Ejiri, KK, Kitazawa (WHOT-QCD), PTEP (2022)
n=Nt m=1

Corresponding to the effective theory discussed in Sect. 4.3
among the Polyakov-type loops, let us assume that
1 0+ˆ
Lm (Nt , n) ≈ Lm (Nt , n) cn,m Re "2
where cn, m is a constant to be determined by a Monte Carlo simu

nmax !
∞
$
% n
1!
0
+mµ/T
∗
−mµ/T
Lm (Nt , n) cn,m e+ cn,m e
κc =
2 n=N
t m=1

The leading-order calculation of Ref. [8] for Nt = 4 correspom ≥ 2 are absent. To judge the magnitude of the effect from hig
the case of the worst convergence with Ux,µ = 1. In this case, b'&
nmax !
∞!
mµ n0κc = L0 (Nt ,Lm (Nt , n) cosh
T
n=N
t m=1

Though L0m (Nt , n) decreases as m increases, cosh (mµ/T) may
Fig. 14. Upper bound of µ/T suchthat higher-m terms
are small,given
Eq. (71).
forinmµ/T> 1, we findApproximatingcosh (mµ/T)≈easmµ/T
(
(
(
(
0
L
(N
,
n)
µ
t
(
(
< ln ( 0 m(( Lm+1 (Nt , conT Non-vanishingn) (κ around the heavy quark limit κ = 0, with M(κ) the Wilson quark kernel.
as thebyconditionthat theof the higher-m
is small. In
tributions to the expansion terms are given
closed loops
ofeffectthe hopping
term B terms= −∂M/∂κ.