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

Lattice Study for Conformal and Walking Dynamics
in Large Nf Gauge Theory
A Potential Interest for Use of New Fermions

Kohtaroh MiuraA , M. LombardoA , E. PallanteB
A. DeuzemanC , and T. SilvaB
Laboratori Nazionali di Frascati - INFNA
Rijksuniversiteit GroningenB
University of BernC

Talk at YIPQS, YITP, Feb. 20, 2011

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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,”
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. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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
General Motivation to Study Conformal Dynamics
Goal and Strategy

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

4Summary and Future Works

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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

Emergence of Conformal Symmetry in Gauge Theory
General Motivation to Study Conformal Dynamics
Goal and Strategy

Table of Contents

1

Introduction
Emergence of Conformal Symmetry in Gauge Theory
General Motivation to Study Conformal Dynamics
Goal and Strategy

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

4Summary and Future Works

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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

Emergence of Conformal Symmetry in Gauge Theory
General Motivation to Study Conformal Dynamics
Goal and Strategy

Gauge Theory with IRFP

S[g]

S[g0]

µ

µ0
B(g , Nc , Nf ) = µ(1)gg∗ Conformal

B(g)
IRFP

g∗

dg
.
dµ

g
µ

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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

Emergence of Conformal Symmetry in Gauge Theory
General Motivation to Study Conformal Dynamics
Goal and Strategy

2-loop beta-function

0.8

Nf= 6: (b0, b1)=(+,+)
Nf=12: (b0, b1)=(+, -)Nf=18: (b0, b1)=(- , -)0.6

beta func.

0.4
0.2
0

g*=(-b0/b1)1/2

-0.2
-0.4
-0.6
-0.8
0

0.5

1

1.5

2

2.5

3

3.5

4

g

ˆ
˜
B(g , Nc , Nf ) = −g 3 b0 (Nc , Nf ) + b1 (Nc , Nf )g 2

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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

Emergence of Conformal Symmetry in Gauge Theory
General Motivation to Study Conformal Dynamics
Goal and Strategy

Nf − Nc Phase Diagram

Nf

ke

Li
D

16.5

b0 =0

??
L
A

M
E
Q FOR gN lkin
O
C Wa

8.05

-b0 /b1 =Nc /6C2 [R]
IRFP chiral

b1=0
BZ-IR
appears

QCD-Like
0
2

3

Nc

c.f. Dietrich and Sannino (’07)

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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

Emergence of Conformal Symmetry in Gauge Theory
General Motivation to Study Conformal Dynamics
Goal and Strategy(New) Strong Dynamics for Electroweak Phase Transition

Composite Higgs: To avoid Higgs mass hierarchy problem.
Walking Dynamics: To make masses of SM fermions (63 ν)
without large FCNC current.
»Z ΛETC–„«γ[g∗2 ]ˆ˜ ConformalhΨ̄Ψi|ETC
ΛETC
= exp
.
d(log µ) γ g 2 (µ) −−−−−→
ΛTChΨ̄Ψi|TCΛTC(2)

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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

Emergence of Conformal Symmetry in Gauge Theory
General Motivation to Study Conformal Dynamics
Goal and Strategy(New) Strong Dynamics for Electroweak Phase Transition

Composite Higgs: To avoid Higgs mass hierarchy problem.
Walking Dynamics: To make masses of SM fermions (63 ν)
without large FCNC current.
»Z ΛETC–„«γ[g∗2 ]ˆ˜ ConformalhΨ̄Ψi|ETC
ΛETC
= exp
.
d(log µ) γ g 2 (µ) −−−−−→
ΛTChΨ̄Ψi|TCΛTC(2)

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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

Emergence of Conformal Symmetry in Gauge Theory
General Motivation to Study Conformal Dynamics
Goal and Strategy(New) Strong Dynamics for Electroweak Phase Transition

Composite Higgs: To avoid Higgs mass hierarchy problem.
Walking Dynamics: To make masses of SM fermions (63 ν)
without large FCNC current.
»Z ΛETC–„«γ[g∗2 ]ˆ˜ ConformalhΨ̄Ψi|ETC
ΛETC
= exp
.
d(log µ) γ g 2 (µ) −−−−−→
ΛTChΨ̄Ψi|TCΛTC(2)

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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

Emergence of Conformal Symmetry in Gauge Theory
General Motivation to Study Conformal Dynamics
Goal and Strategy

Summary-Table I: Lattice Studies for Conformal Window

Fundamental Reps.
Nf = 16: Damgaard et al.(’97, Meson Mass Analyses), A.Hasenfratz (’10,
MCRG), Fodor et al.(’09, Mass Scaling Low)
Nf = 12: Appelquist-Fleming-Neil(’08, Step-Scaling, Schrodinger-Functional),
Fodor et al.(’09,’11), A.Hasenfratz(’10), Itou et al.(PoS2010, TPL),
Deuzeman-Lombardo-Pallante(’08, Bulk-Trans., Const. Phys.), Mawninney etal.(’09), Kogut-Sinclair (’90).
Nf = 8: Appelquist-Fleming-neil(’08), A.Hasenfratz(’10),
Deuzeman-Lombardo-Pallante (’08, Finite T , a(βc )/a(βc0 ) = Nτ0 /Nτ ). Ohki etal.(PoS2010, TPL, Nc = 2).
Nf = 6: LSF Collaboration (’10, Enhancement of PBP/Fπ3 ) ,Miura-Lombardo-Pallante (Present, Finite T ).

