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title. Universality of the Nf=2 Running Coupling in the Schrödinger Functional Scheme. Tsukuba Univ 　　　　　　　　　　　 K. Murano, S. Aoki, Y.　Taniguchi, Humboldt-Universität zu Berlin S. Takeda for PACS-CS collaboration. contents. ● Introduction.

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  1. title Universality of the Nf=2 Running Coupling in the Schrödinger Functional Scheme Tsukuba Univ 　　　　　　　　　　　 K. Murano,S. Aoki, Y.　Taniguchi, Humboldt-Universität zu Berlin S. Takeda for PACS-CS collaboration

  2. contents ●Introduction ○ Nf=2 running coupling in SF by Alpha collaboration(Nucl.Phys.B713:378-406,2005) ○purpose of our study ●review about SF scheme ○finite size scheme ○definition of running coupling (SF scheme) ●our result ○set up ○concludion

  3. Running coupling (result) Running coupling (SF scheme) non perturbative result(Nf=2 dynamical) hep-lat /0411025 (Alpha) -- PT one and non-PT one are same each other(?)

  4. Beta function (Nf=2) Non-perturbative beta function (SF scheme) hep-lat /0411025 * Nf=2 non-PT beta become be apart from PT one in strong coupling region. *Nf=2 beta is passing the Nf=0 one.

  5. Why Schrödinger Functional ? Advantage １：　solve large lattice problem ●renormarization scale ●cut off scale ●reduce finite size effect ●calculation on the PT scale Restriction of Lattice size SF scheme can reduce this restriction.

  6. 有限サイズscheme Why Schrödinger Functional ? Advantage １：　solve large lattice problem ※ call “scheme” include Finite size effect ●renormarization scale finite size scheme ●cut off scale ●reduce finite size effect ●calculation on the PT scale Restriction of Lattice size (Not significant)

  7. 展開パラメータ(予備) Ex) & ※ obserbable remain unchanged. SF scheme include finite size effect ----- do you feel it strange ? point: Running coupling is only expansion parameter Physical obserbable It doesn’t matter whether this part depend on Box size. we can shift it finite value

  8. SF scheme（予備） (Alpha’92) Definition of running coupling ・ temporal： dirichlet BC Spatial: Twisted BC Back ground field : normarization factor Effective Action

  9. 問題２ ×25 If calculate at same Lattice spacing ・・・ SF scheme solve the Large scale problem If calculate over large scale, enormous lattice size is needed. … …

  10. SSF 定義 Step scaling function ※）　like one integrate beta-function with Initial value u to twice box size. （Ｓ＝２） ・・・ Possible to follow Running

  11. SSF 測定法 （1-node） β a a : tune beta tune a’ a’ a ※use large for large box Calculation of step scaling function N=2 ※we can choose any lattice size you like. β→β’ • Tune beta for running coupling eqal to u0. • Calculate with twice Lattice (same beta) • (and get = u1) 3. Tune beta for running coupling equal to u1. 4. Get from calculation in twice Lattice size

  12. tune tune a’ a’ Continuum limit Take N large with Tune beta to make β’→β’’ Constant. N=2 N’ β’’→β’’’ N’’

  13. Gauge action Purpose of our study: Purpose of our study： Check with different Action （especially strong regime） Action of gauge field: Iwasaki Action Plaquete Action

  14. Set up (others) Set up ●Fermion action: Clover action (Nf=2) ●Csw: Non-PT ●Tuning of : (mass independent scheme) Uncertainty in from mismatch of m is estimated by Perturbatively. ●algorithm：　　HMC Machine: cluster machine kaede in academic computing & Communication center Tsukuba Univ (60 cpu )

  15. Set up (gauge) Set up Boundary O(a) improvement Boundary O(a) improved coeffcient ct. (Only these was not given by non-PT.) plaquete 0 1-loop 2-loop 0 1-loop 1-loop iwasaki 0 1-loop (Alpha’ 92, 96,00) gauge fermion (Takeda’ 04)

  16. Running coupling 測定点 Running coupling (SF scheme) non perturbative result(Nf=2 dynamical) (Alpha’05) We calculated running coupling In Weak and strong coupling region. Calculate in weak coupling point and strong coupling point

  17. Weak coupling result weak coupling region Continuum extrapolation of running coupling Compare Iwasaki action withPlaquete action(Alpha’ 04) Plaquete action Iwasaki action;1-loop Iwasaki action; tree Weak coupling (u=0.9793) Iwasaki : β~6 plaquete: β~9 ● weak coupling Result from Iwasaki action is consistent with that from plaquete action. a/L

  18. strong coupling region Long auto correlation (by unsemble of “semi-stable”) in strong coupling region (and large Lattice size) is reported by alpha. Plaquete gauge action ( quenched ) Destribution of 1/g number

  19. strong coupling region Dstribution of Ex) strong coupling: L=16^4 Beta= 2.61192 Kappa= 0.13363 Number 631 traj Distribution looks reasonable. In the case of Iwasaki action, We didn’t find long auto-correlation.

  20. Strong coupling Why ? ：small In Iwasaki Bare coupling may be too big to calculate perturbatively result strong coupling region Continuum extrapolation of running coupling Compare Iwasaki action withPlaquete action(Alpha’ 04) Plaquete action Iwasaki action;1-loop scaling violation is large Strong coupling (u=3.3340) Iwasaki: β~2 plaquete: β~5 ∑(u, a/L) a/L Coupling boundary

  21. Scaling (quenched)(予備) Scaling of Iwasaki action ( quenched) Nucl.Phys.Proc.Suppl.129:408-410,2004 S.Takeda, S.Aoki, K.Ide Strong point one-loop (same with one we used) Choice B one-loop Choice A tree Tree ct is better than PT one. It seems be able to extrapolate.(?)

  22. We calculated that with tree again. : tree Another set up is same with before one.

  23. As before, We didn’t find long auto-correlation in Iwasaki action. Dstribution of Ex) strong coupling: L=16^4 Beta= 2.755 Kappa= 0.1334 Number 11161 traj

  24. Strong coupling Iwasaki action; tree result weak coupling regime Plaquete action Iwasaki action;1-loop Weak coupling (u=0.9793) Iwasaki : β~6 plaquete: β~9 The result remains consistent with one from plaquette action . a/L

  25. Strong coupling Iwasaki action; tree result strong coupling region Plaquete action Iwasaki action;1-loop Strong coupling (u=3.3340) Iwasaki: β~2 plaquete: β~5 Large scaling violation in the case of 1-loop ct is much Improved by tree ct. a/L O(a) behavior become very good and we got result consistent with result from plaquete action.

  26. まとめ conclusion Purpose calculate SF running coupling in Weak and strong with Iwasaki action And compare result with earlier study. Tree impposible ct result We confirmed that the result is consistent each other within error bar. ( in SF scheme) Beta functionbehave differently from perturbative expectation in strong coupling regime.

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