格子シミュレーションによる固定点の探索

伊藤　悦子　(工学院大学)

arXiv : 0902.3768 (hep-lat) and Work inprogress

2009/5/18 seminar@中央大学

Erek Bilgici a Antonino Flachi b Masafumi Kurachi c C. –J. David Lin d Hideo Matsufuru e Hiroshi Ohki b, f

Tetsuya Onogi b Takeshi Yamazaki ｇ

CollaboratorsNumerical simulation was carried out onthe vector supercomputer NEC SX-8in YITP, Kyoto Universityand RCNP, Osaka University

a : University of Grazb : YITP, Kyoto Universityc : Los Alamos National Laboratoryd : National Chio-Tung University, and National Center for Theoretical Sciencee : KEKf : Department of Physics, Kyoto Universityｇ: Tsukuba University

Large flavor QCD理論に非自明赤外固定点が存在するか？

格子シミュレーションで非摂動論的なrunning coupling constantをみてみよ

うそのためには、色々な工夫が必要。計算に適したくりこみスキーム、解析方法など。

今日の発表の流れ 動機と関連した研究

Wilson loop schemeの定義

running coupling constantをみるには？

Quenched QCDでの結果 シミュレーションのパラメータ

Technical steps シミュレーションの結果

12 flavor QCDの結果

step scalingの方法

動機と関連した研究

Fixed point・・・universality, symmetry Examples of fixed point　　scalar theory　　　2-dim. NLSM : Gaussian（asymptotic free）　　　3-dim. LSM : IR (Wilson-Fischer) fixed point　　　3-dim. NLSM : non-trivial UV fixed point　　gauge theory　　　4-dim. QCD : Gaussian（asymptotic free）　　　4-dim. large flavor SU(N) theory : Non-trivial IR fixed point？？

IR fixed point of large flavor SU(3)theoryThe existence of conformal window (Caswell 1974)

perturbative 2-loopThe phase structure (Banks-Zaks 1982) 　　　　　　　expansion

A rich phase structure

Assuming the existence of this fixed point walking technicolor large anomalous dimension state Electroweak sym. breaking unparticle?

Simulation steps:1.generate the gauge configurations2.measure some physical quantities

Lattice gauge theory:Consider a discrete space-time, and sum all paths in path integralIn the continuum limit, we get the usual quantum field theory

Gauge field on the link Fermion field on the site

Plaquet action

parameters:

bare coupling constant

Nf=8 Nf=12

The running coupling constant in Schrodinger functionalscheme.

There is no edivence of fixed point. There is a flat region in low energyscale.

Existence of the fixed point should not depend on scheme

Previous studies in lattice QCDDamgaard, Heller, Krasnitz, Olesen: Phys.Lett.B400:169Iwasaki et al: Phys.Rev.D76:034504Appelquist, Fleming and Neil: Phys.Rev.Lett.100:171607 arXiv:0901:3766

Practical conditions of the renormalized coupling on the lattice:Luscher et.al.Nucl.Phys.Proc.Suppl.30:139-148,1993.

1. i is non-perturbatively defined

2. can be computed through numerical simulation

3. A perturbative computation is possible. (In principle)

• Schrodinger functional scheme

• Wilson loop scheme (New scheme!)

• Polyakov loop scheme

Examples of scheme:

This talk (no O(a/L) discretazationerror)

More

Wilson loop schemeの定義

: box size (spatial)

: box size (temporal)

: size of the Wilson loop (spatial)

: size of the Wilson loop (temporal)

: lattice spacing

: bare coupling

The Wilson loop

Perturbative expansion

Non-perturbative definition of the running couplingconstant

In our simulation, we take,then

More

We choose the renormalization scheme:

is estimated by calculating the Creutz ratio.

Renormalized coupling in “Wilson loopscheme”

No O(a/L) contribution!!

fix the free parameter in the renormalization condition

take the continuum limit is the scale which defines the running coupling

constant of step scaling

renormalized coupling

To take the continuum limit, we have to set the scale “ ”.It corresponds to tuning to keep a certain input physical parameterconstant.

How to take the continuum limit

input output

running coupling constantをみるには？

Choose a value ofrenormalized couplingconstant at energy scale ()

Step scaling

renormalized coupling

Tuning the beta (barecoupling) for small latticesize

Step scaling

Do the simulation for thelarge lattice size

Take the continuum limit(energy scale)

Step scaling

Tune the beta at small latticesize

Step scaling

Take the continuum limit(energy scale)

Step scaling

Step scaling

Tune the beta at small latticesize

Take the continuum limit(energy scale)

Step scaling

Step scaling

Step scaling

We obtain the scaling ofthe running coupling.

