spectrum and quark masses in 2+1 ßavor qcd -results from ......dynamical up, down and strange...
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Spectrum and quark masses in 2+1 flavor QCD
-results from CP-PACS and JLQCD-
Tomomi IshikawaCenter for Computational Sciences, Univ. of Tsukuba
CP-PACS and JLQCD Collaborations
KEK Theory Meeting 2006“Particle Physics Phenomenology”
2nd-4th, March 20061
Mathematically fine definition of NP QCDRegularization is naturally introduced by lattice spacings.
finite number of d.o.f.
Monte Carlo simulation is applicable.
Hadronic observable can be obtained directly from “first principle” of QCD.
Discretization error can be controlled theoretically.
Lattice QCD
2
quark field ψ
link variable (gauge field) Uµ
lattice spacing a
〈O[Uµ, ψ̄, ψ]〉
=1Z
∫DUµDψDψ̄O[Uµ, ψ̄, ψ]
× exp(−Sgauge[Uµ]− Squark[Uµ, ψ̄, ψ]
)
Applicationshadron spectroscopy
fundamental parameters in QCD
• quark masses
•
weak matrix elements
• B physics, CP violation, CKM matrix
topology, confinement
finite temperature, finite density
3
αs, ΛQCD
::
main topic in this talk
Difficulties of the lattice QCD simulationlarge number of d.o.f.
Increase in numerical cost as quark mass is reduced
Systematic errors and approximationfinite lattice spacings continuum extrapolation
finite volume infinite volume extrapolation
large quark mass chiral extrapolation
quenched approximation (not systematic approximation)
• ignore vacuum polarization of quarks (sea quark effect)• numerical cost is largely reduced
4
e.g. # of d.o.f. ∼ (323 × 64)× (8× 4 + 4× 3) ∼ O(108)
∫DψDψ̄ exp
(∫d4xψ̄/D[Uµ]ψ
)= det /D[Uµ] =1
Quenched (Nf=0)• CP-PACS : Plaquette gauge + Wilson quarks, continuum limit
Phys.Rev.D67(2003)034503, Phys.Rev.Lett.84(2000)238.
• CP-PACS : RG improved gauge + clover quarks with PT , continuum limit Phys.Rev.D65(2002)054505.
Dynamical up and down quarks (Nf=2)
• JLQCD : Plaquette gauge + clover quarks eith NPT Phys.Rev.D68(2003)054502.
• CP-PACS : RG improved gauge + clover quarks with PT , continuum limit Phys.Rev.D65(2002)054505, Phys.Rev.Lett.85(2000)4674.
Large scaled simulations in Nf=0 and Nf=2 QCD by CP-PACS and JLQCD
5
strange : quenched
cSW
cSW
cSW
meson spectrumNf=0
• Systematic deviation from experiment is O(5-10%).
Nf=2
• The deviation is considerably reduced.
Nf=0 and Nf=2 results
6
0.950
1.000
1.050
0.8700.8800.8900.900
0.885
0.890
0.895
0.900
mes
on m
asse
s [G
eV]
0 0.1 0.2a [fm]
0.500
0.550
Nf=0(RG+PT clover)Nf=2(RG+PT clover)experiment
φ (K-input)
K* (K-input)
K* (φ-input)
K (φ-input)RG+clover (Phys.Rev.D65(2002)054505)
cSW : perturbationNf = 0 : a = 0.11− 0.2 fm, La = 2.7− 3.2 fmNf = 2 : a = 0.086− 0.215 fm, La = 2.0− 2.5 fm
the effect of dynamical up and down quarks
Baryon spectrumNf=0
• large volume
• systematic deviation from experiment (5-10%)
Nf=2
• small volume
• for heavier baryons
• for lighter baryons
7
RG+clover (Phys.Rev.D65(2002)054505)
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
mas
s [G
eV]
experimentNf=0 (K-input)Nf=0 (φ-input)Nf=2 (K-input)Nf=2 (φ-input)
PSspin 0
Vectorspin 1
Octetspin 1/2
Decupletspin 3/2
K
NK*
φ
ΛΣ
Ξ Δ
Σ∗
Ξ∗
Ω
cSW : perturbationNf = 2 : a = 0.086− 0.215 fm, La = 2.0− 2.5 fm
PLQ+Wilson (Phys.Rev.D67(2003)034503)
Nf = 0 : a = 0.05− 0.10 fm, La = 3.08− 3.26 fm
good agreement with exp.
