Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future Works

Thermodynamic Study for (Pre-) Conformal and Dynamics
in Many Flavor QCD
Kohtaroh MiuraA , M. LombardoA , E. PallanteB
A. DeuzemanC , and T. SilvaB
Laboratori Nazionali di Frascati - INFNA
Rijksuniversiteit GroningenB
University of BernC

Talk at KMI-SCGT, Nagoya Univ., March. 20, 2012

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future Works

References

References
K. Miura, M. P. Lombardo and E. Pallante, “Chiral phase transition at
finite temperature and conformal dynamics in large Nf QCD,”
To appear in Phys. Lett. B, arXiv:1110.3152 [hep-lat].
K. Miura, M. P. Lombardo and E. Pallante, “Thermodynamic Study for
Conformal Phase in Large Nf Gauge Theory,” PoS Lattice 2011,arXiv:1111.1098 [hep-lat].
Tools
MILC Code: http://www.physics.utah.edu/˜detar/milc/milc qcd.html
Argolithm: Rational Hybrid Molecular-Dynamics with Omelyan-Integrator
Computers: IBM-sp6 in CINECA, SP16000 in YITP, andItalian-Grid-Infrastructures

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future Works

Table of Contents

1

Introduction
Emergence of Conformal Symmetry in Gauge Theory
FRG Study for Finite T Chiral Trans.

2

Results: Six Flavor QCD at Finite Temperature
Critical Coupling Determination in Nf = 6 QCD-Like Theory
Asymptotic Scaling

3

Discussion: Nf Dependence of Tc
Nf Dependence of Tc /ΛL
Nf Dependence of Tc /M: Use of UV Reference Scale M
Miransky-Yamawaki Diagram

4

Summary and Future Works

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future Works

Emergence of Conformal Symmetry in Gauge Theory
FRG Study for Finite T Chiral Trans.

Table of Contents

1

Introduction
Emergence of Conformal Symmetry in Gauge Theory
FRG Study for Finite T Chiral Trans.

2

Results: Six Flavor QCD at Finite Temperature
Critical Coupling Determination in Nf = 6 QCD-Like Theory
Asymptotic Scaling

3

Discussion: Nf Dependence of Tc
Nf Dependence of Tc /ΛL
Nf Dependence of Tc /M: Use of UV Reference Scale M
Miransky-Yamawaki Diagram

4

Summary and Future Works

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future Works

Emergence of Conformal Symmetry in Gauge Theory
FRG Study for Finite T Chiral Trans.

Scaling Property near IRFP

g*
g

dg 2
d log(µ/ΛUV )=θ(Nf ) (g 2 − g∗2 (Nf )) ,

µχ

∼

h
i1/θ(Nf )A(Nf ) gχ2 (Nf ) − g∗2 (Nf )
,

Nf∗

:

Tχ (Nf )∼(1)gχ2 (Nf∗ ) = g∗2 (Nf∗ ) ,Lower Bound of CW.
˛
˛1/θ(N ∗ )fµχ (Nf ) ∼ ˛Nf − Nf∗ ˛

(2)
(3)

(Braun-Gies(’11)) . (4)Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future Works

Emergence of Conformal Symmetry in Gauge Theory
FRG Study for Finite T Chiral Trans.

Scaling Property near IRFP

g*
g

dg 2
d log(µ/ΛUV )=θ(Nf ) (g 2 − g∗2 (Nf )) ,

µχ

∼

h
i1/θ(Nf )A(Nf ) gχ2 (Nf ) − g∗2 (Nf )
,

Nf∗

:

Tχ (Nf )∼(1)gχ2 (Nf∗ ) = g∗2 (Nf∗ ) ,Lower Bound of CW.
˛
˛1/θ(N ∗ )fµχ (Nf ) ∼ ˛Nf − Nf∗ ˛

(2)
(3)

(Braun-Gies(’11)) . (4)Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future Works

Emergence of Conformal Symmetry in Gauge Theory
FRG Study for Finite T Chiral Trans.

