Lattice Study for Conformal and Walking Dynamics in Large N f Gauge - - PowerPoint PPT Presentation

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 Theo

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, and Italian-Grid-Infrastructures

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

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

4

Summary and Future Works

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

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

4

Summary and Future Works

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

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[g′]

µ µ′

B(g, Nc, Nf ) = µ dg dµ . (1) B(g) g g µ

IRFP

g∗

Conformal

g∗

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

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
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 0.5 1 1.5 2 2.5 3 3.5 4 beta func. g g*=(-b0/b1)1/2 Nf= 6: (b0, b1)=(+,+) Nf=12: (b0, b1)=(+, -) Nf=18: (b0, b1)=(- , -)

B(g, Nc, Nf ) = −g 3ˆ 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 Theo

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

N Nf

c

2 3 8.05 16.5 b =0

1

BZ-IR appears

• b /b

b =0

CONFORMAL??

=N /6C [R] c

2

chiral IRFP

1

QCD-Like Walking QED-Like

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 Theo

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 (∋ ν) without large FCNC current. ¯ ΨΨ|ETC ¯ ΨΨ|TC = exp »Z ΛETC

ΛTC

d(log µ) γ ˆ g 2(µ) ˜–

Conformal

− − − − − → „ΛETC ΛTC «γ[g2

∗]

. (2)

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

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 (∋ ν) without large FCNC current. ¯ ΨΨ|ETC ¯ ΨΨ|TC = exp »Z ΛETC

ΛTC

d(log µ) γ ˆ g 2(µ) ˜–

Conformal

− − − − − → „ΛETC ΛTC «γ[g2

∗]

. (2)

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

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 (∋ ν) without large FCNC current. ¯ ΨΨ|ETC ¯ ΨΨ|TC = exp »Z ΛETC

ΛTC

d(log µ) γ ˆ g 2(µ) ˜–

Conformal

− − − − − → „ΛETC ΛTC «γ[g2

∗]

. (2)

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

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 et al.(’09), Kogut-Sinclair (’90). Nf = 8: Appelquist-Fleming-neil(’08), A.Hasenfratz(’10), Deuzeman-Lombardo-Pallante (’08, Finite T, a(βc)/a(β′

c) = N′ τ/Nτ). Ohki et

al.(PoS2010, TPL, Nc = 2). Nf = 6: LSF Collaboration (’10, Enhancement of PBP/F 3

π) ,

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 Theo

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 Theo

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

Nf Tc 50 100 150 200 2 4 6 8 10 12 12.9 Tc Nf = 25 MeV Critical Exp. Analyses Braun-Gies ('06)

Tc ∝ ˛ ˛Nf − N∗

f

˛ ˛2.54 (Braun-Gies(’06)) , (3) c.f .Miransky-Yamawaki Scaling (’97) . (4)

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

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

Nf Tc 50 100 150 200 2 4 6 8 10 12 12.9 Tc Nf = 25 MeV Critical Exp. Analyses Braun-Gies ('06)

Tc ∝ ˛ ˛Nf − N∗

f

˛ ˛2.54 (Braun-Gies(’06)) , (3) c.f .Miransky-Yamawaki Scaling (’97) . (4)

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

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 transitions in 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 and study 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 − Nf phase diagram.

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

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 transitions in 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 and study 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 − Nf phase diagram.

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

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 transitions in 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 and study 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 − Nf phase diagram.

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

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 transitions in 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 and study 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 − Nf phase diagram.

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

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

4

Summary and Future Works

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

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

0.1 0.2 0.3 0.4 0.5 0.6 4 4.2 4.4 4.6 4.8 5 5.2 5.4 βL 6 Flavors, am=0.02 on 163 x 4 PBP PLOOP

β c

L = 4.65 ??

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

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

0.1 0.2 0.3 0.4 4.6 4.8 5 5.2 5.4 βL 6 Flavors, am=0.02, 243 x 8 PBP 8 x PLOOP

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

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

ˆ ma ∼ σa σ · σδ · F(t σ−1/β) (5) σ = √σaσa , t = Tc − T T , (6) χ−1

ab

≡ ∂ ˆ ma ∂σb ∼ “ δab − σaσb σ ” χ−1

π + σaσb

σ χ−1

σ

, (7) Rπ ≡ χσ χπ = 8 > < > : (Broken) δ−1 (T ≃ Tc) 1 (Restored) . (8)

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

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

ˆ ma ∼ σa σ · σδ · F(t σ−1/β) (5) σ = √σaσa , t = Tc − T T , (6) χ−1

ab

≡ ∂ ˆ ma ∂σb ∼ “ δab − σaσb σ ” χ−1

π + σaσb

σ χ−1

σ

, (7) Rπ ≡ χσ χπ = 8 > < > : (Broken) δ−1 (T ≃ Tc) 1 (Restored) . (8)

