Update on the joint project
CP-PACS/JLQCD
Nf=2+1 program
Tomomi Ishikawa (CCS, Univ. of Tsukuba)
for CP-PACS/JLQCD collaboration
tomomi@ccs.tsukuba.ac.jp
The 2nd International Lattice Field Theory Network Workshop
“From Actions to Experiment”
March 07-10, 2005,
National e-Science Center, Edinburgh
1
Introduction
Light hadron spectrum
Direct test of QCD at low energy scale
Determination of fundamental parameters
quark masses, QCD coupling, ....
Systematic studies by CP-PACS and JLQCD
quenched QCD (continuum limit)
RG-improved gauge
+ clover quark ( tad.imp. cSW )
(CP-PACS, 2001)
plaquette + Wilson
(CP-PACS, 2001)
Systematic deviation
from experiment (5-10%)
artifact due to the quenched approx.
2
1.03
2-flavor QCD (continuum limit)
ud : dynamical
s : quenched
φ
experiment
1.02
meson mass [GeV]
1.01
RG-improved gauge
+ clover quark
(tad.imp. cSW )
(CP-PACS, 2001)
Nf=
1
2
0.99
0.98
Nf=0
0.97
0.96
0.95
K-input, RG+clover(PT)
0
0.05
0.1
a [fm]
0.15
0.2
0.25
0.58
deviation is reduced
0.57
φ-input, RG+clover(PT)
meson mass [GeV]
0.56
Next Step :
3-flavor full QCD
ud : dynamical
s : dynamical
Nf=0
0.55
0.54
0.53
2
=
f
N
0.52
0.51
0.5
K
experiment
0.49
0.48
3
0
0.05
0.1
a [fm]
0.15
0.2
0.25
Contents
Introduction
Strategy
Simulation parameters
Analysis method
Light meson spectrum
Light meson decay constant
Light quark masses
Summary
The results in this talk are preliminary.
4
Strategy
Lattice action
gauge : RG improved action (Iwasaki, ‘83)
quark : non-perturbatively O(a) improved Wilson action
csw is non-perturbatively determined. (K-I. Ishikawa, ‘03)
Algorithm
with degenerate up and down quarks and strange quark
We employ HMC algorithm for up and down quarks.
Polynomial HMC (Forcrand and Takaishi, ‘97, K-I.Ishikawa et.al., ‘02)
Polynomial approx. of inv. of Dirac matrix
D−1 ∼ P2Npoly [D] = T̄ [D]T [D]
Metropolis test for correction factor det[P2Npoly [D]D]
exact algorithm
We employ PHMC algorithm for strange quark.
5
Simulation points and statistics
Production: on-going
Production: already finished
β = 2.05 (a ∼ 0.07 fm)
β = 1.83 (a ∼ 0.122 fm)
L3 × T = 283 × 56
cSW = 1.628
L3 × T = 163 × 32
cSW = 1.761
stat. = 3000 − 5000 traj.
0
1
2
fixed physical
volume
3
∼ (2.0 fm)
1
stat. = 6000 − 9000 traj.
3
2
β = 1.90 (a ∼ 0.10 fm)
L3 × T = 203 × 40
cSW = 1.715
2
a
stat. = 5000 − 8000 traj.
Production: already finished
Scaling study
6
Machines
Earth Simulator
@JAMSTEC
SR8000/F1
@KEK
CP-PACS
SR8000/G1
VPP5000
@CCS, Univ. of Tsukuba @CCS, Univ. of Tsukuba @ACCC, Univ. of Tsukuba
7
Simulation parameters
K parameters
5 ud quark parameters
2 s quark parameters
mps /mV ∼ 0.60 − 0.78
(0.18 : exprriment)
mps /mV ∼ 0.7
(0.68 : 1-loop ChPT)
3
3
80
physical point
50
40
VWI
Ks=0.13710
0
VWI
mud
50
[MeV]
physical point
70
Ks=0.13640
40
Ks=0.13510
70
60
Ks=0.13540
50
50
100
physical point
80
60
Ks=0.13760
3
90
[MeV]
80
Ks=0.13580
VWI
90
[MeV]
90
60
3
β=2.05, L xT=28 x56
100
mS
[MeV]
100
mS
VWI
100
70
3
β=1.90, L xT=20 x40
mS
3
β=1.83, L xT=16 x32
0
VWI
mud
8
50
[MeV]
100
40
0
VWI
mud
50
[MeV]
100
Simulation parameters at β=1.83
β = 1.83(a ∼ 0.122fm), L3 × T = 163 × 32, cSW = 1.761
δτ Npoly
traj.