Scaling Property near IRFP

g*
g

dg 2
d log(µ/ΛUV )=θ(Nf ) (g 2 − g∗2 (Nf )) ,

µχ

∼

h
i1/θ(Nf )A(Nf ) gχ2 (Nf ) − g∗2 (Nf )
,

Nf∗

:

Tχ (Nf )∼(1)gχ2 (Nf∗ ) = g∗2 (Nf∗ ) ,Lower Bound of CW.
˛
˛1/θ(N ∗ )fµχ (Nf ) ∼ ˛Nf − Nf∗ ˛

(2)
(3)

(Braun-Gies(’11)) . (4)Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future Works

Emergence of Conformal Symmetry in Gauge Theory
FRG Study for Finite T Chiral Trans.

Scaling Property near IRFP

g*
g

dg 2
d log(µ/ΛUV )=θ(Nf ) (g 2 − g∗2 (Nf )) ,

µχ

∼

h
i1/θ(Nf )A(Nf ) gχ2 (Nf ) − g∗2 (Nf )
,

Nf∗

:

Tχ (Nf )∼(1)gχ2 (Nf∗ ) = g∗2 (Nf∗ ) ,Lower Bound of CW.
˛
˛1/θ(N ∗ )fµχ (Nf ) ∼ ˛Nf − Nf∗ ˛

(2)
(3)

(Braun-Gies(’11)) . (4)Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future Works

Emergence of Conformal Symmetry in Gauge Theory
FRG Study for Finite T Chiral Trans.

T − Nf Phase Diagram: Functional Renormalization Group
Braun-Gies ('06)
200

Tc= 25 MeV
Nf

150
Tc

100
Critical Exp.
Analyses

50
0

2

4

6

8

10

12 12.9

Nf˛˛2.54
Tc ∝ ˛Nf − Nf∗ ˛(Braun-Gies(’06,’09,’11)) ,(5)»–
B(Nf∗ )
c.f .Miransky-Yamawaki Scaling (’97): Tc ∼ exp − p ∗. (6)|Nf − Nf |Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future Works

Emergence of Conformal Symmetry in Gauge Theory
FRG Study for Finite T Chiral Trans.

T − Nf Phase Diagram: Functional Renormalization Group
Braun-Gies ('06)
200

Tc= 25 MeV
Nf

150
Tc

100
Critical Exp.
Analyses

50
0

2

4

6

8

10

12 12.9

Nf˛˛2.54
Tc ∝ ˛Nf − Nf∗ ˛(Braun-Gies(’06,’09,’11)) ,(5)»–
B(Nf∗ )
c.f .Miransky-Yamawaki Scaling (’97): Tc ∼ exp − p ∗. (6)|Nf − Nf |Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future Works

Emergence of Conformal Symmetry in Gauge Theory
FRG Study for Finite T Chiral Trans.

Goal

Ultimate Goal
We investigate finite T chiral phase transitionsin Nf = 0, 4, 6, 8, (and 12) cases, (and more in future).
The lower edge of the conformal window is extracted
from a vanishing Tc (Nf ).
Miranski and/or BG Scaling are interesting physics, and useful to find andstudy the walking region.
Six Flavor QCD-Like? Theory
Six flavor QCD is expected to be in the region of walking (pre-conformal)
dynamics (Appelquist (’11)).
The chiral and deconfinement dynamics at finite temperature has not been
investigated, while they provide important data to complete the Tc − Nfphase diagram.

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future Works

Emergence of Conformal Symmetry in Gauge Theory
FRG Study for Finite T Chiral Trans.

Goal

Ultimate Goal
We investigate finite T chiral phase transitionsin Nf = 0, 4, 6, 8, (and 12) cases, (and more in future).
The lower edge of the conformal window is extracted
from a vanishing Tc (Nf ).
Miranski and/or BG Scaling are interesting physics, and useful to find andstudy the walking region.
Six Flavor QCD-Like? Theory
Six flavor QCD is expected to be in the region of walking (pre-conformal)
dynamics (Appelquist (’11)).
The chiral and deconfinement dynamics at finite temperature has not been
investigated, while they provide important data to complete the Tc − Nfphase diagram.

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future Works

Emergence of Conformal Symmetry in Gauge Theory
FRG Study for Finite T Chiral Trans.

