2020/8/24 @ 熱場 online

Improvement of the SFtX method
based on the gradient flowin the study of finite temperature Nf = 2+1 QCD
K. Kanaya*, A. Baba*, S. Ejiri1, M. Kitazawa2, H. Suzuki3, Y. Taniguchi*, T. Umeda4[ *: U. Tsukuba, 1: Niigata U., 2: Osaka U., 3: Kyushu U., 4: Hiroshima U. ]We study thermodynamic properties of lattice Nf =2+1 QCD with improved Wilson quarks, adopting theSFtX method (small flow-time expansion method)
based on the gradient flow. We study therenormalization scale and 2-loop matching coefficientsin the SFtX method in Nf = 2+1 QCD. We find thatthe µ0 renormalization scale of Harlander et al.

1. SFtX method based on GF
Gradient flow: modification of fields in terms of a fictitious time t.
Narayanan-Neuberger(2006), Lüscher(2009–)

Flowed field Bµ ≈ smeared Aµ over a physical range √(8t).
Operators of flowed fields have no UV divergences nor short-distanceLüscher-Weisz (2011)
singularities at t > 0.

SFtX method: a general method to correctly calculate any
renormalized observables on the latticeH. Suzuki, PTEP 2013, 083B03 (2013) [E:2015, 079201]

We are applying the SFtX method to 2+1 flavor QCD withimproved Wilson quarks. The SFtX method removes difficulties
associated with the chiral violation.

2. QCD with heavy u,d and physical s
Taniguchi et al., PRD96, 014509 (2017); D95, 054502 (2017), D99, 059904(E) (2019)

As the first systematic application of the method to QCD with dynamical quarks,
we first studied the case that u and d quarks are heavier than their physicalmasses.
NF =2+1 QCD, Iwasaki gauge + NP-O(a)-improved clover
fine lattice a ≈ 0.07fm (ß = 2.05), slightly heavy ud (mPS/mV ≈ 0.63), ≈phys. s.
CP-PACS+JLQCD T = 0 config. (283x56)
WHOT-QCD T > 0 fixed-scale config.set (323xNt, Nt = 4, 6, …, 16)

➤ EoS
30

EoS by SFtX agrees well with conventional
integration method at T≤300 MeV (Nt ≥10).

gradient flow
T-integration25(e+p)/T420
15

Disagreement at T≥350 MeV attributed to aindep. lattice artifacts of O((aT)2 =1/Nt2) at Nt≤8.

10500100200300

400500

improves the applicability and reliability of the SFtXm e t h o d . O n t h e o t h e r h a n d , n o a p p a re n t
improvements are seen with the 2-loop matchingcoefficients on this lattice, while contamination ofO(1/Nt2) discretization errors is suggested at Nt ≤ 10
in the equation of motion which was assumed in thecalculation of Harlander et al.

3.1 Renormalization scale
The renormalization scale of the matching coefficients in the SFtX
method has a freedom to choose its value µ within a range. Besides
the conventional µd = 1/√(8t), wetest a new scale µ0 = 1/√(2eγt),which may extend signal to larger t.
We find that the µ0-scale• is consistent with the µd-scale,• improves the signal at large t,• extends the stability andapplicability of the SFtX method.

3.2 Two-loop matching coefficients
We have been using 1-loop matching coefficients of Makino andSuzuki (2014). Recently, Harlander et al. calculated 2-loop coeff.’s forEMT, using equation of motion (EoM) for quarks (2018). Use of 2loop coeff.’s may improve data at small intermediate t.
In our test, we find that the 2-loop• is consistent with 1-loop
besides the effects of EoM,
• shows no apparent improvementon our fine lattice,• suggests EoM getsO(1/Nt2) artifacts at Nt ≤ 10.

600➤ Chiral susceptibility (disconnected)
T (MeV)➤ Topological susceptibility
Gluonic and fermionic definitions agree, evenat a≠0 and with Wilson-type quarks!

4. QCD with physical u, d, s quarks
Kanaya et al., PoS Lattice2019, 088 (2020)

Extending the study in 2+1 flavor QCD with heavy u,d quarks, weare generating config.'s at the physical point on a slightly coarser
lattice:
NF =2+1 QCD, Iwasaki gauge + NP-O(a)-improved clover

ing
on-go

Power-low behavior consistent with DIGA.

Consistency among different methods suggests our lattice is fine enough.
The SFtX method works well.

3. Improvement of the SFtX method

Taniguchi et al., PRD102, 014510 (2020)

We find that the µ0-scale improves well.
Our preliminary results at the physicalpoint, using 1-loop matching coeff.'s withthe µ0-scale, suggest:

0.012
0.01
0.008
0.006
0.004
0.002

chi /dof=0.01
1 loop µ000.5

11.5

22e+p ≈ KS (cont. lim.), e-3p ≈ 3 * KS
but with large errors.
u,d show a sharper chiral trans. than s.

25(e+p)/T40t/a

2

20
15
10
5
00100

200

300

400

500

600

T (MeV)

0.00014
ud chiral susceptibility

Besides generating finite-temperature configurations at the physicalpoint, we carried out tests of the renormalization scale and 2-loop
matching coefficients in the SFtX method, using the heavy u, dconfigurations. We found that a proper choice of therenormalization scale extends the stability and applicability of theSFtX method much.