dynamical ud quark effect
large discrepancy from exp.
small Finite Size Effect (FSE)
FSE
ud quark massesNf=0 Nf=2 :
8
0 0.1 0.2a [fm]
2.0
3.0
4.0
5.0
mud
MS(µ
=2G
eV)
[MeV
]
Nf=0,VWI
Nf=0,AWI
Nf=2,AWINf=2,VWI
RG+clover (Phys.Rev.D65(2002)054505)
ud quark mass is reduced.
dynamical ud quark effects
s quark massesNf=0 Nf=2 :
9
a [fm]70
80
90
100
110
120
130
140
150
msM
S(µ
=2G
eV)
[MeV
]
Nf=0,VWI
Nf=0,AWI
Nf=2,AWI
Nf=2,VWI
K-input
a [fm]70
80
90
100
110
120
130
140
150
msM
S(µ
=2G
eV)
[MeV
]
Nf=0,VWI
Nf=0,AWI
Nf=2,AWINf=2,VWI
φ-input
strange quark mass is largely reduced.
dynamical ud quark effects
discrepancy between K- and φ-input vanish.
RG+clover (Phys.Rev.D65(2002)054505)
How is the results in the fully unquenched QCD (Nf=2+1)?
CP-PACS and JLQCD joint projectWilson quark formalism Chiral symmetry is violated at finite lattice spacing. Computational cost is large. Theoretical ambiguity is less.
Dynamical up, down and strange quarks
Related papers :
Nf=2+1 full QCD project
10
Up and down are degenerate.Strange has a distinct mass.
Nucl.Phys.Proc.Suppl.129(2004)188, Nucl.Phys.Proc.Suppl.140(2005)225, PoS LAT2005(2005)057, Phys.Rev.D65(2002)094507, Phys.Rev.D73(2006)034501
S. Aoki, M. Fukugita, S. Hashimoto, K-I. Ishikawa, T. Ishikawa,N. Ishizuka,
Y. Iwasaki, K. Kanaya,T. Kaneko, Y. Kuramashi,M. Okawa,T. Onogi,
Y. Taniguchi, N. Tsutsui, N. Yamada, A. Ukawa and T. Yoshie
11
Nf=2+1 full QCD project members (CP-PACS and JLQCD)
12
Recent Nf=2+1 simulationsMILC collaboration
RG(Symanzik)+Improved staggered fermion(Asqtad)
• • •
Simulation is very fast, but there is theoretical ambiguity (fourth root trick)
good agreement with experiment
RBC and UKQCD collaborationRG(Iwasaki,DBW2)+Domain wall fermion
chiral symmetry on finite lattice spacing
computationally very demanding
a ≈ 0.125 fm (coarse), 0.09 fm (fine), 0.06 fm (super fine)volume ≥ (2.5fm)3
mud = 10− 50 MeV
Lattice actiongauge : RG improved action (Iwasaki,1985)
quark : non-perturbatively improved Wilson action
is non-perturbatively determined. (Phys.ReV .D73(2006)034501)
Algorithm standard HMC algorithm for up and down quarks
• usual pseudo-fermion method
Polynomial HMC (PHMC) algorithm for strange quark
• odd flavor algorithm
• exact algorithm (Phys.Rev.D65(2002)094507)
Strategy
13
cSW
O(a)
Lattice spacing, size, etc.
fixed physical volume
14
∼ (2 fm)3
a2
3
21
1
20
Simulation parameters
β = 2.05(a ∼ 0.070 fm)cSW = 1.628L3 × T = 283 × 56HMC traj.
= 6000− 6500
β = 1.90(a ∼ 0.100 fm)cSW = 1.715L3 × T = 203 × 40HMC traj.
= 5000− 9000
β = 1.83(a ∼ 0.122 fm)cSW = 1.761L3 × T = 163 × 32HMC traj.
= 7000− 8600
Quark mass parameters
15
5 ud quark parameters
mps/mV ∼ 0.60 − 0.78
(0.18 : exprriment)
2 s quark parameters
mps/mV ∼ 0.7
(0.68 : 1-loop ChPT)
0 0.05
a mudVWI
0.02
0.03
0.04
0.05
a m
SV
WI
β=1.83, L3xT=163x32
physical point
Ks=0.13760
Ks=0.13710
0 0.05
a mudVWI
0.02
0.03
0.04
0.05
0.06
a m
SV
WI
β=1.90, L3xT=203x40
physical point
Ks=0.13640
Ks=0.13580
0 0.05
a mudVWI
0.02
0.03
a m
SV
WI
β=2.05, L3xT=283x56
physical point
Ks=0.13540
Ks=0.13510
Gauge configuration generation
16
2003
2004
2005
2006
Production runs
β=1.83
β=1.90
β=2.05
Earth Simulator@JAMSTEC
SR8000/F1@KEK
SR8000/G1@CCS,
Univ. of Tsukuba
VPP5000@ACCC,
Univ. of Tsukuba
CP-PACS@CCS,
Univ. of Tsukuba
ProcedureWe perform the measurements at only unitarity points.
