Universality of the Nf2

1
title
Universality of the Nf2 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
2
contents
? Introduction
? Nf2 running coupling in SF by Alpha
collaboration (Nucl.Phys.B713378-406,2005)
? purpose of our study
? review about SF scheme
? finite size scheme
? definition of running coupling (SF scheme)
? our result
? set up
? concludion
3
Running coupling (result)
Running coupling (SF scheme) non
perturbative result (Nf2 dynamical) hep-lat
/0411025
(Alpha)
-- PT one and non-PT one are same each other(?)
4
Beta function (Nf2)
Non-perturbative beta function (SF
scheme) hep-lat /0411025
Nf2 non-PT beta become be apart from PT one in
strong coupling region.
Nf2 beta is passing the Nf0 one.
5
Gauge action
Purpose of our study
Purpose of our study Check with different
Action (especially strong regime)
Action of gauge field
Iwasaki Action
Plaquete Action
6
Intro
Why Schrödinger Functional ?
purpose non-PT check of QCD
Lattice QCD
QCD (perturbative)
Jet physics
Hadronic input
Compare by Perturbatively
Develop by Non-PT
Low Energy
High Energy
7
Why Schrödinger Functional ?
Advantage 1 solve large lattice problem
? renormarization scale
? cut off scale
? reduce finite size effect
? calculation on the PT scale
Restriction of Lattice size
SF scheme can reduce this restriction.
8
?????scheme
Why Schrödinger Functional ?
Advantage 1 solve large lattice problem
? call scheme include Finite size
effect
? renormarization scale
finite size scheme
? cut off scale
? reduce finite size effect
? calculation on the PT scale
Restriction of Lattice size
(Not significant)
9
???????(??)
SF scheme include finite size effect
----- do you feel it
strange ?
point Running coupling is only expansion
parameter
Physical obserbable
It doesnt matter whether this part depend on Box
size.
we can shift it finite value
10
SF scheme
(Alpha 92)
Definition of Running coupling (SF scheme)
define running coupling as coeff of
Effective Action
11
SF scheme(??)
(Alpha92)
Definition of running coupling
temporal dirichlet BC
Spatial Twisted BC
Back ground field
normarization factor
Effective Action
12
??2
SF scheme solve the Large scale problem
If calculate over large scale, enormous lattice
size is needed.


13
SSF ??
Step scaling function
?) like one integrate beta-function with
Initial value u to twice box size.
(S2)

Possible to follow Running
14
SSF ???
Calculation of step scaling function
N2
? we can choose any lattice size you like.
ß?ß

• Tune beta for running coupling eqal to u0.

• 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
15
Continuum limit
Take N large with Tune beta to make
ß?ß
Constant.
N
ß?ß
N
16
Gauge action
Purpose of our study
Purpose of our study Check with different
Action (especially strong regime)
Action of gauge field
Iwasaki Action
Plaquete Action
17
Set up (others)
Set up
? Fermion action Clover action (Nf2)
? Csw Non-PT
? Tuning of
(mass
independent scheme)
Uncertainty in from mismatch of m
is estimated by Perturbatively.
?algorithm HMC
Machine cluster machine kaede in academic
computing Communication
center Tsukuba Univ (60 cpu )
18
Set up (gauge)
Set up
Boundary O(a) improvement
Boundary O(a) improved coeffcient ct.
(Only these was not given by non-PT.)
plaquete
2-loop
1-loop
0
0
1-loop
1-loop
1-loop
0
iwasaki
(Alpha 92, 96,00)
gauge
fermion
(Takeda 04)
19
Running coupling ???
Running coupling (SF scheme) non
perturbative result (Nf2 dynamical) (Alpha0
5)
We calculated running coupling In
Weak and strong coupling region.
Calculate in weak coupling point and strong
coupling point
20
Weak coupling
Continuum extrapolation of running coupling
Preliminary
Iwasaki action ? Plaquete action (Alpha 04)
???????
Iwasaki action 1-loop
Plaquete action
Iwasaki action tree
Weak coupling (u0.9793) Iwasaki ß6
plaquete ß9
?(u, a/L)
?(u, a/L)
a/L
? weak coupling plaquete action ? Iwasaki
action ??????????
21
Weak coupling
result
weak coupling region
Continuum extrapolation of running coupling
Compare Iwasaki action with Plaquete action
(Alpha 04)
Iwasaki action 1-loop
Plaquete action
Iwasaki action tree
Weak coupling (u0.9793) Iwasaki ß6
plaquete ß9
? weak coupling Result from Iwasaki action is
consistent with that from plaquete action.
a/L
22
strong 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

23
strong coupling region
Dstribution of
Ex)
strong coupling L164 Beta 2.61192 Kappa
0.13363
Number
631 traj
Distribution looks reasonable.
In the case of Iwasaki action, We didnt find
long auto-correlation.
24
Strong coupling
result
strong coupling region
Continuum extrapolation of running coupling
Compare Iwasaki action with Plaquete action
(Alpha 04)
Iwasaki action 1-loop
Plaquete action
scaling violation is large
Strong coupling (u3.3340) Iwasaki
ß2 plaquete ß5
?(u, a/L)
a/L
Coupling boundary
25
Scaling (quenched)(??)
Scaling of Iwasaki action ( quenched)
Nucl.Phys.Proc.Suppl.129408-410,2004 S.Takeda,
S.Aoki, K.Ide
Strong point
one-loop
(same with one we used)
Choice B
one-loop
Choice A
tree
Tree ct is better than PT one.
It seems be able to extrapolate.(?)
26
We calculated that with tree again.
tree
Another set up is same with before one.
27
Strong coupling
result
Plaquete action
Iwasaki action 1-loop
Weak coupling (u0.9793) Iwasaki ß6
plaquete ß9
Strong coupling (u3.3340) Iwasaki
ß2 plaquete ß5
a/L
a/L
conclusiton
28
As before, We didnt find long auto-correlation
in Iwasaki action.
Dstribution of
Ex)
strong coupling L164 Beta 2.755 Kappa
0.1334
Number
11161 traj
29
Strong coupling
result
weak coupling regime
Plaquete action
Iwasaki action 1-loop
Weak coupling (u0.9793) Iwasaki ß6
plaquete ß9
The result remains consistent with one from
plaquette action .
a/L
30
Strong coupling
result
strong coupling region
Plaquete action
Iwasaki action 1-loop
Strong coupling (u3.3340) Iwasaki
ß2 plaquete ß5
Large scaling violation in the case of 1-loop ct
is much Improved by tree ct.
a/L
O(a) behavior become very good and we got result
consistent with result from plaquete action.
31
???
conclusion
Purpose
calculate SF running coupling in Weak and strong
with Iwasaki action And compare result with
earlier study.
Tree impposible ct
result
We confirmed that the result is consistent each
other within error bar.
( in SF scheme) Beta function behave differently
from perturbative expectation in strong
coupling regime.
32
?????(??)
hep-lat/0110088 Nf3 (clover)
Iwasaki action
Plaquete action
? Plaquete action ?? ???????? ? Iwasaki action
?? ???????????
33
Scaling (quenched)(??)
Scaling of Iwasaki action ( quenched) (Takeda04)
Weak point
Strong point
34
Nf2 data(??)
Nf2 Step Scaling Function
35
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