0.13655
1/80
80
7000
0.7773(14)
0.7523(15)
0.13710
1/85
80
7000
0.7518(21)
0.7518(21)
1/100
100
7000
0.7079(19)
0.7416(17)
0.13800
1/120
110
8000
0.6623(23)
0.7363(17)
0.13825
1/140
120
8000
0.6214(25)
0.7343(16)
0.13655
1/90
110
7000
0.7766(15)
0.7230(19)
0.13710
1/100
110
8600
0.7445(14)
0.7126(17)
1/110
120
8000
0.7035(18)
0.7035(18)
0.13800
1/120
130
8100
0.6518(22)
0.6936(19)
0.13825
1/150
150
8100
0.6085(33)
0.6887(22)
Kud
0.13760
0.13760
Ks
0.13710
0.13760
δτ, Npoly
mP S /mV (LL) mP S /mV (SS)
PHM C ! 85%, PGN P ! 90%
9
Simulation parameters at β=1.90
β = 1.90(a ∼ 0.1fm), L3 × T = 203 × 40, cSW = 1.715
Kud
Ks
δτ Npoly
traj.
mP S /mV (LL) mP S /mV (SS)
0.13580
1/125
5000
0.7674(16)
0.7674(16)
0.13610
1/125
6000
0.7427(18)
0.7640(17)
7500
0.7206(19)
0.7687(14)
0.13640
0.13580
1/140
110
0.13680
1/160
8000
0.6710(28)
0.7677(17)
0.13700
1/180
8000
0.6384(22)
0.7687(15)
0.13580
1/125
5200
0.7666(17)
0.7210(22)
0.13610
1/125
8000
0.7448(15)
0.7187(17)
9000
0.7153(18)
0.7153(18)
0.13640
0.13640
1/140
140
0.13680
1/160
9000
0.6642(22)
0.7139(18)
0.13700
1/180
8000
0.6237(28)
0.7100(20)
δτ, Npoly
PHM C ! 85%, PGN P ! 90%
10
Simulation parameters at β=2.05
β = 2.05(a ∼ 0.071fm), L3 × T = 283 × 56, cSW = 1.628
Kud
Ks
0.13470
δτ Npoly
1/175
traj.
1500
mP S /mV (LL) mP S /mV (SS)
0.7715(59)
0.7224(67)
Production is
now on-going.
0.13510
0.13540
1/195
0.13510
1/225
200
1500
0.7335(46)
0.7335(46)
1500
0.6915(111)
0.7472(92)
0.13550
1/235
1500
0.6549(121)
0.7329(69)
0.13560
1/250
1500
0.6305(68)
0.7372(58)
0.13470
1/175
1500
0.7761(58)
0.6768(63)
0.13510
1/195
1500
0.7353(66)
0.6795(82)
1500
0.6889(96)
0.6889(96)
0.13540
0.13540
1/225
250
0.13550
1/235
1500
0.6614(175)
0.6896(138)
0.13560
1/250
1500
0.6320(102)
0.6797(88)
δτ, Npoly
PHM C ! 85%, PGN P ! 90%
11
Measurement
Physical volume ∼ (2.0 fm)3 is not large to calculate
baryon.
We focus on the meson sector.
Measurements are performed at every 10 trajectories.
Smearing
For meson masses and quark masses:
Doubly smeared source - local sink is used.
For decay constants:
Doubly smeared source - smeared sink is also used.
12
Analysis method
Chiral extrapolation
Polynomial fit functions in quark masses are used.
include up to quadratic terms
interchanging symmetry among 3 sea quarks
among 2 valence quarks
Chiral fits are made to
light-light (LL), light-strange (LS) and strange-strange (SS)
simultaneously.
ignoring correlations among LL, LS and SS
Application of WChPT is future task.
