Determination of running coupling αs for N f = QCD with Schrödinger functional scheme. Yusuke Taniguchi for PACS-CS collaboration

Transcription

1 Determination of running coupling αs for N f = 2 + QCD with Schrödinger functional scheme Yusuke Taniguchi for PACS-CS collaboration

2 Purpose of this project Determine the fundamental parameter of N f = 2 + QCD L = 4g 2F µν a F µν a + ψ ( ) i γµ D µ + m i ψi Strong coupling g Low energy input m π, m K, m Ω are measured by PACS-CS Quark masses m i Bare quark masses are measured by PACS-CS We adopt low energy hadron masses as input. Comparison with high energy input. Need RG running from low to high energy region. 2

3 Goal of this project Evaluate α S (M Z ) by input of hadron masses (m π, m K, m Ω ). NP renormalization factor of quark mass (Z m ). Method Schrödinger functional scheme (Lüscher et al, Alpha) for non-perturbative renormalization L T=L Finite volume of L 4 Appropriate boundary condition Renormalization scale /L Covers from low to high energy region. 3

4 Step Scaling Function Renormalization group flow g(l) g(2l) when one changes the renormalization scale L 2L 4 implemented easily on lattice gauge coupling energy scale (GeV) Follow the renormalization group flow in discretized way. 4

5 Numerical setup Iwasaki gauge action β = tree level boundary improvement Wilson fermion with clover term. non-perturbative c SW one loop boundary improvement θ = π/5 RHMC/HMC algorithm for 3rd/two flavour(s) CPS++ code (RBC Collaboration) Machines PC cluster kaede at Tsukuba: ( 80 PU) SR000 at Tokyo: ( 64 PU) PACS-CS: (256 PU) month RSCC at Riken (28 PU) T2K at Tsukuba: (2560 cores) T2K at Tokyo: (28 cores) 5

6 Configurations Take the continuum limit by three box sizes. L/a L/a Σ(u, a/l) = g 2 (2L) g 2 (L)=u,m=0 Tuning of β and κ for fixed physical box size. g K 70K 230K 70K 20K 320K 00K K 40K 86K 34K 50K 74K 380K K 50K 70K 70K 0K 44K 20K K 5K 48K 35K 40K 34K 9K K 86K 22K 98K 74K 22K 34K 6 4 3K 52K 42K 23K 40K 5K 54K 6

7 Perturbative improvement Deviation from the continuum perturbative SSF Σ (u, a/l) σ PT (u) σ PT (u) = δ (a/l)u + δ 2 (a/l)u 2 + L/a δ Perturbative behavior of + δ(a/l): (high β simulation). PT one loop quadratic fit with fixed d. PT one loop quadratic fit with fixed d. PT one loop quadratic fit with fixed d u u u 7

8 SSF Two loops improvement ( Σ (2) u, a ) Σ (u, a/l) = L + δ (a/l)u + d 2 (a/l)u 2 6 SSF for coupling Σ () (u,a/l) Const fit w/ L/a=6, 8 Const fit w/ L/a=4, 6, 8 Linear fit w/ L/a=4, 6, 8 g 2 (2L)/g 2 (L).5 L/a=4 L/a=6 L/a=8 Const fit w/ 6, 8 PT 3 loops polynomial fit a/l g 2 (L) 8

9 Polynomial fit of NP SSF σ(u) = u + s 0 u 2 + s u 3 + s 2 u 4 + s 3 u 5 + s 4 u 6, s 2 = , s 3 = , s 4 = Introduction of physical scale Lattice spacing as an intermediate scale: CP-PACS Collaboration (m π 500 MeV) β a (fm) L max /a /L max (MeV) g 2 (L max ) (6) (5.4) 5.565(54) (9) (9.7) 4.695(23) (26) 6 480(8) 4.740(79) PACS-CS Collaboration (m π 55 MeV) β a (fm) L max /a /L max (MeV) g 2 (L max ) (3) (7.8) 4.695(23) 9

10 α MS (M Z ) NP running from L max to /µ = 2 5 L max by SSF. 2-loops matching between SF and MS scheme at µ. 4-loops β-fn. in MS scheme. 3-loops matching between α (3) MS (m c) and α (4) MS (m c). 3-loops matching between α (4) MS (m b) and α (5) MS (m b) CP PACS Constant fit with three β Constant fit with β=2.05,.90 Linear fit β=.90 (PACS CS) a/l CP-PACS: α s (M Z ) = (8)( 48)( 73) 3-loops vs. 2-loops matching const. vs. linear extrapolation PACS-CS: α s (M Z ) = 0.225(4)( 5) 0

