On gauge theory phase diagrams at zero and finite temperature K. - PowerPoint PPT Presentation

On gauge theory phase diagrams at zero and finite temperature K. Tuominen University of Jyväskylä & Helsinki Institute of Physics SCGT 2012, Nagoya.

Outline: SU( N c ) gauge + fund rep. matter x f x f ≡ N f Β H Λ L N c 7 6 1. Vacuum phase diagrams Λ Β H Λ L 5 Λ 4 Β H Λ L Λ 3 2 Β H Λ L Λ 1 6 N c 2 3 4 5 SU H N L gauge theory, massless fermions T Deconfining Conformal 2. Finite T phase diagrams Quasi - Confining conf. Conformal x f x c x * 1 2 3 4 5

1. Vacuum Phase diagrams x f ≡ N f x f N c Β H Λ L 7 6 Λ Β H Λ L 5 Λ 4 Β H Λ L Λ 3 Β H Λ L 2 Λ 1 6 N c 2 3 4 5 α ∗ = − β 0 Fixed point from 2-loop betaf. β 1 Critical coupling for chiral π α c = breaking from SD-equ. 3 C 2 ( R ) α ∗ ≤ α c Conformal window:

1. Vacuum Phase diagrams Higher representations: (Sannino, Tuominen ’04) x f ≡ N f x f N f N c Β H Λ L 7 15 F 6 Λ Β H Λ L 5 Λ 2AS 10 4 Β H Λ L Λ 3 Β H Λ L 5 2 Λ 2S 1 6 N c 6 N c 2 3 4 5 2 3 4 5 α ∗ = − β 0 Fixed point from 2-loop betaf. β 1 Critical coupling for chiral π α c = breaking from SD-equ. 3 C 2 ( R ) α ∗ ≤ α c Conformal window:

Higher representations: (Sannino, Tuominen ’04) N f 15 F -Walking with less flavors: phenomenologically viable 2AS 10 Technicolor models - Study on the lattice 5 2S 6 N c 2 3 4 5

Higher representations: (Sannino, Tuominen ’04) N f 15 F -Walking with less flavors: phenomenologically viable 2AS 10 Technicolor models - Study on the lattice 5 2S 6 N c 2 3 4 5 Lots of efforts during last 4...5 years. SU(2) adjoint: (Catteral et al., Hietanen et al., Del Debbio et al.,...) SU(2) fundamental: (Del Debbio et al., Karavirta et al.,...) SU(3) fundamental: (Appelquist et al., Kuti et al.,...) SU(3) sextet: (De Grand et al.,...)

Some history: SU(2) gauge + 2 adjoint Wilson fermions on the lattice Lattice phase diagram and spectrum (Hietanen, Rantaharju, Rummukainen, Tuominen, JHEP 0905 (2009). Strong coupling boundary at β L ∼ 2 Seems volume independent: Lattice artifact? Non-QCD like continuum physics?

Some history: SU(2) gauge + 2 adjoint Wilson fermions on the lattice Lattice phase diagram and spectrum (Hietanen, Rantaharju, Rummukainen, Tuominen, JHEP 0905 (2009). Strong coupling boundary at β L ∼ 2 Seems volume independent: Lattice artifact? Non-QCD like continuum physics? At β ≥ 2 m π ∼ m ρ ∼ m q Conformal?

Measure the coupling at β L ≥ 2 (Hietanen, Rummukainen, Tuominen, PRD 81 (2009).) SU(2) with 2 fundamentals as a control case: fundamental rep. � L =2.2 12 � L =2.5 � L =2.8 � L =3.1 � L =3.5 9 2 g 6 3 0 0 5 10 15 20 L/a

Measure the coupling at β L ≥ 2 (Hietanen, Rummukainen, Tuominen, PRD 81 (2009).) SU(2) with 2 fundamentals SU(2) with 2 adjoints: as a control case: � L =2.05 fundamental rep. 6 � L =2.2 � L =2.2 12 � L =2.5 � L =2.5 � L =3 � L =2.8 5 � L =3.5 � L =3.1 � L =4.5 � L =3.5 � L =8 9 4 2 2 g g 3 6 2 = 2.2 g * 2 3 1 0 0 5 10 15 20 0 L/a 4 8 12 16 20 24 L/a 2 = 2.2, � = 7) fit ( g * 1-loop 0.08 2-loop 3-loop MS 4-loop MS 0.04 � Similar behavior observed also in 0 Del Debbio et al., De Grand et al. -0.04 0 2 4 6 8 2 g