13
Takeda’s talk
Chiral fit function for mesons
Pseudo-Scalar
Vector
m2P S (mud , ms ; mval1 , mval2 )
mV (mud , ms ; mval1 , mval2 )
= BSP S (2mud + ms )
+BVP S (mval1 + mval2 )
= AV + BSV (2mud + ms )
+BVV (mval1 + mval2 )
PS
+DSV
(2mud + ms )
×(mval1 + mval2 )
V
+DSV
(2mud + ms )
×(mval1 + mval2 )
PS
+CS1
(2m2ud + m2s )
PS
+CS2
(m2ud + 2mud ms )
V
+CS1
(2m2ud + m2s )
V
+CS2
(m2ud + 2mud ms )
+CVP 1S (m2val1 + m2val2 )
+CVV 1 (m2val1 + m2val2 )
+CVP 2S mval1 mval2
+CVV 2 mval1 mval2
14
Chiral fit results
β=1.83
3
3
3
β=1.83, L xT=16 x32
3
β=1.83, L xT=16 x32
1
χ2 /d.o.f. = 0.32
χ2 /d.o.f. = 0.66
0.5
0.9
0.4
a mV
(a mPS)
2
0.8
0.3
0.7
0.2
0.1
0
0.6
Ks=0.13710
Ks=0.13760
0
0.5
0.01 0.02 0.03 0.04 0.05 0.06
a
VWI
mud
Ks=0.13710
Ks=0.13760
0
0.02
0.04
a
15
VWI
mud
0.06
β=1.90
3
3
3
β=1.90, L xT=20 x40
3
β=1.90, L xT=20 x40
0.4
0.8
χ2 /d.o.f. = 1.09
χ2 /d.o.f. = 0.93
0.7
a mV
(a mPS)
2
0.3
0.2
0.1
0.6
0.5
Ks=0.13580
Ks=0.13640
0
0
0.01
0.02
a
0.03
0.04
VWI
mud
0.05
Ks=0.13580
Ks=0.13640
0.4
0.06
0
0.01
0.02
a
16
0.03
0.04
VWI
mud
0.05
0.06
β=2.05
3
3
3
β=2.05, L xT=28 x56
3
β=2.05, L xT=28 x56
0.5
χ2 /d.o.f. = 1.38
χ2 /d.o.f. = 0.20
0.45
a mV
(a mPS)
2
0.1
0.4
0.05
0.35
Ks=0.13510
Ks=0.13540
0
0
0.01
0.02
a
0.03
VWI
mud
0.04
Ks=0.13510
Ks=0.13540
0.3
0.05
0
0.01
0.02
a
17
0.03
VWI
mud
0.04
0.05
Physical point
Inputs to fix the quark masses and the lattice spacing
K-input
mπ , m ρ , m K
mπ , m ρ , m φ
φ-input
lattice spacings
β=1.83
β=1.90
β=2.05
K-input
φ-input
1.611(23) GeV
1.968(37) GeV
2.51(16) GeV
1.596(27) GeV
1.965(35) GeV
2.49(17) GeV
K-input and φ-input give consistent results.
18
Light meson spectrum
Continuum extrapolation
non-perturbatively O(a) improved Wilson quark
Meson spectrum should scale as
a2 .
extrapolation function :
m(a) = A + Ba2
2 point extrapolation (β=1.90 + β=1.83)
The statistics at β=2.05 is not enough at
present.
19
1.05
1
φ-input
K-input
φ
K*
meson mass [GeV]
1
meson mass [GeV]
0.9
experiment
0.95
0.8
experiment
0.7
0.6
K*
0.9
K
0.5
0.85
0
0.005
2
0.01
2
0.015
0.4
0.02
a [fm ]
0
0.005
2
0.01
2
0.015
0.02
a [fm ]
In the continuum limit
In the continuum, meson spectrum is consistent with
experiment.
But the error in the continuum limit is large at this stage.
20
Comparison with Nf=2, 0
1.05
1
1
meson mass [GeV]
meson mass [GeV]
0.9
φ
Nf=
2
1
φ-input
K-input
Nf=
2+
Nf=0
0.95
experiment
K*
0.8
0.7
experiment
0.6
K*
0.9
0.5
0.85
0
0.05
0.1
0.15
a [fm]
0.2
0.4
0.25
Nf=2
Nf=0
K
Nf=2+1
0
0.05
0.1
0.15
a [fm]
0.2
We observe that meson spectrum in Nf=2+1 seems
slightly closer to experiment than that in Nf=2.
21
0.25
Light meson decay constant
PS decay constants
fP S
!0|A4 |P S" = fP S mP S
Vector decay constants
fV
!0|Vµ |V " = !µ fV mV
Improvement and renormalization
Tad-pole-improved 1-loop cA , ZA , bA , ZV , bV
Improvement of the vector current is not performed at the
present.