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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

Emergence of Conformal Symmetry in Gauge Theory
General Motivation to Study Conformal Dynamics
Goal and Strategy

Summary-Table II: Lattice Studies for Conformal Window

Other Representations: Nf = 2
Symm.: Degrand-Shamir-Svetitsky (’10, Nc = 3 Schrodinger-Functional),
Fodor-Holland-Kuti-Nogradi-Schroeder (’09, Nc = 3
Random-Matrix-Therory)
Adjoint:√ Debbio-Lucini-Patella-Pica-Rago (’10, Nc = 2,
MPS / σ =Large), Hietanen-Rummukainen-Tuominen (’09, Nc = 2,
Schrodinger-Functional), Catterall-Giedt-Sannino-Schneible (’08, Nc = 2,
Meson Mass and String Tension)

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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

Emergence of Conformal Symmetry in Gauge Theory
General Motivation to Study Conformal Dynamics
Goal and Strategy

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
Tc

∝

˛
˛
˛Nf − Nf∗ ˛2.54(Braun-Gies(’06)) ,c.f .Miransky-Yamawaki Scaling (’97) .

(3)
(4)

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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

Emergence of Conformal Symmetry in Gauge Theory
General Motivation to Study Conformal Dynamics
Goal and Strategy

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
Tc

∝

˛
˛
˛Nf − Nf∗ ˛2.54(Braun-Gies(’06)) ,c.f .Miransky-Yamawaki Scaling (’97) .

(3)
(4)

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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

Emergence of Conformal Symmetry in Gauge Theory
General Motivation to Study Conformal Dynamics
Goal and Strategy

Goal

Ultimate Goal
We investigate finite T chiral phase transitionsin Nf = 0, 4, 6, 8, and 12 cases, (and more and more in future).
The lower edge of the conformal window is extracted
from a vanishing Tc (Nf ).
Miranski scaling and beyond 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. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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

Emergence of Conformal Symmetry in Gauge Theory
General Motivation to Study Conformal Dynamics
Goal and Strategy

Goal

Ultimate Goal
We investigate finite T chiral phase transitionsin Nf = 0, 4, 6, 8, and 12 cases, (and more and more in future).
The lower edge of the conformal window is extracted
from a vanishing Tc (Nf ).
Miranski scaling and beyond 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. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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

Emergence of Conformal Symmetry in Gauge Theory
General Motivation to Study Conformal Dynamics
Goal and Strategy

Goal

Ultimate Goal
We investigate finite T chiral phase transitionsin Nf = 0, 4, 6, 8, and 12 cases, (and more and more in future).
The lower edge of the conformal window is extracted
from a vanishing Tc (Nf ).
Miranski scaling and beyond 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. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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

Emergence of Conformal Symmetry in Gauge Theory
General Motivation to Study Conformal Dynamics
Goal and Strategy

Goal

Ultimate Goal
We investigate finite T chiral phase transitionsin Nf = 0, 4, 6, 8, and 12 cases, (and more and more in future).
The lower edge of the conformal window is extracted
from a vanishing Tc (Nf ).
Miranski scaling and beyond 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. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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
General Motivation to Study Conformal Dynamics
Goal and Strategy

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

4Summary and Future Works

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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

PBP and PLOOP as a Function of βL

0.6

6 Flavors, am=0.02 on 163 x 4
PBP
PLOOP

0.5
0.4
0.3
0.2
0.1
0
4

4.2

4.4

4.6

4.8
βL

5

5.2

5.4

βLc = 4.65 ??