Measure the coupling at β L ≥ 2 (Hietanen, Rummukainen, Tuominen, PRD 81 (2009).) SU(2) with 2 fundamentals SU(2) with 2 adjoints: as a control case: � L =2.05 fundamental rep. 6 � L =2.2 � L =2.2 12 � L =2.5 � L =2.5 � L =3 � L =2.8 5 � L =3.5 � L =3.1 � L =4.5 � L =3.5 � L =8 9 4 2 2 g g 3 6 2 = 2.2 g * 2 3 1 0 0 5 10 15 20 0 L/a 4 8 12 16 20 24 L/a Wilson fermions: 2 = 2.2, � = 7) fit ( g * 1-loop 0.08 Large ( O ( a )) lattice artifacts 2-loop 3-loop MS 4-loop MS 0.04 � Similar behavior observed also in 0 Del Debbio et al., De Grand et al. -0.04 0 2 4 6 8 2 g

Measure the coupling at β L ≥ 2 (Hietanen, Rummukainen, Tuominen, PRD 81 (2009).) SU(2) with 2 fundamentals SU(2) with 2 adjoints: as a control case: � L =2.05 fundamental rep. 6 � L =2.2 � L =2.2 12 � L =2.5 � L =2.5 � L =3 � L =2.8 5 � L =3.5 � L =3.1 � L =4.5 � L =3.5 � L =8 9 4 2 2 g g 3 6 2 = 2.2 g * 2 3 1 0 0 5 10 15 20 0 L/a 4 8 12 16 20 24 L/a Wilson fermions: 2 = 2.2, � = 7) fit ( g * 1-loop 0.08 Large ( O ( a )) lattice artifacts 2-loop 3-loop MS 4-loop MS Need improved actions. 0.04 � Similar behavior observed also in 0 Del Debbio et al., De Grand et al. -0.04 0 2 4 6 8 2 g

Measuring the coupling using the Schrödinger functional (=background field) Chromoelectric background field from fixed boundaries: U k ( x ) | ( x 0 = L ) = exp( aC 0 U k ( x ) | ( x 0 =0) = exp( aC k ( η )) , k ( η )) C k = i k = i C 0 L diag( φ 0 1 ( η ) , . . . , φ 0 L diag( φ 1 ( η ) , . . . , φ n ( η )) n ( η )) g 2 � ∂ S cl. g 2 = ∂ S Coupling defined as response to changes of the background field: 0 ∂η ∂η µ ∼ 1 /L

Measuring the coupling using the Schrödinger functional (=background field) Chromoelectric background field from fixed boundaries: U k ( x ) | ( x 0 = L ) = exp( aC 0 U k ( x ) | ( x 0 =0) = exp( aC k ( η )) , k ( η )) C k = i k = i C 0 L diag( φ 0 1 ( η ) , . . . , φ 0 L diag( φ 1 ( η ) , . . . , φ n ( η )) n ( η )) g 2 � ∂ S cl. g 2 = ∂ S Coupling defined as response to changes of the background field: 0 ∂η ∂η SU(2): µ ∼ 1 /L φ 0 φ 1 = − η Fundamental rep. 1 = η − ρ η = π / 4 , φ 0 ρ = π 2 = ρ − η φ 2 = η

Measuring the coupling using the Schrödinger functional (=background field) Chromoelectric background field from fixed boundaries: U k ( x ) | ( x 0 = L ) = exp( aC 0 U k ( x ) | ( x 0 =0) = exp( aC k ( η )) , k ( η )) C k = i k = i C 0 L diag( φ 0 1 ( η ) , . . . , φ 0 L diag( φ 1 ( η ) , . . . , φ n ( η )) n ( η )) g 2 � ∂ S cl. g 2 = ∂ S Coupling defined as response to changes of the background field: 0 ∂η ∂η SU(2): µ ∼ 1 /L φ 0 φ 1 = − η Fundamental rep. 1 = η − ρ η = π / 4 , φ 0 ρ = π 2 = ρ − η φ 2 = η SU(3): φ 0 φ 1 = η − ρ 1 = − φ 1 − 4 ρ Fundamental rep. φ 0 2 = − φ 3 + 2 ρ φ 2 = η ( ν − 1 / 2) η = 0 , ρ = π / 3 , φ 0 ν = 0 3 = − φ 2 + 2 ρ φ 3 = − η ( ν + 1 / 2) + ρ