Physical volume is not large to calculate baryon masses.
Measurements are performed at every 10 HMC trajectories.
We mainly focus on the meson sector.
17
∼ (2 fm)3
Measurement of the meson masses
mvalence = msea
Effective mass plot
18
0 10 20T
0.22
0.225
0.23
effe
ctiv
e m
ass
effective mass plotbeta=2.05, Kud=0.13560, Ks=0.13540
π (smeared source - point sink)
0 10 20T
0.34
0.35
0.36
0.37
0.38
0.39
effe
ctiv
e m
ass
effective mass plotbeta=2.05, Kud=0.13560, Ks=0.13510
ρ (smeared source - point sink)
ameff (t) = − lnG(t + 1)
G(t)
G(t) : correlation function
Fitting procedurePolynomial 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) meson simultaneously.
• ignoring correlations among LL, LS and SS
Chiral extrapolation
19
• Polynomial fit forms using the VWI quark mass definition
20
mq =12
(1
Kq− 1
KC
)mud,ms : sea quarks
mval1,mval2 : valence quarks
# of fit parameters = 16
m2PS = BPS
S (2mud + ms) + BPSV (mval1 + mval2)
+ DPSSV (2mud + ms)(mval1 + mval2)
+ CPSS1 (2m2
ud + m2s) + CPS
S2 (m2ud + 2mudms)
+ CPSV 1 (m2
val1 + m2val2) + CPS
V 2 mval1mval2
mV = AV + BVS (2mud + ms) + BV
V (mval1 + mval2)+ DV
SV (2mud + ms)(mval1 + mval2)+ CV
S1(2m2ud + m2
s) + CVS2(m
2ud + 2mudms)
+ CVV 1(m
2val1 + m2
val2) + CVV 2mval1mval2
21
0 0.01 0.02 0.03 0.04 0.05 0.06
a mudVWI
0
0.1
0.2
0.3
0.4
(a m
PS)2
Ks=0.13580Ks=0.13640
β=1.90, L3xT=203x40, quadratic
χ2/d.o.f.=0.94
0 0.01 0.02 0.03 0.04 0.05 0.06
a mudVWI
0.4
0.5
0.6
0.7
0.8
a m
VKs=0.13580Ks=0.13640
β=1.90, L3xT=203x40, quadratic
χ2/d.o.f=0.90
Fitting form from ChPTchiral log behavior
Wilson ChPT (WChPT)
• include explicit chiral symmetry breaking effect of Wilson quark
• Nf=2+1 version
fitting is very difficult due to highly non-linear form (in progress)
22
m2π lnm2
π
(Phys.Rev.D73(2006)014511), hep-lat/0601019
Physical point and lattice spacingInputs to fix the quark masses and the lattice spacing
K-input
φ-input
lattice spacings
K-input and φ-input give consistent results.
23
mπ, mρ, mK
mπ, mρ, mφ
mρ
β κc χ2/d.o.f.1.83 0.138685(16) 0.391.90 0.137657(32) 0.942.05 0.13631(11) 1.18
Table 1: Chiral fit results (pseudo-scalar)
β χ2/d.o.f.1.83 0.661.90 0.902.05 1.72
Table 2: Chiral fit results (vector)
K-input φ-inputβ a−1[GeV] a[fm] a−1[GeV] a[fm]
1.83 1.612(22) 0.1222(17) 1.598(26) 0.1233(20)1.90 1.983(38) 0.0993(19) 1.980(37) 0.0995(19)2.05 2.84(11) 0.0693(26) 2.84(10) 0.0695(26)
Table 3: Lattice spacings from mρ (2006/02/26)
κud κs traj. mπ/mρ mηs/mφ κud κs traj. mπ/mρ mηs/mφ
0.13580 5000 0.7673(15) 0.7673(15) 0.13580 5200 0.7667(16) 0.7210(21)0.13610 6000 0.7435(18) 0.7647(17) 0.13610 8000 0.7443(15) 0.7182(17)0.13640 0.13580 7600 0.7204(19) 0.7687(15) 0.13640 0.13640 9000 0.7145(16) 0.7145(16)0.13680 8000 0.6701(27) 0.7673(17) 0.13680 9200 0.6630(21) 0.7127(17)0.13700 8000 0.6389(21) 0.7693(15) 0.13700 8000 0.6241(28) 0.7101(20)
Table 4: Statistics for the measurement (β = 1.90)
to set s quark mass
Prediction
Continuum extrapolationWe use the non-perturbatively improved Wilson quark, thus meson spectrum should scale as .
