Non-perturbative renormalization of

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1 Non-perturbative renormalization of N f = 2 + QCD with Schrödinger functional scheme Yusuke Taniguchi for PACS-CS collaboration

2 Our ultimate purpose 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: target of this talk Low energy input r is measured by PACS-CS (Namekawa) Quark masses m i Bare quark masses are measured by PACS-CS (Kadoh, Kuramashi, Ukita) NP renormalization factor will be needed. We adopt input of low energy experimental values. Comparison with high energy input (estimation of systematic error) Need calculation from weak to strong coupling region. 2

3 Plan of this project Evaluate α S (M Z ) by an input of low energy observable (r ). NP renormalization factor of quark mass. as a by product of α S (M Z ) in this talk (inhomogeneous BC at t =, T ) Method Non-perturbative renormalization with Schrödinger functional L T=L Finite volume of L 4 Appropriate boundary condition Renormalization scale /L Good compatibility with lattice. Covers from low to high energy region. 3

4 Schrödinger functional scheme (Lüscher et al, Alpha) Dirichlet boundary condition at t =, T. U k (x) x = = 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 = g 2 Fµν 2 Γ [B µ ] = Γ[B µ ] = gr 2 (L)k[B µ] Mass renormalization factor P (t = L/2) Oboundary Z m (L) = lattice P (t = L/2) Oboundary tree 4

5 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 3 2 energy scale (GeV) Follow the renormalization group flow in discretized way. 5

6 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. 6

7 Numerical setup Iwasaki gauge action β = tree level boundary improvement inhomogeneous DBC, θ = π/5 Wilson fermion with clover term. non-perturbative c SW one loop boundary improvement RHMC/HMC algorithm for 3rd/two flavour(s) CPS++ code Machines PC cluster kaede at Tsukuba: ( 8 PU) SR at Tokyo: ( 64 PU) PACS-CS: (256 PU) month RSCC at Riken (28 PU) T2K at Tsukuba: (256 cores) days T2K at Tokyo: (28 cores) 7

8 Current status Take the continuum limit by three box sizes. L/a L/a Tuning of β and κ is finished for fixed physical box size. g K 7K 23K 7K 2K 32K K 8 4 4K 4K 86K 34K 5K 74K 38K K 5K 5K 7K K 44K 2K K 5K 42K 35K 28K 2K 38K K 86K 22K 98K 74K 22K 22K K 8K 6K 3.2K 4K 5.K 3.2K Now performing simulation for larger box of 2L L/a = 8 L/a = 6: Now going on 8

9 Distribution of data Distribution of S/ η /g 2 (2 4 at strong coupling) 5 6 "plaq.dat" using "plaq.dis" using : S/ η vs τ Its distribution 9

10 SSF (preliminary) σ(u) = g 2 (2L) u=g 2 (L) SSF for coupling g 2 (2L) L/a=4 L/a=6 L/a=8 PT, Nf=3, 3 loop PT, Nf=2 PT, Nf= Nf=2 (Alpha) g 2 (L)

11 SSF (preliminary) σ(u)/u SSF for coupling g 2 (2L)/g 2 (L).5 L/a=4 L/a=6 L/a=8 PT, Nf=3, 3 loop PT, Nf=2 PT, Nf= Nf=2 (Alpha) g 2 (L)

12 SSF (preliminary) σ(u)/u SSF for coupling g 2 (2L)/g 2 (L).5 L/a=4 L/a=6 L/a=8 3 loop PT NPT fit of L/a= g 2 (L) Polynomial fit σ(u) = u + s u 2 + s u 3 + s 2 u 4 + f 3 u 5 + f 4 u 6 χ 2 /dof.8 2

13 Introduction of scale r at vanishing PCAC mass (m u = m d = m s ) β (r /a) m= (26) (27)(34) (66) L 4 β g 2 m AWT (23).39(28) (7).(3) (7).8(2) Λ = µ (b g) b ( 2b 2 exp ) ( g exp 2b g Λ MS = 2.62Λ SF dg ( β + b g 3 b )) b 2 g 3

14 SSF of Z m (preliminary) SSF for quark mass SSF for quark mass Z P (2L)/Z P (L).9.8 L/a=4 L/a=6 L/a=8 3 loop PT Z P (2L)/Z P (L).9.8 L/a=4 PT, 3 loop g 2 (L) g 2 (L) Scaling behaviour is not good for L/a = 4. May be able to take the continuum limit by two data points. PT improvement may not work for 4 4 but may be for

15 Conclusion Calculation for the running coupling is going on Scaling behaviour seems to be good. Scaling behaviour of quark mass SSF is not so good. We may need perturbative improvement. Future work Take the continuum limit Take data for L/a = 6 Much expectation on T2K machine! Adopt appropriate setup for Z m and repeat the calculation Homogeneous BC, θ =.5 5

16 Distribution of data (8 4 at strong coupling) 4 "plaq.dat" using "plaq.dis" using : (6 4 at strong coupling) 6 "plaq.dat" using "plaq.dis" using :

17 Scaling of SSF For one loop improved gauge boundary term (condition B) SSF at strong coupling 2 5 SSF 5..2 a/l 7