Smearing
For the calculation, smeared source - smeared sink is
also used.
22
Results
320
190
experiment
180
170
310
fK
290
280
fV [MeV]
fPS [MeV]
160
150
140
fπ
130
270
fρ
260
250
240
230
experiment
220
120
110
fφ
300
K-input
0
0.005
0.01
2
2
a [fm ]
0.015
K-input
210
200
0.02
0
0.005
0.01
2
a [fm ]
2
0.015
0.02
Consistent with experiment in the continuum limit
But the error in the continuum limit is very large at this stage.
O(a) contributions in the decay constants remain (?)
23
Light quark masses
VWI quark mass
WI
mVquark
1
=
2
!
"
1
1
−
K
Kc
mP S,LL (Kud , Ks )|Kud =Ks =Kc = 0
VWI quark mass can be negative value due to lack of
chiral symmetry of the Wilson quark.
AWI quark mass
!∆4 A4 (t)P (0)"
=
2 !P (t)P (0)"
The scaling violation is small in Nf=2 case.
I
mAW
quark
Renormalization
−1
µ
=
a
MF-improved 1-loop matching with MS at
.
4-loop running to µ = 2 GeV
24
Continuum extrapolation (using β=1.83 and 1.90)
15
4
K-input
(µ=2GeV) [MeV]
10
MS
3.5
3
mud
mud
MS
(µ=2GeV) [MeV]
AWI, K-input
5
AWI
0
-5
-10
-15
VW
-20
I
2.5
0
0.005
2
0.01
2
0.015
-25
0.02
a [fm ]
0
0.005
2
0.01
2
0.015
0.02
a [fm ]
We assume that the scaling is a2 .
For ud quark VWI definition gives a positive value in the
continuum.
25
110
AWI and VWI quark mass
are consistent each other
in the continuum limit.
100
K-input and φ-input also
give consistent results.
φ-input
But the error in the
continuum limit is very
large at this stage.
90
ms
MS
(µ=2GeV) [MeV]
VWI
AWI
K-
80
inp
0
0.005
2
ut
0.01
2
0.015
0.02
a [fm ]
26
Comparison with
Nf=2, 0
ud quark mass
Nf=0,VWI
MS
(µ=2GeV) [MeV]
4.5
mud
In the continuum limit,
ud quark masses in
Nf=2+1 are almost
same as in Nf=2.
5
4
Nf=
0,A
WI
3.5
Nf=2,A
3
WI
+1
=2
Nf
2.5
27
0
0.05
,V
W
I
I
W
2
,A
K-input
Nf
=2
0.1
0.15
a [fm]
0.2
0.25
120
150
ms (K-input)
Nf=0,VWI
(µ=2GeV) [MeV]
Nf=
0,A
W
I
100
WI
+1
WI
0
0.05
Nf=
0,A
WI
120
0.1
0.15
a [fm]
VW
,
2
=
f
110
N
0.2
80
0.25
I
I
W
2,A
=
f
N
100
Nf=2+1,AWI
90
1,V
,A
W
I
WI
2+
80
70
Nf=2,A
130
Nf=
Nf
=2
ms
90
Nf=0,VWI
MS
,V
2
Nf=
ms
MS
(µ=2GeV) [MeV]
110
140
ms (φ-input)
0
0.05
0.1
Nf=2
+1,V
WI
0.15
a [fm]
0.2
0.25
Strange quark mass
In the continuum limit, Strange quark masses in Nf=2+1
are almost same as in Nf=2.
28
Summary
Nf=2+1 project
RG-improved gauge action
+ non-perturbatively O(a) improved Wilson quark action
Exact Nf=2+1 algorithm (PHMC)
Production run
Production at β=1.83 and 1.90 has been already
finished.
Production at β=2.05 is on-going.
(It will be finished in 1 year.)
29
Analysis of meson spectrum and quark masses
Encouraging results in the continuum limit are obtained.
2
Assuming that the scaling is a ...
Meson spectrum is consistent with experiment.
All definitions and inputs of quark masses are
consistent.
However large error remains at this stage.
Required task
Production and analysis at β=2.05
Present statistics is too small.
Non-perturbative determination of
cΓ , ZΓ , bΓ
Application of WChPT for the chiral extrapolation
(in progress)
Takeda’s talk
30