Nf=12 QCD の非摂動的 running coupling

伊藤　悦子 ( 工学院大学 )arXiv:0910.4196 and Work in progress

A01 meeting @ KEK 2010/2/15

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

Tetsuya Onogi g Eigo Shintani g Takeshi Yamazaki h

CollaboratorsNumerical simulation was carried out on the vector supercomputer NEC SX-8 in YITP, Kyoto University and RCNP, Osaka UniversitySR and BlueGene in KEK

a : University of Grazb : YITP, Kyoto Universityc : Tohoku Universityd : National Chio-Tung University, and National Center for Theoretical Sciencee : KEKf : Department of Physics, Kyoto Universityg: Osaka Universityh: Tsukuba University

Plan to talk

Introduction

Twisted Polyakov loop scheme

Simulation (quenched QCD)

Nf=12 case (TPL scheme)

Summary and Future directions

Introduction

Gardi and Karliner Nucl.Phys.B529:383-423,1998. Ncw~12 from resummation method of 3 terms of the beta fn

Miransky and Yamawaki Phys.Rev.D55:5051-5066,1997 Ncw=11.9 based on the same idea of Gardi et al.

Ryttov and Sannino Phys.Rev.D78:065001,2008 Ncw=8.25 from calclulation of the anomalous dimension

large flavor SU(3) gauge theories:

Two-loop perturbative beta function:

The signs of C1 and C2 depend on Nf.

Previous and Recent studies in lattice QCD with Nf=12Appelquist, Fleming and Neil: Phys.Rev.Lett.100:171607(2008), PRD79:076010

The running coupling constant in Schrodinger functional scheme.

Nf=8 There is no edivence of fixed point. Nf=12 : a flat region in low energy scale.

Deuzeman, Lombordo and Pallante arXiv:0904.4662 [hep-ph] there is a conformal phase...Fodor, Holland, Kuti, Nogragi, and Schroeder arXiv:0809.4890, 0911:2463 [hep-lat] Nf=12 is below the conformal window. (stout-smeared staggered fermion)

Difficulty to study low energy behavior comes from a large discretization error . (Appelquist et al. neglect O(a/L) error.)To conclude the existence of IR fixed point in the case of Nf=12 case, we have to precisely estimate O(a/L) discretization error.

Nonperturbative renormalized coupling constant

In SU(3) lattice gauge theory, the VEV of operator O depends on

Lattice size (L0)Lattice spacing (a)bare coupling constant

L0

a

If , the above equation gives the nonperturbative coupling constant

On the lattice study, we can calculate nonperturbatively the VEV on the right hand side.

Schrodinger functional (SF) scheme Wilson loop scheme Twisted Polyakov loop scheme

Some renormalization schemes

no O(a/L) discretization error

Twisted Polyakov loop scheme

de Divitiis et al.:NPB422:382

To satisfy the gauge inv. and translation inv.

Polyakov loop in Twisted direction

Twisted Polyakov loop (TPL) scheme

Previous works of SU(2) gauge theory

At tree level, is proportional to bare coupling.

In our works,We measure the ratio of Polykov loop correlators.We have to check the at IR fixed point which should be smaller than 32.

TPL coupling:

This is a lattice artifact of the TPL coupling.

simulation (quenched QCD)

1. Tune the value of beta (bare coupling) for a small lattice size, which gives the renormalized coupling “u”.

2. Carry out the simulation for the twice size of lattice.

3. Take the continuum limit (energy scale )

Step scaling

Notation ：input renormalized coupling :output renormalized coupling :

quenched QCD (pure Yang-Mills theory)

Simulation lattice size

parameter

statistics 200,000 – 400,000 sweeps for each parameter

pseudo-heatbath algorithm and overrelaxation (1:5)

step scaling with s=1.5

Interpolation using following fit fn.

(n=3,4)

inputoutput

In practice, we use the fit function to obtain the tuned value of beta and the output values of renormalized coupling.

The leading order of the discretization error is O(a2).

IR limit:

In high energy region,

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

TPL scheme can control both statistical and systematic errors.

Comparison of Running of the renormalized coupling constants in Quenched QCD

Capitani et.al:NPB B544 : 669-698

TPL schemeWloop scheme

SF scheme

Phys. Rev. D 80, 034507 (2009)

The value of running itself has scheme dependence. The behaviors of the coupling constant should be scheme independent except for lattice artifacts.

