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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.