2-4 February 2004 Tsukuba-Univ.

“Nonperturbative improvement and Renormalization”

Tetsuya Onogi ( YITP, Kyoto-Univ. )

Hideo Matsufuru ( CRC, KEK )

Takashi Umeda (YITP, Kyoto-Univ.)

Hidenori Fukaya (YITP, Kyoto-Univ.)

Tsukuba Workshop / 2-4 Feb. 2004

created by

Nonperturbative improvement

on anisotropic lattices

Contents

1

1) Introduction

Our motivation, Present status

2) Nonperturbative O(a) improvement

on anisotropic lattices

Strategy, Numerical simulation

3) Summary & Outlook

Anisotropic lattices for heavy quark physics

Tsukuba Workshop / 2-4 Jan. 2004

Introduction

1

Tsukuba Workshop / 2-4 Jan. 2004

Our motivation:

to provide hadronic matrix elements for flavor physics

with high precision

Recent experimental developments in flavor physics

--- B factories, Charm factory

need precise theoretical predictions (~2%)

of hadron matrix elements

decay constants, bag parameters, form factors,

quark masses, light-cone wave functions, etc.

Why anisotropic lattice ?

1

Tsukuba Workshop / 2-4 Jan. 2004

Problem in lattice QCD calculations:

for heavy quarks ( c and b ), large error

We need a framework in which has

(i) continuum limit

(ii) systematic improvement

such as nonperturbarive renormalization technique ( for )

(iii) modest size of computation

To achieve calculations with ~2% precision, we need

yet another approach which satisfies above conditions (i)-(iii).

our proposal: Anisotropic lattice

O



m

q

a



m

q

=

0



=

a

s

/

a

t

OK

OK

next page

Our expectation

1

Tsukuba Workshop / 2-4 Jan. 2004

For (not necessarily )

mass dependences of parameters in the action are so small

that the tuned parameters for massless quark are also

applicable in such a quark mass region

Then, systematic improvement is possible !

( performed at ~ 0 )

To be justified numerically, and in perturbation theory.

Tree level : OK

O(a ) improved version : OK (2% accuracy)

High precision : in progress

m

q

≪

1

/

a

t

m

q

≪

1

/

a

s

m

q

0

Quark action

1

S

F

=

∑

x

,

y







x



K



x

,

y







y



K



x

,

y



=



x

,

y

−



t

[



1

−



4



U

4



x





x





4,

y





1





4



U

4



x

−



4





x

−



4,

y

]

−



s

∑

i

[



r

−



i



U

i



x





x





i

,

y





r





i



U

i



x

−



i





x

−



i

,

y

]

−



s

c

E

∑

i



4

i

F

4

i



x





x

,

y



r



s

c

B

∑

i



j



i

,

j

F

i

,

j



x





x

,

y

T.Umeda et al., Int. J. Mod. Phys. A16 (2001) 2215.

J.Harada et al., Phys. Rev. D64 (2001) 074501.

constructed following the Fermilab approach

El-Khadra et al., Phys. Rev. D55 (1997) 3933.

(action retains explicit Lorentz invariant form)

must be tuned

r

=

1

/



Tsukuba Workshop / 2-4 Jan. 2004



F

,

c

E

,

c

B

Our previous results

1

Tsukuba Workshop / 2-4 Jan. 2004

One-loop perturbative calculation shows

that dependence under control

Harada et al., Phys. Rev. D64 (2001) 074501.

Numerical simulation ( in quenched approx., tadpole tree & )

Quark mass dependence of is small for

Matsufuru et al., Phys. Rev. D64 (2001) 114503.

Relativity relation of heavy-light meson well holds for

using massless tuned

Harada et al., Phys. Rev. D66 (2002) 014509.