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

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

ˆ ma ∼ σa σ · σδ · F(t σ−1/β) (5) σ = √σaσa , t = Tc − T T , (6) χ−1

ab

≡ ∂ ˆ ma ∂σb ∼ “ δab − σaσb σ ” χ−1

π + σaσb

σ χ−1

σ

, (7) Rπ ≡ χσ χπ = 8 > < > : (Broken) δ−1 (T ≃ Tc) 1 (Restored) . (8)

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

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

0.2 0.4 0.6 0.8 1 4 4.2 4.4 4.6 4.8 5 5.2 5.4 βL 6 Flavors, am=0.02, 163 x 4 PBP Rπ

β c

L = 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 Theo

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 = 6

0.2 0.4 0.6 0.8 1 4 4.2 4.4 4.6 4.8 5 5.2 5.4 βL 6 Flavors, am=0.02, 163 x 6 PBP Rπ

β c

L = 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 Theo

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

0.2 0.4 0.6 0.8 1 4.6 4.8 5 5.2 5.4 βL 6 Flavors, am=0.02, 243 x 8 PBP Rπ

β c

L = 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 Theo

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

0.2 0.4 0.6 0.8 1 4.8 5 5.2 5.4 5.6 βL 6 Flavors, am=0.02, 243 x 12 PBP Rπ

β c

L = 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 Theo

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 β c

L 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 \Nt 4 6 8 12 6 4.65 ± 0.05 5.05 ± 0.05 5.2 ± 0.05 5.45 ± 0.15 ΛLa(βc) = „ β 2Ncb0 «(b1/(2b2

0))

exp » − βc 4Ncb0 – , (2-Loop) , (9) 1 Nτ = Tc ΛL · ΛLa(βc) . (10)

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

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 β c

L 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 \Nt 4 6 8 12 6 4.65 ± 0.05 5.05 ± 0.05 5.2 ± 0.05 5.45 ± 0.15 ΛLa(βc) = „ β 2Ncb0 «(b1/(2b2

0))

exp » − βc 4Ncb0 – , (2-Loop) , (9) 1 Nτ = Tc ΛL · ΛLa(βc) . (10)

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

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

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.0001 0.0002 0.0003 0.0004 1/Nt ΛL a(βL

c)

Fit for Nt=6, 8, 12 Lattice Data

1 Nτ = Tc ΛL · ΛLa(βc) . (11)

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

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

4

Summary and Future Works

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

Introduction Results: Six Flavor QCD at Finite Temperature Discussion: Nf Dependence of Tc Summary and Future Works Nf 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 \Nt 4 6 8 12

• 7.88 ± 0.05
• 4
• 5.89 ± 0.03
• 6

4.675 ± 0.025 5.025 ± 0.025 5.225 ± 0.025 5.45 ± 0.05 8

• 4.1125 ± 0.0125
• 4.34 ± 0.04

1 Nτ = Tc ΛL (Nf ) · ΛLa(βc(Nf )) . (12)

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

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

Nf Dependence of Tc/ΛL

250 500 750 1000 1250 1500 1750 2000 2250 2 4 6 8 Tc/ΛL Nf

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

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

Nf Dependence of Tc/ΛL

250 500 750 1000 1250 1500 1750 2000 2250 2 4 6 8 Tc/ΛL Nf

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

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

g 2Nc/g2 Mchiral Energy Lambda

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

2 2 ..

Tc ~

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

Introduction Results: Six Flavor QCD at Finite Temperature Discussion: Nf Dependence of Tc Summary and Future Works Nf 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 obtain M(βref) × a(βc) = b2 b1 βc + 2Ncb1/b0 βref + 2Ncb1/b0 !b1/(2b2

0)

exp " −βc − βref 4Ncb0 # . (13)

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

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

Reference-Scale M II

g 2Nc/g2

4.0 0.63

MUV Mchiral Energy

Appelquist et.al. ('88)

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

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

Tc/M as a function of Nf

0.5 1 1.5 2 2.5 3 3.5 2 4 6 8 R(Nf)/R(0) Nf am=0.02 R=Tc/MUV R=Tc/Mchiral R=Tc/ΛL

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

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

Critical Flavor Number N∗

f

0.05 0.1 0.15 0.2 0.25 4 6 8 10 12 R(Nf)/R(0) Nf am=0.02 Fit, exp=2.54 Nf

*

βref=4.0

Tc(Nf ) = K|Nf − N∗

f |2.54 ,

(Braun-Geis (’11)) (14) N∗

f

= 10.5 ± 0.4 , (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 Theo

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

4

Summary and Future Works

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

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 the

• nset of the walking dynamics.

We have also discussed the generalized ratio Tc/MUV , and the lower edge

• f the conformal window has been estimated to be N∗

f = 10.5 ± 0.4 by

assuming 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 Theo

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 Theo