Improvement: Luscher, Narayanan, Weisz, Wolff (hep-lat/9207009) Developed and tested for QCD (fundamental rep. fermions). ψ ( x ) i ¯ X S impr = S 0 + a 5 c sw 4 σ µ ν F µ ν ( x ) ψ ( x ) + δ S G,b + δ S F,b x ˜ c t c t Two counterterms due to nontrivial boundaries

Improvement: Luscher, Narayanan, Weisz, Wolff (hep-lat/9207009) Developed and tested for QCD (fundamental rep. fermions). ψ ( x ) i ¯ X S impr = S 0 + a 5 c sw 4 σ µ ν F µ ν ( x ) ψ ( x ) + δ S G,b + δ S F,b x ˜ c t c t c sw = 1 Two counterterms due to c t = 1 + c (1) t g 2 nontrivial boundaries 0 c (1) t g 2 c t = 1 + ˜ ˜ 0

Improvement: Luscher, Narayanan, Weisz, Wolff (hep-lat/9207009) Developed and tested for QCD (fundamental rep. fermions). ψ ( x ) i ¯ X S impr = S 0 + a 5 c sw 4 σ µ ν F µ ν ( x ) ψ ( x ) + δ S G,b + δ S F,b x ˜ c t c t c sw = 1 Two counterterms due to c t = 1 + c (1) t g 2 nontrivial boundaries 0 c (1) t g 2 c t = 1 + ˜ ˜ 0 Perturbative stepscaling: Σ ( u, s, L/a ) = g 2 ( g 0 , sL/a ) | g 2 ( g 0 ,L/a )= u = u + [ Σ 1 , 0 ( s, L/a ) + Σ 1 , 1 ( s, L/a ) N f ] u 2

Improvement: Luscher, Narayanan, Weisz, Wolff (hep-lat/9207009) Developed and tested for QCD (fundamental rep. fermions). ψ ( x ) i ¯ X S impr = S 0 + a 5 c sw 4 σ µ ν F µ ν ( x ) ψ ( x ) + δ S G,b + δ S F,b x ˜ c t c t c sw = 1 Two counterterms due to c t = 1 + c (1) t g 2 nontrivial boundaries 0 c (1) t g 2 ˜ c t = 1 + ˜ 0 Perturbative stepscaling: Σ ( u, s, L/a ) = g 2 ( g 0 , sL/a ) | g 2 ( g 0 ,L/a )= u = u + [ Σ 1 , 0 ( s, L/a ) + Σ 1 , 1 ( s, L/a ) N f ] u 2 δ i = Σ 1 ,i (2 , L/a ) , i = 0 , 1 2 b 0 ,i ln 2 b 0 , 0 = 11 Nc/ (48 π 2 ) b 0 , 1 = N f T R / (12 π 2 )

Improvement: Luscher, Narayanan, Weisz, Wolff (hep-lat/9207009) Developed and tested for QCD (fundamental rep. fermions). ψ ( x ) i ¯ X S impr = S 0 + a 5 c sw 4 σ µ ν F µ ν ( x ) ψ ( x ) + δ S G,b + δ S F,b x ˜ c t c t c sw = 1 Two counterterms due to c t = 1 + c (1) t g 2 nontrivial boundaries 0 c (1) t g 2 c t = 1 + ˜ ˜ 0 1.7 Perturbative stepscaling: 1.6 1.5 SU2 Improved Σ ( u, s, L/a ) = g 2 ( g 0 , sL/a ) | g 2 ( g 0 ,L/a )= u SU2 Unimproved 1.4 SU3 Improved δ 1 SU3 Unimproved 1.3 SU4 Improved SU4 Unimproved = u + [ Σ 1 , 0 ( s, L/a ) + Σ 1 , 1 ( s, L/a ) N f ] u 2 1.2 1.1 1 0.04 0.06 0.08 0.1 0.12 0.14 0.16 a/L δ i = Σ 1 ,i (2 , L/a ) , i = 0 , 1 SU(2) gauge + N f = 2 fundamental 2 b 0 ,i ln 2 b 0 , 0 = 11 Nc/ (48 π 2 ) b 0 , 1 = N f T R / (12 π 2 )

Nontrivial calculation of the counterterms required for higher representations. (T. Karavirta et al. JHEP 1106 (2011), 1101.0154 T. Karavirta et al. PRD85 (1012), 1201.1883)

1. Need to include all coefficients consistently: 1.6 SU(3) gauge 1.5 N f = 2 sextet 1.4 Improved δ 1 1.3 Unimproved c sw =1, c t =0 1.2 1.1 1 0.04 0.06 0.08 0.1 0.12 0.14 0.16 a/L