Goal

Ultimate Goal
We investigate finite T chiral phase transitionsin Nf = 0, 4, 6, 8, (and 12) cases, (and more in future).
The lower edge of the conformal window is extracted
from a vanishing Tc (Nf ).
Miranski and/or BG Scaling are interesting physics, and useful to find andstudy the walking region.
Six Flavor QCD-Like? Theory
Six flavor QCD is expected to be in the region of walking (pre-conformal)
dynamics (Appelquist (’11)).
The chiral and deconfinement dynamics at finite temperature has not been
investigated, while they provide important data to complete the Tc − Nfphase diagram.

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future Works

Emergence of Conformal Symmetry in Gauge Theory
FRG Study for Finite T Chiral Trans.

Goal

Ultimate Goal
We investigate finite T chiral phase transitionsin Nf = 0, 4, 6, 8, (and 12) cases, (and more in future).
The lower edge of the conformal window is extracted
from a vanishing Tc (Nf ).
Miranski and/or BG Scaling are interesting physics, and useful to find andstudy the walking region.
Six Flavor QCD-Like? Theory
Six flavor QCD is expected to be in the region of walking (pre-conformal)
dynamics (Appelquist (’11)).
The chiral and deconfinement dynamics at finite temperature has not been
investigated, while they provide important data to complete the Tc − Nfphase diagram.

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future Works

Critical Coupling Determination in Nf = 6 QCD-Like Theory
Asymptotic Scaling

Table of Contents

1

Introduction
Emergence of Conformal Symmetry in Gauge Theory
FRG Study for Finite T Chiral Trans.

2

Results: Six Flavor QCD at Finite Temperature
Critical Coupling Determination in Nf = 6 QCD-Like Theory
Asymptotic Scaling

3

Discussion: Nf Dependence of Tc
Nf Dependence of Tc /ΛL
Nf Dependence of Tc /M: Use of UV Reference Scale M
Miransky-Yamawaki Diagram

4

Summary and Future Works

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future Works

Critical Coupling Determination in Nf = 6 QCD-Like Theory
Asymptotic Scaling

Critical Coupling Determination from Rπ , Nt = 4
1

6 Flavors, am=0.02, 163 x 4
PBP
Rπ

0.8
0.6
0.4
0.2
0
4

Rπ

4.2

4.4

4.6

≡

χσ
→
χπ

4.8
βL

(
0
1

5

5.2

5.4

(Broken)(Restored).

(7)

βLc = 4.65 ± 0.05 .

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future Works

Critical Coupling Determination in Nf = 6 QCD-Like Theory
Asymptotic Scaling

Critical Coupling Determination from Rπ , Nt = 61

6 Flavors, am=0.02, 163 x 6
PBP
Rπ

0.8
0.6
0.4
0.2
0
4

4.2

4.4

4.6

4.8
βL

5

5.2

5.4

βLc = 5.05 ± 0.05 .

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future Works

Critical Coupling Determination in Nf = 6 QCD-Like Theory
Asymptotic Scaling

Critical Coupling Determination from Rπ , Nt = 8

1

6 Flavors, am=0.02, 243 x 8
PBP
Rπ

0.8
0.6
0.4
0.2
0
4.6

4.8

5

5.2

5.4

βL

βLc = 5.2 ± 0.05 .

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future Works

Critical Coupling Determination in Nf = 6 QCD-Like Theory
Asymptotic Scaling

Critical Coupling Determination from Rπ , Nt = 12

1

6 Flavors, am=0.02, 243 x 12
PBP
Rπ

0.8
0.6
0.4
0.2
0
4.8

5

5.2
βL

5.4

5.6

βLc = 5.45 ± 0.15 .

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future Works

Critical Coupling Determination in Nf = 6 QCD-Like Theory
Asymptotic Scaling

From βLc To A Physical Quantity, Tc /ΛL

Table: The summary table of βc at Nf = 6 with ma = 0.02.

