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 title

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

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

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

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

purpose ： non-PT check of QCD Lattice QCD QCD (perturbative) Hadronic input Develop by Non-PT Jet physics Compare by Perturbatively Low EnergyHigh Energy Intro

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

● renormarization scale finite size scheme ※ call “scheme” include Finite size effect 有限 サイ ズ sche me ● reduce finite size effect ● cut off scale ● calculation on the PT scale Advantage １： solve large lattice problem Restriction of Lattice size (Not significant)

we can shift it finite value Ex)& It doesn’t matter whether this part depend on Box size. 展開パ ラメー タ ( 予備 ) Physical obserbable ※ obserbable remain unchanged. point: Running coupling is only expansion parameter

SF scheme define running coupling as coeff of Effective Action Definition of Running coupling （ SF scheme) (Alpha’ 92)

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

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

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

Calculation of step scaling function β→β’ tune N=2 （ 1- node ） ※ we can choose any lattice size you like. aaa a’ ※ use large for large box β : tune beta a’ SSF 測定法 1.Tune beta for running coupling eqal to u0. 2.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

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

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

● Tuning of : (mass independent scheme) Set up ● Fermion action: Clover action (Nf=2) ● Csw: Non-PT ●algorithm ： HMC Set up (others) Uncertainty in from mismatch of m is estimated by Perturbatively. beta L (2L)4 (8)6 (12)8 (16)4 (8)6 (12) 8 (16) Machine: cluster machine kaede in academic computing & Communication center Tsukuba Univ (60 cpu )

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

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

Preliminary Continuum extrapolation of running coupling ● weak coupling plaquete action と Iwasaki action で、結果は一致した。 Iwasaki action と Plaquete action (Alpha’ 04) との結果の比較 a/L ∑(u, a/L) Weak coupling (u=0.9793) Iwasaki : β~6 plaquete: β~9 Iwasaki action ;1-loop Plaquete action Iwasaki action ; tree Weak coupling

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

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

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

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

We calculated that with tree again (8)6(12)8(16) (8)6(12)8(16) : tree Another set up is same with before one.

a/L Strong coupling a/L Plaquete action Iwasaki action ;1-loop Iwasaki action ; tree Weak coupling (u=0.9793) Iwasaki : β~6 plaquete: β~9 Strong coupling (u=3.3340) Iwasaki: β~2 plaquete: β~5 conclusiton

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

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

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

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

● Plaquete action には 一次相転移がある ● Iwasaki action では 相転移が弱められている hep-lat:/ Nf=3 (clover) Iwasaki action Plaquete action 一次相転移 ( 予備 )

Scaling of Iwasaki action ( quenched) (Takeda’04) Weak point Strong point Scaling (quenched)( 予 備 )

Nf=2 Step Scaling Function Nf=2 data （予備）