Heavy-light decay constant

consistent with previous result & with O(10%) accuracy

Matsufuru et al., hep-lat/0209090

Encouraging results for further development !!

m

q

a

t

≪

1



F

m

a

a

t

≪

1



F

c

E

c

B

m

q

a

t

For high precision study

1

Tsukuba Workshop / 2-4 Jan. 2004

Calibrations of parameters to 0.2% level of accuracy

in quenched approximation

Applicable to a few percent calculation of matrix elements

Gauge field:

O(0.2%) calibration is possible Matufuru et al, Lattice2003

Quark field the latter half of this talk

Test in heavy quark region

Applications to heavy-light matrix elements

Strategy for the improvement

1

Tsukuba Workshop / 2-4 Jan. 2004

Five parameters should be determined

Calibration step

(1) Schrödinger functional method

tuning of , ( )

(2) Spectroscopy in coarse and fine directions on lattice with T,L>2fm

Physical isotropy condition for

tuning of ,

(3) Schrödinger functional method

determine ,

(4) Check of systematic error

Light hadron spectrum and dispersion relation,

Taking the continuum limit





F

,

c

E

,

c

B

,

c

A

,



c



c

E

c

B

c

B



F

c

A



c

m

PS

,

m

V

Determination of

1

Tsukuba Workshop / 2-4 Jan. 2004

c

E

Improvement condition

at finite L/a with classical background field

determination at high

determination at low



M





according to the Alpha collaboration's program





1,



2,



3



=

1

6



−



,

0,









1

'

,



2

'

,



3

'



=

1

6



−

5



,

2



,

3





T

=

2

L



,



k

=

0

Test on isotropic lattice

1

Tsukuba Workshop / 2-4 Jan. 2004

consistent with Alpha collaboration result

is well determined by a background field, is not so

c

E

c

B

Result at high on anisotropic lattice

1

Tsukuba Workshop / 2-4 Jan. 2004

on anisotropic lattice

gluonic anisotropy from

Klassen, NPB

mean-field from

plaquette in 1-loop calc.

tadpole tree

nonperturbative improved is close to tree level value

is almost independent of



=

9.5

8

3

×

64



=

4

c

E

=

1.13

c

B



M



M

c

E



1

Tsukuba Workshop / 2-4 Jan. 2004

on anisotropic lattice

gluonic anisotropy from

Matsufuru et al, PRD

tadpole tree



=

6.1

8

3

×

64



=

4

c

E

=

1.26

nonperturbative improved is determined on a practical lattice

is not so sensitive to

c

B



M

c

E

Result at = 6.1 on anisotropic lattice



a

s

−

1

=

2.0

GeV



M

Determination of

1

Tsukuba Workshop / 2-4 Jan. 2004



F

is determined from meson mass ratio in fine to coarse direction



F

test on 136config.



=

5.75,

12

2

×

24

×

96,

a

s

−

1

=

1.1

GeV

is roughly determined

this results is consistent with from dispersion relation of meson

dependence is mild



F



F

c

B

Determination of

1

Tsukuba Workshop / 2-4 Jan. 2004

is determined from

meson mass splitting ratio in fine to coarse direction

meson mass splitting is too noisy

dependence is mild (?)

c

B

c

B

c

B

Effective mass plot

1

Tsukuba Workshop / 2-4 Jan. 2004

using local operators

each meson mass is fitted with t=30~48, z=8~12

using smeared operators

Schrödinger Functional method with coarse temporal direction

for improvement

Summary

1

Tsukuba Workshop / 2-4 Jan. 2004

We are developing anisotropic lattices for precise computation

of heavy-light matrix elements.

Results obtained so far are encouraging for further development.

We have started precise calibrations of gauge and quark fields

in quenched approximation.

Nonperturbative improvement of anisotropic quark action

is well determined by Schrödinger functional method

is determined by meson mass ratio

in large volume simulation

suffers from large statistical noise

c

E

c

B



F

Outlook

1

Tsukuba Workshop / 2-4 Jan. 2004

How can we verify that parameters in suffices

for heavy quark mass ?

Mass dependent tuning for bottom quark region

Applications to heavy-light matrix elements

Extension to dynamical QCD

Nonperturbative improvement of anisotropic quark action

determination of

Smeared operators

Schrödinger functional with coarse time direction

c

B

m

q

=

0

Improvement condition

1

Tsukuba Workshop / 2-4 Jan. 2004



M

L T

8 16 1 2.77

10 20 1 2.19

8 64 4 0.0690

10 80 4 0.0441





M

×

10

4

in a classical background field is determined numerically



M

=



M

∣

M

=

0,

c

E

=

1,



F

=



at

L

/

a

s

=

8

Takashi Umeda

2003-09-04T17:47:02

Takashi Umeda