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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

PBP and PLOOP as a Function of βL
3

6 Flavors, am=0.02, 24 x 8
PBP
8 x PLOOP

0.4

0.3

0.2

0.1

0
4.6

4.8

5

5.2

5.4

βL

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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

Hyperscaling Hypothesis

m̂a

∼

σ

=

χ−1
ab

≡

Rπ

≡

σa
· σ δ · F (t σ −1/β )σ√
Tc − Tσa σa , t =
,
T
“
”
∂ m̂a
σa σb −1 σa σb −1χπ +∼ δab −
χσ ,
∂σb
σ
σ
8
>
<0 (Broken)χσ.
= δ −1 (T ' Tc )>χπ:
1 (Restored)

(5)
(6)
(7)

(8)

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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

Hyperscaling Hypothesis

m̂a

∼

σ

=

χ−1
ab

≡

Rπ

≡

σa
· σ δ · F (t σ −1/β )σ√
Tc − Tσa σa , t =
,
T
“
”
∂ m̂a
σa σb −1 σa σb −1χπ +∼ δab −
χσ ,
∂σb
σ
σ
8
>
<0 (Broken)χσ.
= δ −1 (T ' Tc )>χπ:
1 (Restored)

(5)
(6)
(7)

(8)

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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

Hyperscaling Hypothesis

m̂a

∼

σ

=

χ−1
ab

≡

Rπ

≡

σa
· σ δ · F (t σ −1/β )σ√
Tc − Tσa σa , t =
,
T
“
”
∂ m̂a
σa σb −1 σa σb −1χπ +∼ δab −
χσ ,
∂σb
σ
σ
8
>
<0 (Broken)χσ.
= δ −1 (T ' Tc )>χπ:
1 (Restored)

(5)
(6)
(7)

(8)

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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

4.2

4.4

4.6

4.8
βL

5

5.2

5.4

βLc = 4.65 ± 0.05 .

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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. c.f. Bulk Transition at Nf = 12
(Deuzaman et al, (’10)).

Nf \Nt644.65 ± 0.05

65.05 ± 0.05„«(b1/(2b02 ))ΛL a(βc )

=

1
Nτ

=

β
2Nc b0Tc· ΛL a(βc ) .
ΛL

85.2 ± 0.05

12
5.45 ± 0.15

»
–
βc
,
exp −
4Nc b0(2-Loop) ,

(9)
(10)

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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. c.f. Bulk Transition at Nf = 12
(Deuzaman et al, (’10)).

Nf \Nt644.65 ± 0.05

65.05 ± 0.05„«(b1/(2b02 ))ΛL a(βc )

=

1
Nτ

=

β
2Nc b0Tc· ΛL a(βc ) .
ΛL

85.2 ± 0.05

12
5.45 ± 0.15

»
–
βc
,
exp −
4Nc b0(2-Loop) ,

(9)
(10)

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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

0.4

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
Tc=· ΛL a(βc ) .
Nτ
ΛL

(11)

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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

Table of Contents

1

Introduction
Emergence of Conformal Symmetry in Gauge Theory
General Motivation to Study Conformal Dynamics
Goal and Strategy

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

4Summary and Future Works

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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

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.

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
-

1
Tc
=
(Nf ) · ΛL a(βc (Nf )) .
Nτ
ΛL

12
5.45 ± 0.05
4.34 ± 0.04

(12)

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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

Nf Dependence of Tc /ΛL

2250
2000
1750

Tc/ΛL

1500
1250
1000
750
500
250
0
0

2

4
Nf

6

8

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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

Nf Dependence of Tc /ΛL

2250
2000
1750

Tc/ΛL

1500
1250
1000
750
500
250
0
0

2

4
Nf

6

8

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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

Nf Dependence of Tc /ΛL , In Terms of 2-Loop beta-func.
Tc ~ Mchiral

Energy2Nc/g2

Lambda0

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

g

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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

Reference-Scale M: Renomalization Scale

Integrating 2-loop beta function 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)

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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

Reference-Scale M II
Energy2Nc/g2MchiralMUVAppelquist et.al. ('88)
4.0

0.63

g

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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

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. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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

Critical Flavor Number Nf∗
am=0.020.25
Fit, exp=2.54
N f*βref=4.0

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∗=

10.5 ± 0.4 ,(Braun-Geis (’11))(14)(Presnt Work with βref =4.0)(15)

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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
General Motivation to Study Conformal Dynamics
Goal and Strategy

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

4Summary and Future Works

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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

Summary

Summary
We have reported the developments of the project for a thermodynamic
study on the (Pre-)Conformal (walking) dynamics.
We have investigated the chiral phase transition (crossover) in the
QCD-Like theor with six flavors, based on the hyperscaling and asymptotic
scaling analyses.
The ratio Tc /ΛL has started increasing at Nf = 6, which would imply theonset of the walking dynamics.
We have also discussed the generalized ratio Tc /MUV , and the lower edgeof the conformal window has been estimated to be Nf∗ = 10.5 ± 0.4 byassuming the IR exponents proposed by FRG.

Kohtaroh MiuraA , M. LombardoA , E. PallanteB A. DeuzemanC , and T. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th

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 and complete T − Nf Phase Diagram:
The potential measurement is on 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. SilvaBLattice Study for Conformal and Walking Dynamics in Large Nf Gauge Th