Light meson spectrum
24
m(a) = A + Ba2
a2
O(a)
K-input → mK∗ ,mφ
φ-input → mK ,mK∗
The calculation of is in progress.mη,mη′
disconnected diagram is needed.
ResultsIn the continuum, meson spectrum is consistent with experiment.The statistical error in the continuum limit is large.
25
In the Nf=2,0 case, extrapolation function is m(a)=A+Ba, because clover action is perturbatively O(a) improved.
1.000
1.050
0.890
0.895
0 0.01 0.02 0.03 0.04 0.05
a2 [fm
2]
0.870
0.880
0.890
0.900
mes
on m
asse
s [G
eV]
0 0.01 0.02 0.03 0.04 0.05
a2 [fm
2]
0.500
0.550
Quenched(RG+PT clover)Nf=2(RG+PT clover)Nf=2+1(RG+NPT clover)experiment
φ (K-input)
K* (K-input)
K* (φ-input)
K (φ-input)
VWI quark mass
VWI quark mass can be negative due to lack of chiral symmetry of the Wilson quark.
AWI quark mass
The scaling violation is smaller than that of VWI in Nf=2 case.
Renormalizationtad-pole improved 1-loop matching with at
4-loop running to
Light quark masses
26
MS
mV WIq =
12
(1K− 1
Kc
)
Kc ←− mπ(Kud,Ks)|Kud=Ks=Kc = 0
mAWIq =
〈∆4A4(t)P (0)〉2〈P (t)P (0)〉
mLATq (a−1) −→ mMS
q (µ = 2 GeV)
µ = a−1
µ = 2 GeV
up and down quarkWe assume that the contribution is small and use the extrapolation function which is same as in the meson spectrum (linear in ).
In the continuum limit, the VWI definition gives a positive value.
27
NOTE. Nf=2, 0 : RG+clover(PT)
(AWI, combined K with φ-input)
0 0.005 0.01 0.015 0.02
a2 [fm
2]
-20
-10
0
10
mud
MS(µ
=2G
eV)
[MeV
]
VWI
AWI
Nf=2+1 (K-input)
0 0.01 0.02 0.03 0.04 0.05
a2 [fm
2]
2.0
2.5
3.0
3.5
4.0
4.5
5.0
mud
MS(µ
=2G
eV)
[MeV
] Quenched,VWI
Quenched,AWI
Nf=2,AWINf=2,VWI
Nf=2+1,AWI (K-input)
mMSud (µ = 2GeV) = 3.49(15) MeV
O(a)
a2
strange quarkAll definitions of strange quark mass gives consistent results in the continuum limit.
The difference between Nf=2+1 and Nf=2 is invisible in the continuum limit as in the ud quark.
28
(AWI, combined K with φ-input)
NOTE. Nf=2, 0 : RG+clover(PT)
80
90
100
110
120
msM
S(µ
=2G
eV)
[MeV
]
0 0.01 0.02 0.03 0.04 0.05
a2 [fm
2]
70
80
90
100
110
120
130
140
msM
S(µ
=2G
eV)
[MeV
]
Nf=0,VWI
Nf=0,AWI
Nf=2,AWI
Nf=2,VWI
Nf=2+1,AWI
K-input
Nf=2+1,VWI
φ-input Nf=0,VWI
Nf=2,VWI Nf=0,AWI
Nf=2,AWI
Nf=2+1,VWI
Nf=2+1,AWI
mMSs (µ = 2GeV) = 90.9(3.7) MeV
Baryon spectrum
29
0.8
1
1.2
1.4
1.6
mas
s [G
eV]
experimentNf=2+1 (K-input)Nf=0 (K-input)
octet decupletspin 1/2 spin 3/2
N
ΛΣ
Ξ Δ
Σ∗
Ξ∗
Ω
Signal is not good.Pattern of the spectrum is reproduced.But the spectrum and FSE is unclear.