11 Λ parameter Λ = µ (b 0 g) b ( 2b 2 0 exp ) ( g exp 2b 0 g 0 dg ( β + b 0 g 3 b )) b 2 0 g NP running from L max to /µ = 2 5 L max by SSF. 3-loops β-fn. above µ. 3-loops matching at threshold m c and m b to get Λ (5) MS CP PACS Constant fit with three β Constant fit with β=2.05,.90 Linear fit β=.90 (PACS CS) a/l CP-PACS: Λ (5) = 239(0)( 6)( 22) MeV MS 3-loops vs. 2-loops matching const. vs. linear extrapolation PACS-CS: = 266(20)( 7) MeV Λ (5) MS

12 Non-perturbative β-function β ( σ(u) ) = β( u) u σ(u) Polynomial fit (fixed to 3-loop coeff.) σ(u) u. σ(u) = u + s 0 u 2 + s u 3 + s 2 u 4 + s 3 u 5 + s 4 u 6, s 3 = , s 4 = β(g)/g Nf=3 NPT Nf=2 NPT Nf=3 PT 3 loops Nf=2 PT 3 loops α=g 2 /4π 2

13 SSF of Z P ( Σ P u, a ) L = Z P (g 0, 2L/a) Z P (g 0, L/a) g 2 (L)=u,m=0 SSF for quark mass 0.95 Σ P (u,a/l) Const fit w/ L/a=6, 8 Z P (2L)/Z P (L) L/a=4 L/a=6 L/a=8 Const. fit w/ 6, 8 PT 2 loops NPT fit a/l g 2 (L) Scaling behavior is not good for L/a = 4. Continuum limit with two data points 6 and 8. 3

14 Non-perturbative renormalization factor Z m Renormalization factor for the RGI mass. M = Z M (g 0 )m 0 (g 0 ), Z A (g 0 ) Z M (g 0 ) = Z P (g 0, a/l max ) }{{} m(2 n /L max ) m(/l max ) }{{} M m(2 n /L max ) }{{} Z m (SF) (g 0, a/l max ) NP run in SF 2 loops run For PACS-CS result at β =.90 (example) L max = 544.0(7.8) MeV n = 5 M/m(/L max ) =.305(33) Z P (g 0, a/l max ) = (28) Z A (m = 0.04) = (60), Z A (m = 0.0) = (60) Z M (g 0 ) =.705(45) Running back from M with 4-loops RG fn. in MS scheme. Zm MS (µ = 2 GeV, β =.90) =.353(39) (Preliminary!) (cf. Zm tad =.383).388(39) 4

15 Non-perturbative running mass m(µ) m(/l n ) M = n i= σ P (u i ) m(/l max ) M }{{} }{{} SSF of Z P Input ( m(/l n ) ( ) d M = 2b 0 g 2 0 g(ln ) 2b (L n ) 0 exp dg 0 ( τ (g) β(g) d 0 b 0 g ) ) 0.8 SF scheme Nf=3 2/3 loops m(µ)/m µ/λ 5

16 α s Running coupling α s (M Z ) from hadron mass input. CP-PACS: α s (M Z ) = (8)( 48)( 73) PACS-CS: α s (M Z ) = 0.225(4)( 5) Scaling behavior seems to be good. PT behavior of the deviation Σ (u, a/l) σ PT (u) is non-trivial Two loops coeff. is comparable to one loop. Iwasaki action may not be a good setup for SF scheme. Z m Zm MS (µ = 2 GeV, β =.90) =.353(39) (Preliminary!) Zm PT,tad =.383 Scaling behavior of quark mass SSF seems not so good. We may need perturbative improvement. Future work May adopt more suitable gauge action Better PT behavior (Lüscher-Weisz action) 6

17 Schrödinger functional scheme (Lüscher et al, Alpha) Dirichlet boundary condition at t = 0, T. U k (x) x0 =0 = exp (ac k ), C k = i ϕ ϕ 2 L ϕ 3 Unique global minimum at background field B µ. Mass gap in fermionic mode (quark mass can be set to zero). Renormalized coupling S 0 = g0 2 Fµν 2 Γ 0[B µ ] = Γ[B µ ] = gr 2 (L)k[B µ] Mass renormalization factor Z m (L) = P (t = L/2) Oboundary k tree Oboundary 7

18 Step scaling function The point is: To take continuum limit for every step of RG flow. L 2L, g(l, a) g(2l, a) Obtain the RG flow g(l) g(2l) in the continuum. 8