Nf \Nt644.65 ± 0.05

65.05 ± 0.05

85.2 ± 0.05

12
5.45 ± 0.15

Comment
We have used a single lattice mass ma = 0.02.
Thermal scaling property indicates that Nf = 6 looks out of conformal
window (c.f. Bulk Transition at Nf = 12 (Deuzaman et al, (’10))).
„ΛL a(βc )

=

1
Nτ

=

β
2Nc b0«(b1/(2b02 ))Tc· ΛL a(βc ) .
ΛL

»
–
βc
exp −
,
4Nc b0(2-Loop) ,

(8)
(9)

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future Works

Critical Coupling Determination in Nf = 6 QCD-Like Theory
Asymptotic Scaling

From βLc To A Physical Quantity, Tc /ΛL

Table: The summary table of βc at Nf = 6 with ma = 0.02.

Nf \Nt644.65 ± 0.05

65.05 ± 0.05

85.2 ± 0.05

12
5.45 ± 0.15

Comment
We have used a single lattice mass ma = 0.02.
Thermal scaling property indicates that Nf = 6 looks out of conformal
window (c.f. Bulk Transition at Nf = 12 (Deuzaman et al, (’10))).
„ΛL a(βc )

=

1
Nτ

=

β
2Nc b0«(b1/(2b02 ))Tc· ΛL a(βc ) .
ΛL

»
–
βc
exp −
,
4Nc b0(2-Loop) ,

(8)
(9)

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future Works

Critical Coupling Determination in Nf = 6 QCD-Like Theory
Asymptotic Scaling

From βLc To A Physical Quantity, Tc /ΛL

Table: The summary table of βc at Nf = 6 with ma = 0.02.

Nf \Nt644.65 ± 0.05

65.05 ± 0.05

85.2 ± 0.05

12
5.45 ± 0.15

Comment
We have used a single lattice mass ma = 0.02.
Thermal scaling property indicates that Nf = 6 looks out of conformal
window (c.f. Bulk Transition at Nf = 12 (Deuzaman et al, (’10))).
„ΛL a(βc )

=

1
Nτ

=

β
2Nc b0«(b1/(2b02 ))Tc· ΛL a(βc ) .
ΛL

»
–
βc
exp −
,
4Nc b0(2-Loop) ,

(8)
(9)

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future Works

Critical Coupling Determination in Nf = 6 QCD-Like Theory
Asymptotic Scaling

Asymptotic Scaling and Its Violation

0.3

Fit for Nt=6, 8, 12
Lattice Data

0.25

1/Nt

0.2
0.15
0.1
0.05
0
0

0.002

0.004

0.006

0.008

0.01

0.012

c

ΛL a(βL )1
Tc=· ΛL a(βc ) .
Nτ
ΛL

(10)

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future Works

Critical Coupling Determination in Nf = 6 QCD-Like Theory
Asymptotic Scaling

Asymptotic Scaling and Its Violation II

0.4

Fit for Nt=6, 8, 12
Lattice Data

0.35
0.3

1/Nt

0.25
0.2
0.15
0.1
0.05
0
0

0.00010.00020.00030.0004cΛL a(βL )

1
Nτ

=

Tc
· ΛL a(βc ) ,ΛL

β = 10/g02 → 6/g 2 .

(11)

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future WorksNf Dependence of Tc /ΛL
Nf Dependence of Tc /M: Use of UV Reference Scale M
Miransky-Yamawaki Diagram

Table of Contents

1

Introduction
Emergence of Conformal Symmetry in Gauge Theory
FRG Study for Finite T Chiral Trans.

2

Results: Six Flavor QCD at Finite Temperature
Critical Coupling Determination in Nf = 6 QCD-Like Theory
Asymptotic Scaling

3

Discussion: Nf Dependence of Tc
Nf Dependence of Tc /ΛL
Nf Dependence of Tc /M: Use of UV Reference Scale M
Miransky-Yamawaki Diagram

4

Summary and Future Works

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future WorksNf Dependence of Tc /ΛL
Nf Dependence of Tc /M: Use of UV Reference Scale M
Miransky-Yamawaki Diagram

Collection of βc , for several Nf
Table: The summary table of βc . The values are obtained by using the same action
except the number of flavors. Blue: Deuzeman-Lombardo-Pallante (’08).