2T=183 MeV (Nt=12) (chi /dof=1.0)
1 loop µ0
1 loop µd0.014

3530

In the application of the SFtX method to QCD at physical u, d, squark masses (physical point QCD) and other more complicatedobservables, we found that we need to improve the method.

semi-fine lattice a ≈ 0.09fm (ß = 1.9), physical u,d,s
PACS-CS T = 0 config. (323x64), fine-tuned to phys.pt.
WHOT-QCD T > 0 fixed-scale configs. (323xNt, Nt = 4, 5, …, 18,T≈122-549MeV)
0.016
ud quark subtracted chiral condensate

Clear peak at Tpc≈190MeV as expected from
Polyakov loop, in spite of the chiral violation
due to the Wilson quark action.

T≈122-157 MeV may be criticalchi2/dof=0.1
u quark 1 loop µ00.00012

Tpcphys < 157 MeV(cf.) Tpcphys = 156.5 ± 1.5 MeV

0.0001

Bazavov et al. PLB795, 15 (2019), HISQ

8x10-56x10-54x10-52x10-500100

200300400500600

Need more statistics at low T's.
Need continuum extrapolation too.

T (MeV)

Tomonaga Center for the History of the Universe, University of Tsukuba

2020/8/24 @ 熱場 online

Improvement of the SFtX method
based on the gradient flowin the study offinite temperature Nf = 2+1 QCD
WHOT-QCD Collaboration:
K.Kanaya*, A.Baba*, S.Ejiri1, M.Kitazawa2, H.Suzuki3, Y.Taniguchi*, T.Umeda4*Univ.Tsukuba, 1Niigata Univ., 2Osaka Univ., 3Kyushu Univ., 4Hiroshima Univ.

SFtX method based on GF

tBµ(t,x)

Gradient Flow: a gauge-inv. smearing of fields in terms of a fictitious "time" t.
Narayanan-Neuberger (2006), Lüscher (2010-)

Operators of flowed fields arefree from UV divergences nor short-distance singularities.

t=0

Small Flow-time eXpansion (SFtX) methodAµ(x)a general method to correctly calculate any renormalized observables non-perturbatively on the latticeH. Suzuki, PTEP 2013, 083B03 (2013)
Y. Taniguchi et al., Phys.Rev. D 96, 014509 (2017)

physical obs. we wantcontinuum
t=0
a→0

latticet=0, a>0

GF
t→0

continuumt>0t → 0 by linear window

GF

finite,physically well-defineda → 0 avoiding small-t singularities at a > 0latticet>0, a>0

evaluate their values
non-perturbatively

construct corresponding operator on the lattice

Because we can construct a lattice operator directly from the continuum operator,
this method is applicable also to observables whose base symmetry is broken on the lattice(Poincaré inv., chiral sym., etc.)➯ energy-momentum tensor➯ QCD with Wilson-type quarks, to cope with problems due to chiral violation.Our first test of SFtX with dynamical quarks
physical point

14 1210 98

Nt= 16 14121076Taniguchi et al., Phys. Rev. D 96, 014509 (2017); Phys. Rev. D 95, 054502 (2017)5(β=1.90)mπ/mρ~0.6(β=2.05)
100

200

6

➤

T[MeV]➤8300

400500➤Tpc ≈190 MeV

EoS

30

12

gradient flow
T-integration25

gradient flow
T-integration

10Reproduce the results of conventional
integral method.

8(e-3p)/T420(e+p)/T42+1 flavor QCD with Inon-perturbatively improved Wilson quarks.
Slightly heavy ud quarks (m𝝅/m𝝆 ≈0.63) with ≈physical s quark (m𝜼ss/m𝝓 ≈0.74).
Fine lattice (a≈0.07fm with improved action).

151064205-2-400100

200300400500600

0100T (MeV)

200300

400

500

600

Disagreement at T≥350 MeV due toO((aT)2 =1/Nt2) lattice artifact at Nt <~ 8.

T (MeV)chiral cond. / chiral susceptibility
Crossover around Tpc≈190 MeV.
Peak higher with decreasing mq.

topological susceptibility
gluonic(T/Tpc)-7.2(0.9)
fermionic(T/Tpc)-7.3(1.7)
DIGA

χt

1×10-3

1×10-4Gluonic and fermionic definitions agree well.

1×10-5
00.5

11.5
T/Tpc

ctlyeridotssuelbaneiraldhcohhtteiwm
The SFtX observables, even e quarks!
measure due to Wilson-typviolation

22.5

Power low consistent with a prediction of Dilute Instanton Gasmodel.

Improvement of the SFtX method
Taniguchi et al., Phys. Rev. D 102, 014510 (2020)

We need to improve the method for application on a coarser lattice and/or more complicated operators.
Here, we test improvements in the matching coefficients, which matches flowed operators at t > 0 to thetarget observable at t = 0 using perturbatively known properties and thus makes the t → 0 extrapolationsmoother.

Choice of the renormalization scale
Conventional choice: μd(t) ≡18t

A new proposal:

We find that the µ0-scale• is consistent with the µd-scale• improves the signal at large t,• extends the stability andapplicability of the SFtX method.

2-loop matching coefficients
Harlander-Kluth-Lange, EPJC 78:944 (2018)2-loop matching coefficients are now available for EMT.
Equation of Motion (EoM) for quarks used in their calculation to reduce # of mixing operators.

We find that the 2-loop results• are consistent with 1-loop besides the effects of EoM,
• show no apparent improvement on our fine lattice,• suggest: EoM gets O(1/Nt2) artifacts at Nt ≤ 10.

Details as well as application to the physical point QCD are discussed in the poster.