0 5 10 15 20T
0.7
0.75
0.8
effe
ctiv
e m
ass
effective mass plotbeta=1.90, Kud=0.13700, Ks=0.13580
N (octet)
0 10 20 30T
0.6
effe
ctiv
e m
ass
effective mass plotbeta=2.05, Kud=0.13540, Ks=0.13510
Δ (decuplet)
(preliminary)
PS decay constant
renormalizationtad-pole improved one-loop renormalization factor
definitions for the calculationWe test two different definitions.
PS meson decay constants
30
〈0|A4|PS〉 = fPSmPS
fPS
〈PP (t)PP (0)〉 ∼ CPPP e−mP St
〈PP (t)PS(0)〉 ∼ CPSP e−mP St
〈PS(t)PS(0)〉 ∼ CSSP e−mP St
〈AP4 (t)PS(0)〉 ∼ CPS
P e−mP St
fPS = CP SA
CP SP
√2CP P
PmP S
(method 1)
fPS = CPSA
√2
mP SCSSP
(method 2)
31
0 0.01 0.02
a2 [fm
2]
0.1
0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.2
f PS [G
eV]
0 0.01 0.02
a2 [fm
2]
0.1
0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.2
π
K
π
K
π
K
π
K
K-input, method 1 K-input, method 2
Two methods are consistent each other.
But huge statistical error.
Statistics might not be enough to calculate decay constants (?)
(preliminary)
Static quark potential
Sommer scale
In our calculation is obtained through
32
Sommer scale
V (r) = V0 −
α
r+ σr
r2
dV (r)
dr
∣
∣
∣
∣
r=r0
= 1.65
R0 = ar0 ! 0.5 fm
r0 =
√
1.65 − α
σ
r0
r0 0 1 2 3r / r0
-4
-3
-2
-1
0
1
2
3
4
[ V(r
) -
V(r
0) ] r
0
string model ( α=π/12 )
β = 1.83
0 1 2 3r / r0
-4
-3
-2
-1
0
1
2
3
4
[ V(r
) -
V(r
0) ] r
0
string model ( α=π/12 )
β = 1.90
0 1 2 3r / r0
-4
-3
-2
-1
0
1
2
3
4
[ V(r
) -
V(r
0) ] r
0
string model ( α=π/12 )
β = 2.05
phenomenological model
Chiral extrapolation
Continuum limit of using the lattice spacing from
33
1
r0
= A + B
(
2
κud
+1
κs
)
R0
mρ
NOTE. Nf=2, 0 : RG+clover(PT)
10.95 11 11.05(2/Kud+1/Ks)/2
0.21
0.22
0.23
0.24
0.25
1/r 0
datafit line
0 0.01 0.02 0.03 0.04 0.05
a2 [fm
2]
0.45
0.5
0.55
0.6
0.65
a r 0 [f
m]
Nf=2+1 (K-input)Nf=2+1 (φ-input)Nf=2quenched
consistent with 0.5 fm
R0 = 0.516(21) fm
(preliminary)
CP-PACS/JLQCD Nf=2+1 projectRG-gauge + non-PT clover (Wilson quark formalism)Gauge configuration generation has been already finished.
Analysis of spectrum and quark massesEncouraging results in the continuum limit are obtained.
Assuming that the scaling is ,
• Meson spectrum is consistent with experiment.
• All definitions and inputs of quark masses gives consistent results.
• Quark masses in the continuum :
Baryon spectrum, decay constants
Summary
34
mMSud (µ = 2GeV) = 3.49(15) MeV
mMSs (µ = 2GeV) = 90.9(3.7) MeV
a2
signal is not so good(?), large statistical error
Required task Non-perturbative determination of renormalization factor
Calcurations in progressη’ meson mass
Heavy meson quantities using a relativistic heavy quark action
35
cΓ, ZΓ, bΓ
(Aoki,Kuramashi,Tominaga,2001)
PACS-CS collaborationclover quark with domain decomposed HMC
PACS-CS (CCS, Univ. of Tsukuba)
cluster, 2560 nodes, 14.3 TFLOPS
JLQCD collaborationdynamical overlap fermion
IBM Blue Gene (KEK)
10 rack (10240 node), 57.3 TFLOPS
36
cΓ, ZΓ, bΓ
Next direction - lighter quark mass -
ambiguity in the chiral extrapolation will be removed