Nf \Nt046
8

44.675 ± 0.025-

6
7.88 ± 0.05
5.89 ± 0.03
5.025 ± 0.025
4.1125 ± 0.0125

8
5.225 ± 0.025-

12
5.45 ± 0.05
4.34 ± 0.04

Comment
We have used a single lattice mass ma = 0.02.
We have used the same action (upto Nf ) and the same (MILC) code to
get the above collection.
1Tc=(Nf ) · ΛL a(βc (Nf )) .
Nτ
ΛL

(12)

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future WorksNf Dependence of Tc /ΛL
Nf Dependence of Tc /M: Use of UV Reference Scale M
Miransky-Yamawaki Diagram

Collection of βc , for several Nf
Table: The summary table of βc . The values are obtained by using the same action
except the number of flavors. Blue: Deuzeman-Lombardo-Pallante (’08).

Nf \Nt046
8

44.675 ± 0.025-

6
7.88 ± 0.05
5.89 ± 0.03
5.025 ± 0.025
4.1125 ± 0.0125

8
5.225 ± 0.025-

12
5.45 ± 0.05
4.34 ± 0.04

Comment
We have used a single lattice mass ma = 0.02.
We have used the same action (upto Nf ) and the same (MILC) code to
get the above collection.
1Tc=(Nf ) · ΛL a(βc (Nf )) .
Nτ
ΛL

(12)

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future WorksNf Dependence of Tc /ΛL
Nf Dependence of Tc /M: Use of UV Reference Scale M
Miransky-Yamawaki Diagram

Nf Dependence of Tc /ΛL
2250
2000
1750

Tc/ΛL

1500
1250
1000
750
500
250
0
0

2

4
Nf

6

8

Enhancement of Tc /ΛL
Vanishing IR-Scale ΛL with closer to the conformal window.
Onset of Walking?

c.f. S.Guputa(’01) and Appelquist(’10).

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future WorksNf Dependence of Tc /ΛL
Nf Dependence of Tc /M: Use of UV Reference Scale M
Miransky-Yamawaki Diagram

Nf Dependence of Tc /ΛL
2250
2000
1750

Tc/ΛL

1500
1250
1000
750
500
250
0
0

2

4
Nf

6

8

Enhancement of Tc /ΛL
Vanishing IR-Scale ΛL with closer to the conformal window.
Onset of Walking?

c.f. S.Guputa(’01) and Appelquist(’10).

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future WorksNf Dependence of Tc /ΛL
Nf Dependence of Tc /M: Use of UV Reference Scale M
Miransky-Yamawaki Diagram

Reference-Scale M: Renomalization Scale

Integrating 2-loop beta function, dg /dM = −g 3 (b0 + b1 g 2 ),from M(βref ) to a−1 (βc ), We obtainM(βref ) × a(βc ) =b02 βc + 2Nc b1 /b0
b1 βref + 2Nc b1 /b0!b1/(2b02 )"βc − βrefexp −4Nc b0#.

(13)

a−1 (βc ): UV Cutoff, and including lattice output βc .
M(βLref ): Renormalization scale in 2-loop perturbations.
c.f: Braun-Gies (’06), αs (M = 1.7GeV) = 0.322.

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future WorksNf Dependence of Tc /ΛL
Nf Dependence of Tc /M: Use of UV Reference Scale M
Miransky-Yamawaki Diagram

Reference-Scale M: Renomalization Scale

Integrating 2-loop beta function, dg /dM = −g 3 (b0 + b1 g 2 ),from M(βref ) to a−1 (βc ), We obtainM(βref ) × a(βc ) =b02 βc + 2Nc b1 /b0
b1 βref + 2Nc b1 /b0!b1/(2b02 )"βc − βrefexp −4Nc b0#.

(13)

a−1 (βc ): UV Cutoff, and including lattice output βc .
M(βLref ): Renormalization scale in 2-loop perturbations.
c.f: Braun-Gies (’06), αs (M = 1.7GeV) = 0.322.