Nf=12 case (TPL scheme)

Lattice fermion

From Nielsen-Ninomiya theorem, one massless lattice fermion has twice degree of freedom for each direction. In four dimensions, there are 16 degree of freedom.

Wilson fermion adds Wilson term to kill 15 doublers breaks chiral symmetry

staggered fermion 16 = 4 spinors and 4 flavors

Overlap fermion (Neuberger 1998) Ginsparg-Wilson relation (chiral symmetry on the lattice)

Domain-Wa ｌ l fermion (Kaplan 1992, Shamir 1993) 5-dim. formulation takes a limit that symmetry breaking effect goes to zero

Wilson staggered Overlap Domain-wall

1 1/4 100 20

Simulation cost

Nf=12 QCD

Lattice size for step scaling

parameter

Hybrid Monte Carlo algorithm (exact algorithm)

At high energy, the data values of the larger lattice size are larger than that of the small lattice .

This behavior is same as one of the SF scheme.

ratio of Polyakov loop

beta

In low beta region (4.0 < beta < 7.0)

On the other hand, the TPL coupling shows an upper boundary for each lattice size.

(cf. Also, in the case of Nf=16 in Wilson loop scheme: Fodor et al. arXiv:0911:2934,)

SF scheme (Appelquist et.al.) In low beta, the SF scheme shows the inversion of the order of lattice size dependence in the fixed beta.

ratio of Polyakov loop

The region of these data values corresponds to 5< beta < 18.

If there is a fixed point, the beta function can be approximated as a linear function of u around the fixed point .That gives

starting point g2=0.4711

We should be study the large u region (beta<5) to find the fixed point.

The growth of renormalized couplings for each step

The running coupling constant

Consistent with the two-loop result in the high energy region.

We cannot find the flat region of running coupling constant yet.

The accumulated error can be calculated by the error of scaling function.

Scaling function:

Error of running coupling

We fit the scaling function sigma(u): 2-parameter fit

Here, s0 and s1 can be fixed by the perturbative 2-loop beta fn.

ALPHA Collaboration ,Nucl. Phys. B 713:378-406,2005

Non-perturbative beta function can be obtained iteratively the following equation.

At the starting point, we use 2-loop beta function in the right hand side.

Nonperturbative beta function(Preliminary)

fitting resultss2= 4.1(4) ×10-2

s3= -1.8(2) ×10-2

chisq/dof= 1.2

Center value of beta fn.

Nonperturbative beta funtion (Preliminary)

There are large error bars, because the errors are accumulated.

example:

The error of beta fn. also can be calculated by the error of the fitting parameters; s2 and s3.

scaling function

To reduce the magnitude of errors, we added many independent steps.

# of data= 90s2= 3.0(2) ×10-2

s3= -1.2(1) ×10-2

chisq/dof= 0.85

# of data= 37 s2= 4.1(4) ×10-2

s3= -1.8(2) ×10-2

chisq/dof= 1.2

fixed point(?)

the previous beta fn. the beta fn. with improved error

There is a signal of fixed point (?)

SummaryIn low beta, the TPL coupling’s behavior is different from SF scheme, and shows the trend of reaching a plateau for each lattice size.

The TPL running coupling and the nonperturbative beta fn. in the case of Nf=12 shows a signal of IR fixed point.

In our analyses, we estimated the discretization errors in O(a2).

Now, we are working in progress to study the lower energy region.

Future directionsNumerical measurement of the anomalous dimension at the IR point the composite operator of fermions is interesting

Study Nf dependence (for example Nf=8 or 16) to study arbitrary Nf, we need Overlap or Domain-wall fermion (maybe we can do in 5 years…)

The other gauge group and representation of fermions many people are studying…(Sannino, Yamada, Ohki…)

Make some improvements to reduce the discretization errors

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

How to take the continuum limit

input output

Examples of input parameters: , mass, Sommer scale….

To measure the running coupling

Notation:

scaling function

We have to take the continuum limit of measurement values.

In our work, we measure the running coupling constant using the other scheme without O(a/L) discretization error, not SF scheme.The existence of the fixed point should be scheme independent.