シミュレーション(quenched QCD)

pseudo-heatbath algorithm and Over-relaxation Plaquet action parameter sets of the lattice size and bare coupling to

keep the input physical quantities constant

beta

Quenched QCD test of the Wilson loop scheme

Lattice size

Set1 Set2 Set6 Set7beta L0/a

(s=1)L0/a(s=1.5)

beta L0/a(s=1)

L0/a(s=1.5)

beta L0/a(s=1)

L0/a(s=1.5)

beta L0/a(s=1)

L0/a(s=1.5)

8.31 10 15 7.80 10 15 6.207 10 15 5.907 10 15

8.25 12 18 7.83 12 18 6.303 12 18 6.000 12 18

8.27 14 21 7.86 14 21 6.377 14 21 6.087 14 21

8.32 16 24 7.91 16 24 6.463 16 24 6.170 16 24

8.40 18 27 7.97 18 27 6.546 18 27 6.229 18 27

High energy 　　　　　 Lowenergy

Parameter sets of the lattice sizeand bare coupling

Extrapolation to the continuum limit

(Step1)

Data4p linear fit5p linear fit5p quadratic fit

シミュレーションの結果

Continuum limit of

Wilsonscheme2-loop1-loop

Log (L_0/L)

Wilson scheme3-loop (MS bar scheme)2-loop1-loop

Log (L_0/L)

Simulation time:only one day for each step

Twisted Polyakov schemeについて

(de Divitiis et al.:NPB422:382)more than 10 years ago

To satisfy the gauge inv. and translationinv.

Polyakov loop in Twisted direction

non-perturbative definition of renormalized coupling in TPL scheme

At tree level, is proportional to bare coupling.

In our works, we have to check the at IR fixed point which shouldbe smaller than this value

Running of the renormalized coupling constantin Quenched QCD

IR limit:

Capitani et.al:NPB B544 : 669-698

TPL scheme Wloop scheme

SF scheme

arXiv:0902.3768 (hep-lat)

We calculate the running coupling of quenched QCDin “Wilson loop scheme”.

We have shown that smearing drastically reducesthe statistical error.

We found there is a window for the parameters (r,n)which both the statistical and discretization errors areunder control.

We also studied “Twisted Polyakov scheme”, andfound the scheme works well, too.

These methods are promising. We will investigatethe large flavor QCD using these renormalizationscheme.

Summary of the quenched QCD tests

Large flavor QCDのとき

Hybrid Monte Carlo algorithm (exact algorithm) staggered fermion (massless 4N flavor) Polyakov loop scheme

cf: Appelquist et.al.

Our data (Preliminary)

The bulk phase transition problem at

There is mass gap, and it is hard to simulate thesmall beta region.

Use the improved action?

S. Aoki et alPRD72:054510 (2005)

large anomalous dim., mass, sparameter,,,

おまけ

Schrodinger functional scheme

one-loop lattice perturbation theory

bare coupling constant

one-loop approximationO(a) errorR algorithm (time step size error)

We would like to propose a novel scheme to investigate thisproblem.

?

pure gauge action:

Continuum limit exists

Approaching to thecontinuum faster for largerR/L0

Smearing of link variables Interpolation of the Creutz ratios Extrapolation to the continuum limit of the running

coupling

APE Smearing of the link variables Definition:

smearing level : nsmearing parameter:

Technicalsteps

definition :

n r=0.25 r=0.30 r=0.35n=1 L0/a >10 L0/a >8.3 L0/a >7.1n=2 L0/a >18 L0/a >15 L0/a

>12.8n=3 L0/a >26 L0/a>21.6

L0/a>18.5

Table: The lower bound forL0/a

Discretization error should be controlled larger r

Noise (statistical error) should be small smaller r or higher n

Oversmearing should be avoided n should be smaller than R/2,

Conditions for good choice of r andn

Oversmearingfor n=1,2

Optimal choice!!

Interpolation of the Creutz ratios

Fit function :

Fit ranges :

To obtain the value of the Creutz ratios for noninteger R, we have to interpolatethem.

Ex)

L0/a R+1/2 R min R max10 3.0 2 412 3.6 2 414 4.2 2 515 4.5 - -16 4.8 3 518 5.4 4 621 6.3 4 624 7.2 5 727 8.1 6 8

IR limit:

In high energy region,

In low energy region, runs slower than 1-loop approximation.

TP scheme can control both statistical and systematic errors.