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future WorksNf Dependence of Tc /ΛL
Nf Dependence of Tc /M: Use of UV Reference Scale M
Miransky-Yamawaki Diagram

Reference-Scale M: Renomalization Scale

Integrating 2-loop beta function, dg /dM = −g 3 (b0 + b1 g 2 ),from M(βref ) to a−1 (βc ), We obtainM(βref ) × a(βc ) =b02 βc + 2Nc b1 /b0
b1 βref + 2Nc b1 /b0!b1/(2b02 )"βc − βrefexp −4Nc b0#.

(13)

a−1 (βc ): UV Cutoff, and including lattice output βc .
M(βLref ): Renormalization scale in 2-loop perturbations.
c.f: Braun-Gies (’06), αs (M = 1.7GeV) = 0.322.

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future WorksNf Dependence of Tc /ΛL
Nf Dependence of Tc /M: Use of UV Reference Scale M
Miransky-Yamawaki Diagram

Reference-Scale M: Renomalization Scale

Integrating 2-loop beta function, dg /dM = −g 3 (b0 + b1 g 2 ),from M(βref ) to a−1 (βc ), We obtainM(βref ) × a(βc ) =b02 βc + 2Nc b1 /b0
b1 βref + 2Nc b1 /b0!b1/(2b02 )"βc − βrefexp −4Nc b0#.

(13)

a−1 (βc ): UV Cutoff, and including lattice output βc .
M(βLref ): Renormalization scale in 2-loop perturbations.
c.f: Braun-Gies (’06), αs (M = 1.7GeV) = 0.322.

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future WorksNf Dependence of Tc /ΛL
Nf Dependence of Tc /M: Use of UV Reference Scale M
Miransky-Yamawaki Diagram

Reference-Scale M II

EnergyMUV

2Nc/g2

4.0

Tc ~ MchiralLambda0

3C 2 [F] .
=. 0.63
4 (pi)2
Appelquist
et.al. ('88)

g

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future WorksNf Dependence of Tc /ΛL
Nf Dependence of Tc /M: Use of UV Reference Scale M
Miransky-Yamawaki Diagram

Tc /M as a function of Nf
am=0.02
3.5
3

R=Tc/MUV
R=Tc/Mchiral
R=Tc/ΛL

R(Nf)/R(0)

2.5
2
1.5
1
0.5
0
0

2

4
Nf

6

8

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future WorksNf Dependence of Tc /ΛL
Nf Dependence of Tc /M: Use of UV Reference Scale M
Miransky-Yamawaki Diagram

Tc /M as a function of Nf : UV M
1.2

am=0.02, R(Nf)=(Tc/M(βref))(Nf)
βref=3.4
βref=3.6
βref=3.8
βref=4.0

1

R(Nf)/R(0)

0.8
0.6
0.4
0.2
0
0

1

2

3

4
Nf

5

6

7

8

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future WorksNf Dependence of Tc /ΛL
Nf Dependence of Tc /M: Use of UV Reference Scale M
Miransky-Yamawaki Diagram

Critical Flavor Number Nf∗
0.25

am=0.02, R(Nf)=(Tc/M(βref))(Nf)Fit
βref=4.0
N f*

R(Nf)/R(0)

0.2
0.15
0.1
0.05
0
4

6

8

10

12

Nf

Tc (Nf )=K |Nf − Nf∗ |2.54 ,Nf∗=

10.43 ± 0.6 ,(Braun-Geis (’11))(14)(Presnt Work with βref =4.0)(15)

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future WorksNf Dependence of Tc /ΛL
Nf Dependence of Tc /M: Use of UV Reference Scale M
Miransky-Yamawaki Diagram

Critical Flavor Number Nf∗
0.35

am=0.02, R(Nf)=(Tc/M(βref))(Nf)Fit
βref=3.4
N f*

0.3

R(Nf)/R(0)

0.25
0.2
0.15
0.1
0.05
0
4

6

8
Nf

Tc (Nf )=K |Nf − Nf∗ |2.54 ,Nf∗=

11.49 ± 0.9 ,1012(Braun-Geis (’11))(16)(Presnt Work with βref =3.4)(17)

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future WorksNf Dependence of Tc /ΛL
Nf Dependence of Tc /M: Use of UV Reference Scale M
Miransky-Yamawaki Diagram

Miransky-Yamawaki Phase Diagram: Naive Speculation

Nf*

Co Q
nf ua
or si
m
al

D-L
ike

ThermalTransition

Tra
n

QE

Bu
lk

Conformal

Nfs.

Banks-Zaks ('82)
Miransky-Yamawaki ('97)
Deuzeman-Lombardo-Pallante ('10)

ChiralgLa

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future WorksNf Dependence of Tc /ΛL
Nf Dependence of Tc /M: Use of UV Reference Scale M
Miransky-Yamawaki Diagram

Thermal Transition Lines in Miransky-Yamwaki Phase Diagram

8
7
6

Fit Nt=6
Nt=4
Nt=6
Nt=8
Nt=12

Nf

5
4
3
2
1
0
0.85

0.9

0.95

1
1.05 1.1
gc=(2Nc/βLc)1/2

1.15

1.2

1.25

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future Works

Table of Contents

1

Introduction
Emergence of Conformal Symmetry in Gauge Theory
FRG Study for Finite T Chiral Trans.

2

Results: Six Flavor QCD at Finite Temperature
Critical Coupling Determination in Nf = 6 QCD-Like Theory
Asymptotic Scaling

3

Discussion: Nf Dependence of Tc
Nf Dependence of Tc /ΛL
Nf Dependence of Tc /M: Use of UV Reference Scale M
Miransky-Yamawaki Diagram

4

Summary and Future Works

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future Works

Summary

Summary
To clarify the Tc − Nf and the MY phase diagram is important toinvestigate the (pre-) conformal nature of a strongly flavored gauge theory.
We have investigated a finite T chiral phase transition (crossover) in the
QCD-Like theory with six flavors, and observed thermal scaling property of
the critical coupling.
The ratio Tc /ΛL starts increasing from Nf = 6, which implies a change ofdynamics.
The ratio Tc /MUV with a UV renormalization scale MUV turns to be adecreasing function of Nf . This property would be consistent with FRG
investigations with the use of a UV reference scale.

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future Works

Summary

Summary
To clarify the Tc − Nf and the MY phase diagram is important toinvestigate the (pre-) conformal nature of a strongly flavored gauge theory.
We have investigated a finite T chiral phase transition (crossover) in the
QCD-Like theory with six flavors, and observed thermal scaling property of
the critical coupling.
The ratio Tc /ΛL starts increasing from Nf = 6, which implies a change ofdynamics.
The ratio Tc /MUV with a UV renormalization scale MUV turns to be adecreasing function of Nf . This property would be consistent with FRG
investigations with the use of a UV reference scale.

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future Works

Summary

Summary
To clarify the Tc − Nf and the MY phase diagram is important toinvestigate the (pre-) conformal nature of a strongly flavored gauge theory.
We have investigated a finite T chiral phase transition (crossover) in the
QCD-Like theory with six flavors, and observed thermal scaling property of
the critical coupling.
The ratio Tc /ΛL starts increasing from Nf = 6, which implies a change ofdynamics.
The ratio Tc /MUV with a UV renormalization scale MUV turns to be adecreasing function of Nf . This property would be consistent with FRG
investigations with the use of a UV reference scale.

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future Works

Summary

Summary
To clarify the Tc − Nf and the MY phase diagram is important toinvestigate the (pre-) conformal nature of a strongly flavored gauge theory.
We have investigated a finite T chiral phase transition (crossover) in the
QCD-Like theory with six flavors, and observed thermal scaling property of
the critical coupling.
The ratio Tc /ΛL starts increasing from Nf = 6, which implies a change ofdynamics.
The ratio Tc /MUV with a UV renormalization scale MUV turns to be adecreasing function of Nf . This property would be consistent with FRG
investigations with the use of a UV reference scale.

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor

Introduction
Results: Six Flavor QCD at Finite Temperature
Discussion: Nf Dependence of Tc
Summary and Future Works

Future Works

Future Works
To set a scale a−1 from zero T simulatoin and complete T − Nf and MY
Phase Diagram: Work in Progress!
Critical behavior near the IR-Fixed Pt.
The color SU(Nc = 2) with 8 flavors at finite T .

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBThermodynamic Study for (Pre-) Conformal and Dynamics in Many Flavor